text:
11 Pro-Cannabis Politicians Who Changed the Game
David Bienenstock August 29, 2018
For decades, three little words struck terror into the heart of any politician who so much as contemplated voicing support for even the mildest reform of the laws against cannabis.
"Soft on drugs."
And so, for a long dark ages, the government's oppressive, abusive and unjust war on cannabis continually escalated, with almost no substantive debate of its merits—not in Congress, not in the corporate media, not in the medical field, and not in the so-called Justice Department.
For Those of You Who Are "Soft on Drugs"...
Let's Get You to a Dispensary Nearby
Many understood that the war was counterproductive and unwinnable, but they dared not say anything, lest they be declared soft on drugs and banished from the establishment. Take for example former Surgeon General Joycelyn Elders, who simply suggested studying the legalization of cannabis as a harm reduction strategy and received official censure from President Clinton in response, before eventually being pushed out of the job.
Once a few brave states legalized—and demonstrated that the sky hasn't fallen—suddenly everybody's free to admit what was always obvious. That cannabis prohibition is, was, and always will be a dismal failure and an affront to humanity.
Now, with legalization polling at over 60% nationwide, politicians worry about losing votes by backing a highly unpopular prohibition. So we are indeed in a new era of cannabis policy, and we wouldn't have ever gotten here without the efforts of these ground-breaking visionary politicians, who changed the game for good.
Fiorello La Guardia
The push to outlaw cannabis in the United States dates back to the beginning of the 20th Century, when an influx of immigrants and refugees—displaced by the Mexican Revolution—began crossing the border in large numbers, which in turn led to an anti-immigration backlash with overtly racist motivations. Laws against marihuana became a convenient club to criminalize these new arrivals, especially after a coordinated propaganda campaign in the media portrayed cannabis as a clear and present danger to the American way of life.
Released as The La Guardia Report in 1944, the landmark study pointedly debunked pretty much every argument in favor of prohibition.
By 1937, all this Reefer Madness led to passage of the Marijuana Tax Act, which nationalized marijuana prohibition just four years after the end of alcohol prohibition.
In direct response to this new law, New York City mayor Fiorello La Guardia commissioned a blue ribbon panel of leading doctors, psychiatrists, psychologists, pharmacologists, chemists and sociologists, and tasked them with making a thorough investigation of cannabis based on a comprehensive review of all available scientific literature, plus primary research.
Released as The La Guardia Report in 1944, the landmark study earned the endorsement of the prestigious New York Academy of Medicine, while pointedly debunking pretty much every argument in favor of prohibition.
After first making his fortune as the founder of one of New Mexico's largest construction companies, Gary Johnson began his career in politics by running for Governor, and winning. In 1999, shortly after voters gave him a second term by a wide margin, Johnson became one of the highest ranking public officials in the country to openly call for cannabis legalization.
Watch This: Various US Politicians Admit They've Tried Cannabis
While running for President as the Libertarian Party candidate in 2016, he told the press he'd only stopped smoking cannabis a couple of months before entering the race. If he won the election, and moved into the White House, he promised not to take a single puff during his time in office.
He's probably smoking a joint right now.
Barbara Lee has served in Congress since 1998, representing Oakland, California and northern Alameda County. One of the most progressive voices in the federal government, she's long been a staunch supporter of cannabis legalization. In 2008, she won an award from the National Organization for the Reform of Marijuana Laws (NORML) for "outstanding public leadership." And ten years later, she's still a national leader on the issue.
In January, Lee introduced the Refer Act in Congress.
"Due to unequal criminalization rates and disparities in access to capital, people of color are being locked out of the new and thriving legal cannabis trade."
Rep. Barbara Lee
"American voters have sent a clear message—it's time for the government to stop wasting taxpayer money on the failed War on Drugs and take long overdue action to reform cannabis policies." Lee said while introducing the bill, which would protect state cannabis laws from federal intrusion. "One in five Americans resides in a state where cannabis use has been legalized. Congress [must] end the federal government's wrongful targeting of this industry and ensure that the will of the American people is respected."
Lee also introduced a companion bill in the House to the Marijuana Justice Act, which was first introduced in the Senate by Corey Booker, and would remove cannabis from the Controlled Substances Act entirely, while providing a process for expungement of criminal records, and creating a $500 million investment fund to support communities and people disproportionately targeted by the War on Drugs.
Most recently, Lee has introduced the the RESPECT Resolution, the first bill in Congress focused on building equity in the cannabis industry.
"Due to unequal criminalization rates and disparities in access to capital, people of color are being locked out of the new and thriving legal cannabis trade," according Congresswoman Lee. "We need to address the systemic exclusion and discrimination at play. Otherwise, we will be prolonging and encouraging the injustices of the past—where brown men spend their lives in prison for cannabis, while white communities get rich off the industry."
(Eraldo Peres/AP)
Long before he became the 40th President of Uruguay (2010—2015), Jose Mujica was a guerrilla fighter with a group of left wing revolutionaries. He spent 13 years in prison while a military dictatorship ran the country, and then entered politics upon his release, when constitutional democracy was restored.
In 2009, he won the Presidency.
"We've been repressing drugs for 80 years and it doesn't work. Drug trafficking is much worse than the consumption of marijuana."
Jose Mujica, Former President of Uruguay
Known as "the humblest world leader," Mujica eschewed life in a Presidential palace and being driven around in a limousine, instead choosing to live on an old farm reachable only by a dirt road, while donating 90% of his salary to charity, refusing to wear a tie, and driving a 1987 Volkswagen bug.
He also undertook a controversial push for cannabis legalization, arguing that bringing the drug under legal control would improve Uruguay's health while denying drug cartels profits and power.
"We've been repressing drugs for 80 years and it doesn't work," he explained, "Drug trafficking is much worse than the consumption of marijuana."
Opposition to legalization was strong when Mujica first proposed the idea, and it was a bumpy road to implementation, but in 2013 Uruguay became the first country to legalize cannabis at the national level, with retail sales starting in 2017.
In 2013, while running for the leadership of The Liberal Party of Canada, Justin Trudeau voiced his support for cannabis legalization for the first time. He also acknowledged that the last time he'd smoked cannabis had only been only three years prior, when he was already serving as a member of Parliament.
"We had a few good friends over for a dinner party, our kids were at their grandmother's for the night, and one of our friends lit a joint and passed it around," he said in an interview. "I had a puff."
Trudeau Reveals Why He Supports Legal Cannabis in Canada
When Trudeau ran for Prime Minister in 2015, he made legalization a key part of his upstart campaign. And while it's been a rough road to making that happen, Canada's national cannabis legalization legislation finally passed in June, and the nation's first full legal recreational sales will begin on October 17.
Iyah V (Jamaica)
In 2014, in rugged, mountainous West Jamaica, respected Rasta elder Ras Iyah V called to order the first meeting of the Westmoreland Hemp and Ganja Farmers' Association—founded to ensure that those who fought and struggled for decades for the end of cannabis prohibition don't get pushed aside as the Caribbean nation moves towards legalization
"We are not going to take baton licks and brutality, and all of a sudden, now when the legalization aspect come, let rich people take it over—people who used to scoff at the very mention of the herb."
Ras Iyah V
"We will not stand by and watch people outside of Rastafari faith and grass-roots people take over this product," Iyah V declared in his opening remarks, to a gathering that included many attendees who very rarely leave their clandestine ganja gardens out in the bush. "We are saying this loud and clear to the government, we are saying it to society, and we are saying it to the international community. Because we are not going to take baton licks and brutality, and all of a sudden, now when the legalization aspect come [sic], let some rich people take it over—people who used to scoff and scorn at the very mention of the herb."
Iyah V has been working to help organize his fellow outlaw Rastafarian herbsmen into a legally-recognized cannabis cooperative—one capable of proactively protecting their integrity and common interests amid the influx of big money expected to arrive along with legalization.
In March, Jamaica's first cannabis retailer opened for business.
Congressional Cannabis Caucus
Individually, Earl Blumenauer, Don Young, Jared Polis, and Dana Rohrabacher have all made important contributions to the cannabis legalization movement. But what's truly game changing is that in 2017 they joined together to form the Congressional Cannabis Caucus, a bipartisan registered Congressional Member Organization that aims to amplify the power of those in the Legislative Branch pushing for an end to cannabis prohibition at the federal level.
Actually, fuck John Boehner.
Election 2018legalization
David Bienenstock
Veteran cannabis journalist David Bienenstock is the author of "How to Smoke Pot (Properly): A Highbrow Guide to Getting High" (2016 - Penguin/Random House), and the co-host and co-creator of the podcast "Great Moments in Weed History with Abdullah and Bean." Follow him on Twitter @pot_handbook.
View David Bienenstock's articles
More Movers & Shakers
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Thank you for mentioning Barbara Lee, she does not get the credit she deserves and also Fuck John Boehner.
Gary Craig
Yep, Boehner is raking in money while others are in the prison yard raking leaves, for pennies a day.
Michael Halliwell
Yet you fail to mention Bernie Sanders.
Peacefull Solutions Peaceful R
He wasnt a game changer though. Ron Paul supported it before Bernie.
Citium DC
No Ron Paul or Bernie Sanders?
Justin Treuduea from Canada is a joke lol. Good for him on supporting legalization.
Dante-the-cat
How so? Canada will be the first G7 nation to legalize it. Watch for dominoes to fall after that.
I hope I'm wrong but Trump could very well throw a wrench in the works for Canada's legalization plans. We'll see when he addresses the UN General Assembly. By the way Trump is no fan of Canada, or Trudeau.
Yup. He hates us. That much is plain. But he can't give edicts to a sovereign nation not his.
Trump is also no fan of the UN.
Dan F
Trump is taking it off the schedule A, he wants feds out of, what should be a state only thing. Jeff Sessions is stopping it from happening faster. Once Trump rids himself of spineless Sessions and gets democrats off his ass look for nice changes to bad federal laws.
OR governor Kate Brown also deserves mention. Not only has she been okay with weed, but she has no problem with all the more retail weed stores, the better. She also tried to get other OR politicians to make it legal in the state to consume cannabis, while off the clock from a job, but other state politicians didn't show enough support for it.
Fun Please
Competition app angry at Facebook for cannabis moderation –
This article here – disparity this and that
Why am I being moderated here? Seems hypocrisy is rampant.
I'm a combat veteran that believes in cannabis and free speech. What's up?
Trump wants to take Marijuana off schedule A. I trust he will do whats right.
Nope. He won't remember he said anything about it. Besides, since when did he ever do "the right thing"?
No, he doesn't.
https://www.washingtonpost.com/politics/trump-say-he-probably-will-support-bill-to-protect-states-that-have-legalized-marijuana/2018/06/08/23fe0884-6b24-11e8-bea7-c8eb28bc52b1_story.html
His base would eat him alive.
I am his base, I support Adult use. Republicans need medicine too and once democrats stop the witch hunt look for him to take it off the fed list of sched 1 class of drugs. He knows it has medical benefit.
Not to mention the evangelicals, who would go ballistic!
You left out Louis Armstrong, Willie Nelson, Bob Marley, Woody Harrelson, Bill Maher, Seth Rogan, … Oh heck, this list could go on forever!
jack flash
Ras Iyah V is an amazing man and a truly inspirational leader.
Michael Arnall
Earl bloomenour. oregon. Pic. Above showes guy vaping, no smoke should show in bag medical
HalfAssed. All about $$$$ get back to solid info Oregon best maryjane program in country state thinking about it? Check out Oregons. Caly ,
Colo, Wash, Airz, all about the$$$$ wouldnt buy in those states unless really hard up
Living in Maryland, no "permission slip," no connections, I am hard up & must go without. And this state is years away from adult use, if ever. Maryland is always, but always, the last state to do anything. And 60 year olds with 0 connections have a hard time finding anything.

redpajama_set_name: RedPajamaCommonCrawl


This is the updated chart from last Friday's bounce-from-support setup on CAD/JPY's 1-hour forex chart. After bouncing off from support at the 94.50, the pair moved for over a hundred pips with such convincing bullish momentum that price broke through the falling trend line that you can see on the above chart. Moreover, bullish momentum was so strong that price managed to push through the 100 SMA and 200 SMA before consolidating into what looks like a bullish flag pattern. Our directional bias for this forex chart pattern is obviously to the upside, although there is always a possibility for a downside breakout. If this is indeed a flag, and the pair breaks to the upside, then we can probably expect price to move for about a hundred pips since that is the height of the flag and its pole.
Next, we've got a plain vanilla descending channel on CHF/JPY's 1-hour forex time frame. The basic and conservative way to play a descending channel is to look for resistance near the top of the channel. And based on the chart, price currently seems pretty close to the top.
I think resistance will form here for these technical reasons: (1) current price levels around the 128.70 handle have seen significant market interest in the past, (2) the moving averages are in downtrend mode, (3) the 100 SMA is acting as dynamic resistance, (4) stochastic is about to enter overbought territory, so forex traders who are bullish on the pair may be exhausted already.
NZD/JPY has been consolidating in what looks like a massive ascending triangle, which is strange since the overall trend is down, even though the moving averages have been oscillating for the past two weeks. Price seems to have trouble breaking through resistance at the 82.90 handle, but if we apply the Fibonacci tool to the most recent swing, we can see that price seems to be holding at the 38.2% Fibonacci retracement. Stochastic is also already in the oversold region, indicating that bulls may potentially be taking over soon.
In the event that this is a valid ascending triangle and price breaks out to the topside, then we can expect a volatility of roughly 230 pips since that is the height of the triangle. But as a triangle, there is also a chance that price will breakout in the opposing direction, and prudent forex traders should prepare for that scenario as well.


text:
\section{Introduction}
\label{s:introduction}

This paper is the last chapter of a work started in \cite{Abbondandolo:2015lt} and further developed in \cite{Abbondandolo:2014rb, Asselle:2015ij, Asselle:2015sp, Asselle:2016qv} devoted to studying the multiplicity of periodic orbits on generic low energy levels in magnetic Tonelli Lagrangian systems on surfaces. Such a study was based on a generalization of Bangert's waist Theorem \cite[Theorem~4]{Bangert:1980ho}, classically formulated for geodesic flows on $S^2$, to the magnetic Tonelli setting. Roughly speaking, a waist is a non-constant periodic geodesic (resp.\ a periodic orbit in the Tonelli case) which minimizes the length (resp.\ the action) among nearby curves. The original waist Theorem says that a Riemannian 2-sphere possesses infinitely many closed geodesics provided it possesses a waist. Such a statement is a crucial ingredient for the proof that, indeed, every Riemannian 2-sphere possesses infinitely many closed geodesics \cite{Bangert:1993wo, Franks:1992jt, Hingston:1993ou}.

Let us introduce the general setting in which we will work. If $M$ is a closed smooth manifold, a Tonelli Lagrangian $L:\mathrm{T} M\to\mathds{R}$ is a smooth function whose restriction to any fiber of $\mathrm{T} M$ is superlinear with positive definite Hessian, see e.g.\ \cite{Mather:1991uq, Fathi:2008xl, Abbondandolo:2013is}. A magnetic Tonelli system is a pair $(L,\sigma)$, where $L:\mathrm{T} M\to\mathds{R}$ is a Tonelli Lagrangian and $\sigma$ is a closed 2-form on $M$, which we refer to as the magnetic form. If $\pi:\mathrm{T}^* M\to M$ denotes the projection of the cotangent bundle, the pair $(L,\sigma)$ defines a flow on $\mathrm{T} M$ that is conjugated through the Legendre transformation $\partial_vL$ to the Hamiltonian flow on $(\mathrm{T}^*M,\mathrm{d} p\wedge \mathrm{d} q+\pi^*\sigma)$ of the dual Tonelli Hamiltonian $H:\mathrm{T}^*M\to\mathds{R}$, $H(q,p)=\max\{ pv - L(q,v)\ |\ v\in\mathrm{T}_qM\}$, see e.g.\ \cite{Arnold:1961lq, Novikov:1982xy, Asselle:2016qv}.
A particularly relevant special case of this setting is the electromagnetic one, when the Lagrangian $L$ is of the form $L(q,v)=\tfrac12 g_q(v,v)-U(q)$ for some Riemannian metric $g$ and some smooth potential $U:M\to\mathds{R}$. In this situation, the system $(L,\sigma)$ models the motion of a particle on $M$ with kinetic and potential energies described by $L$ and under the further effect of a Lorentz force described by $\sigma$. When the potential $U$ vanishes, the dynamics of the system $(L,\sigma)$ is a so-called magnetic geodesic flow.

In this paper, we will focus on the case where $M=S^2$. The energy function $E:\mathrm{T} M\to\mathds{R}$, $E(q,v):=\partial_v L(q,v)v-L(q,v)$, is preserved along the motion. Therefore it is natural to study the dynamics of a magnetic Tonelli flow on a prescribed energy hypersurface $E^{-1}(e)$, and very different qualitative behaviors appear for different values of the energy $e$, see \cite{Cieliebak:2010zt} and references therein. For our purposes two energy values will bear special significance: $e_0(L)$ and $e_1(L,\sigma)$. The former is the minimal energy $e$ such that the corresponding energy hypersurface $E^{-1}(e)\subset\mathrm{T} S^2$ projects onto the whole $S^2$. We postpone the precise definition of $e_1(L,\sigma)$ to Section \ref{s:action}. For now, we just mention that $e_1(L,\sigma)\geq e_0(L)$, and when $\sigma$ is exact with primitive $\theta$ we have $e_1(L,d\theta)=c_u(L+\theta)$, where $c_u(L+\theta)$ is
the Ma\~ n\'e critical value of the universal cover of $L+\theta$ (see e.g.~\cite{Contreras:1999fm, Abbondandolo:2013is} for the definition of Ma\~ n\'e critical values).

The periodic orbits problem for magnetic geodesics was first studied by Novikov \cite{Novikov:1981ef, Novikov:1982xy} in the early 1980s. The classical least action principle for the periodic orbits with prescribed energy is not directly available in this setting, due to the potential non-exactness of the magnetic 2-form. Novikov showed how to recover the variational principle in the universal cover of the space of periodic curves, and in his celebrated ``throwing out cycles'' method he proposed how to exploit the corresponding deck transformation in order to detect action values of periodic orbits (for the throwing out cycles method, see also \cite{Taimanov:1983uo}). For magnetic geodesics on closed surfaces, waists were first studied by Taimanov in a series of papers \cite{Taimanov:1991el,Taimanov:1992fs,Taimanov:1992sm}. Taimanov's result is that, given a kinetic Lagrangian $L(q,v)=\tfrac12 g_q(v,v)$ and an oscillating magnetic 2-form $\sigma$ on a closed 2-dimensional configuration space, there exists a waist $\alpha_e$ at the energy level $e$, for all $e\in (0,e_1(L,\sigma))$ (see also \cite{Contreras:2004lv} for a different proof). When $\sigma$ is exact, Abbondandolo, Macarini, Mazzucchelli and Paternain \cite{Abbondandolo:2014rb} employed Taimanov's waist $\alpha_e$ on any energy level $e$ belonging to a full measure subset of $(0,e_1(L,\sigma))$ in order to construct a sequence of minmax families giving an infinite number of (geometrically distinct) periodic orbits with energy $e$. Short afterwards, Asselle and Benedetti extended the result to non-exact $\sigma$ on surfaces of genus at least one \cite{Asselle:2015ij, Asselle:2015sp}. The results in \cite{Taimanov:1991el, Taimanov:1992fs, Taimanov:1992sm, Abbondandolo:2014rb, Asselle:2015ij, Asselle:2015sp} have been further extended by Asselle and Mazzucchelli \cite{Asselle:2016qv} to the general magnetic Tonelli setting. In this note we complete the picture by treating the last case remained open for the multiplicity problem: the 2-sphere. Namely, we are going to prove the following result.

\begin{thm}
\label{t:main}
Let $L:\mathrm{T} S^2\to\mathds{R}$ be a Tonelli Lagrangian, and $\sigma$ a 2-form on $S^2$. For almost every $e\in(e_0(L),e_1(L,\sigma))$, the Lagrangian system of $(L,\sigma)$ possesses infinitely many periodic orbits with energy $e$.
\end{thm}

We wish to stress that the existence of infinitely many periodic orbits on \emph{all} energy values in $(e_0(L),e_1(L,\sigma))$ is still an open problem. In Theorem~\ref{t:main}, as well as in \cite{Abbondandolo:2014rb, Asselle:2015ij, Asselle:2015sp}, a negligible subset of energies must be excluded due to a lack of compactness in the variational setting that is employed.   However, energy levels with only finitely many periodic orbits can be found above \cite{Ziller:1983lq, Benedetti:2016} as well as below \cite{Asselle:2016qv} the interval $[e_0(L),e_1(L,\sigma)]$.

For closed surfaces $M$ of genus at least one, any closed 2-form $\sigma$ on $M$ lifts to an exact 2-form on the universal cover of $M$. This allows to define the Ma\~n\'e critical value of the universal cover $c_u(L,\sigma)$ for any Tonelli Lagrangian $L:\mathrm{T} M\to\mathds{R}$. We set $e_1^*(L,\sigma):=\min\{e_1(L,\sigma),c_u(L,\sigma)\}$ if $M$ has positive genus, and $e_1^*(L,\sigma):=e_1(L,\sigma)$ if $M=S^2$. The combination of Theorem~\ref{t:main} together with the above mentioned results in \cite{Asselle:2015ij, Asselle:2015sp, Asselle:2016qv}, yields the following statement about the multiplicity of periodic orbits on general closed surfaces.
\begin{thm}
Let $M$ be a closed surface, $L:\mathrm{T} M\to\mathds{R}$ a Tonelli Lagrangian, and $\sigma$ a 2-form on $M$. For almost every $e\in(e_0(L),e_1^*(L,\sigma))$, the Lagrangian system of $(L,\sigma)$ possesses infinitely many periodic orbits with energy $e$.
\hfill\qed
\end{thm}

\begin{rem}\label{r:oscillating}
The open interval $(e_0(L),e_1^*(L,\sigma))$ is not empty for instance if $M$ is orientable,  $\sigma$ is oscillating,  and the Lagrangian has the form of a kinetic energy $L(q,v)=\tfrac12g_q(v,v)$ for some Riemannian metric $g$ (see~\cite{Asselle:2015sp}); in such case, $e_0(L)=0$ is the minimal energy of the system. We recall that a 2-form $\sigma$ on an orientable surface is oscillating when it satisfies $\sigma_{q_-}<0$ and $\sigma_{q_+}>0$ for some $q_-,q_+\in M$. On non-orientable surfaces, any non-zero 2-form lifts to an oscillating 2-form on the orientation double cover.
\hfill\qed
\end{rem}

This paper is dedicated to the memory of Abbas Bahri. Bahri was interested in the problem of periodic orbits of magnetic geodesic flows. In a joint work with Taimanov \cite{Bahri:1998eu}, he established the existence of periodic magnetic geodesics with prescribed energy on closed configuration spaces of arbitrary dimension under the assumption that the analog of the Ricci curvature for the Lagrangian system is positive.

The paper is organized as follows. In  Section~\ref{s:action} we recall the variational setting for our periodic orbits problem: we provide the definition of the action 1-form $\eta_e$, and of its global primitive $A_e$ on the universal cover of the space of loops; at the end we will review the definition of the energy values $e_0$ and $e_1$, and the notion of a waist for magnetic Tonelli systems. In Section~\ref{s:proof} we provide the proof of Theorem~\ref{t:main}.

\subsection*{Acknowledgments}
A.A. and L.A. are partially supported by the DFG grant AB
360/2-1 ``Periodic orbits of conservative systems below the Ma\~ n\'e critical energy value''. G.B. is partially supported by the DFG grant SFB 878. M.M. is partially supported by the ANR grants WKBHJ (ANR-12-BS01-0020) and COSPIN (ANR-13-JS01-0008-01). Part of this project was carried out while M.M. was visiting the Sobolev Institute of Mathematics in Novosibirsk (Russia), under the Program ``Short-Term Visits to Russia by Foreign Scientists'' of the Dynasty Foundation; M.M. wishes to thank the Foundation and Alexey Glutsyuk for providing financial support, and Iskander A. Taimanov for the kind hospitality.

\section{The primitive of the free-period action form}
\label{s:action}

\subsection{The variational principle}
Let $L:\mathrm{T} S^2\to\mathds{R}$ be a Tonelli Lagrangian with associated energy function $E(q,v)=\partial_vL(q,v)v-L(q,v)$, and $\sigma$ a 2-form on $S^2$.
Since we will be interested in the Euler-Lagrange dynamics on a given energy hypersurface $E^{-1}(e)$, for some fixed $e\in\mathds{R}$, we can modify the Tonelli Lagrangian far from $E^{-1}(e)$ and assume without loss of generality that each restriction $L|_{\mathrm{T}_qM}$ coincides with a polynomial of degree 2 outside a compact set. Let $\mathcal{M}:= W^{1,2}(\mathds{T};S^2)\times(0,\infty)$, where $\mathds{T}:=\mathds{R}/\mathds{Z}$ is the 1-periodic circle. For each energy value $e\in\mathds{R}$, we consider the free-period action 1-form $\eta_e$ on $\mathcal{M}$ given by
\begin{align*}
\eta_e(\gamma,p)(\xi,q)= \mathrm{d} S_e(\gamma,p)(\xi,q)+ \int_{\mathds{T} } \sigma_{\gamma(t)}(\xi(t),\dot\gamma(t))\,\mathrm{d} t,\\ (\gamma,p)\in\mathcal{M},\ \ (\xi,q)\in\mathrm{T}_{(\gamma,p)}\mathcal{M},
\end{align*}
where $S_e:\mathcal{M}\to\mathds{R}$ denotes the free-period action functional
\begin{align*}
S_e(\gamma,p)= p \int_{\mathds{T} } L(\gamma(t),\dot\gamma(t)/p)\,\mathrm{d} t + p\,e.
\end{align*}
By the least action principle, $\eta_e$ vanishes at some $(\gamma,p)\in\mathcal{M}$ if and only if the $p$-periodic curve $\Gamma(t):=\gamma(t/p)$ is an orbit of the magnetic Tonelli system of $(L,\sigma)$, see e.g.\ \cite{Asselle:2014hc} and references therein.

The 1-form $\eta_e$ is not exact if $\sigma$ is not exact. In order to work with a primitive of $\eta_e$, following Novikov \cite{Novikov:1981ef, Novikov:1982xy}, we will lift it to the universal cover of $\mathcal{M}$. We see $S^2$ as the unit sphere in $\mathds{R}^3$, oriented in the usual way, and we fix the point $x_0=(-1,0,0)\in S^2$. We consider the universal cover \[\pi:\widetilde{\M} \to \mathcal{M}.\]
As usual, we realize $\widetilde{\M}$ as the space of homotopy classes relative to the endpoints of continuous paths $u:[0,1]\to\mathcal{M}$ starting at $u(0)=(x_0,1)$. Here, we see $x_0$ as the constant loop at $x_0$. The projection map is given by $\pi([u])=u(1)$.
We have $\pi^*\eta_e=\mathrm{d} A_e$, where the functional \[A_e:\widetilde{\M}\to \mathds{R}\] is defined as follows. Given $[u]\in\widetilde{\M}$, we write $u=(\gamma,p)$, where $\gamma(s)\in W^{1,2}(\mathds{T};S^2)$ and $p(s)\in(0,\infty)$ for all $s\in[0,1]$. We see $\gamma$ as a map of the form $\gamma:[0,1]\times \mathds{T} \to S^2$ by setting $\gamma(s,t):=\gamma(s)(t)$. We then set
\begin{align*}
A_e([u]):= S_e(u(1)) + \int_{[0,1]\times \mathds{T} } \gamma^*\sigma.
\end{align*}

\begin{rem}\label{r:local}
Assume that $U\subsetneq S^2$ is a proper open subset, so that $\sigma|_U$ is exact with some primitive $\theta$. Let $\mathcal{U}\subset\widetilde{\M}$ be a connected component of the open set of those $[u]\in\widetilde{\M}$ such that the periodic curve $u(1)$ is contained in $U$. Up to an additive constant, the restriction $A_e|_{\mathcal{U}}$ is equal to $S_e'\circ\pi|_{\mathcal{U}}$, where $S_e':\mathcal{M}\to\mathds{R}$ is the free-period action functional associated with the Lagrangian $L+\theta$, i.e.
\[
\tag*{\qed}
S_e'(\gamma,p)
 =
p\int_{\mathds{T} } L(\gamma(t),\dot\gamma(t)/p)\,\mathrm{d} t + \int_{\gamma}\theta + p\,e.
\]
\end{rem}

It is well known that the fundamental group of the free loop space $W^{1,2}(\mathds{T};S^2)$ is isomorphic to $\mathds{Z}$, and therefore so is the fundamental group of $\mathcal{M}$. A generator $[z]$ of $\pi_1(\mathcal{M},(x_0,1))$ can be defined as follows. For each $s\in\mathds{T}$, consider the affine plane $\Sigma_s\subset\mathds{R}^3$ orthogonal to the vector $(0,\cos(2\pi s),-\sin(2\pi s))$ and passing through $x_0$. We denote by $\zeta(s)\in W^{1,2}(\mathds{T};S^2)$ the closed curve with constant Euclidean speed whose support is precisely the intersection $\Sigma_s\cap S^2$, its starting point is $\zeta(s)(0)=x_0$, and, for all $s\neq 0$, its orientation is such that the ordered pair $\partial_s\zeta(s)(t),\partial_t\zeta(s)(t)$ agrees with the orientation of $S^2$, see Figure~\ref{f:zeta}.
\begin{figure}
\begin{center}
\begin{small}
\input{zeta.pdf_tex}
\caption{The map $\zeta:\mathds{T}\to W^{1,2}(\mathds{T};S^2)$.}
\label{f:zeta}
\end{small}
\end{center}
\end{figure}
We define
$z:= (\zeta,1):\mathds{T} \to \mathcal{M}$.
The group of deck transformations of the universal cover $\widetilde{\M}$ is generated by
\[Z:\widetilde{\M} \to \widetilde{\M},\qquad Z([u]) = [z*u],\]
where $z*u(s)=z(2s)$ for all $s\in[0,1/2]$, and $z*u(s)=u(2s-1)$ for all $s\in[1/2,1]$. The action $A_e$ varies under such a transformation as
\begin{align}
\label{e:action_shift}
A_e\circ Z([u]) = A_e([u])+ \int_{S^2}\sigma.
\end{align}

\subsection{Iterated curves}
For each $v=(\gamma,p)\in\mathcal{M}$, we denote by $v^m=(\gamma^m,mp)\in\mathcal{M}$ its $m$-fold iterate, where
$\gamma^m(t)=\gamma(mt)$.
The iteration map $\psi^m:\mathcal{M}\to\mathcal{M}$, $\psi^m(v)=v^m$, is smooth. We lift this map to a smooth map of the universal cover, so that the following diagram commutes
\begin{align*}
\xymatrix{
\widetilde{\M} \ar[r]^{\widetilde\psi^m} \ar[d]_{\pi} & \widetilde{\M} \ar[d]^{\pi} \\
\mathcal{M} \ar[r]^{\psi^m}  & \mathcal{M}
}
\end{align*}
For instance, we can set $\widetilde\psi^m([u]):=[u^m]$, where
\begin{align*}
u^m(s) =
\left\{
  \begin{array}{ll}
    (x_0,1+2s(m-1)) & \mbox{if }s\in[0,1/2], \vspace{5pt}\\
    u(2s-1)^m & \mbox{if }s\in[1/2,1].
  \end{array}
\right.
\end{align*}
A remarkable property of the iteration map is given by the non-mountain pass Theorem for high iterates, which was first established for electromagnetic Lagrangians in \cite[Theorem~2.6]{Abbondandolo:2014rb}, and extended to general Tonelli Lagrangians in \cite[Lemma~4.3 and proof of Theorem~1.2]{Asselle:2016qv}. As we explained in Remark~\ref{r:local}, $A_e$ coincides locally with the free-period action functional of a suitable Tonelli Lagrangian, and therefore the non-mountain pass Theorem for high iterates holds for $A_e$ as well.

\begin{thm}[Non-mountain pass Theorem for high iterates]
\label{t:non_mountain}
Let $[v]$ be a critical point of $A_e$ such that, for all $m\in\mathds{N}$, the critical circle of $[v^m]$ is isolated in the set of critical points of $A_e$. There exists $m([v])\in\mathds{N}$ such that, for all integers $m>m([v])$, the following holds. There exists an (arbitrarily small) open neighborhood $\mathcal{W}$ of the critical circle of $[v^m]$ such that, if we set $a:=A_e([v^m])$, the inclusion induces an injective map between path-connected components
\[\tag*{\qed}
\pi_0(\{A_e < a\})\hookrightarrow\pi_0(\{A_e < a\}\cup\mathcal{W}).\]
\end{thm}

\subsection{The critical values of the energy}
Let us single out two significant values of the energy. The first one is
$e_0(L):=\max E(\cdot,0)$,
that is, the minimal energy $e$ such that the corresponding energy hypersurface $E^{-1}(e)$ projects onto the whole $S^2$. The second value $e_1(L,\sigma)\geq e_0(L)$, which depends also on the magnetic form $\sigma$, is defined as the supremum of the energies $e\geq e_0(L)$ verifying the following condition: there exists a finite collection $(\gamma_1,p_1),...,(\gamma_n,p_n)\in\mathcal{M}$ such that the $\gamma_i$'s are smooth pairwise disjoint loops, $E(\gamma_i(\cdot),\dot\gamma_i(\cdot)/p_i)\equiv e$ for all $i=1,...,n$, the multicurve $\gamma_1\cup...\cup\gamma_n$ is the oriented boundary of a positively oriented compact embedded surface $\Sigma\subseteq S^2$, and we have
\begin{align*}
S_e(\gamma_1,p_1)+...+ S_e(\gamma_n,p_n) + \int_\Sigma \sigma < 0 .
\end{align*}
We recall that $e_1(L,\sigma)$ reduces to the classical Ma\~n\'e critical value of $L+\theta$ in case $\sigma$ is exact with primitive $\theta$,  see ~\cite{Asselle:2015sp}.

The proof of Theorem~\ref{t:main} will build on the following existence result, which was originally proved by Taimanov \cite{Taimanov:1991el, Taimanov:1992sm} in the case of electromagnetic Lagrangians (see also \cite{Contreras:2004lv} for an alternative proof), and further extended by Asselle and Mazzucchelli \cite[Theorem~6.1]{Asselle:2016qv} to the general case of magnetic Tonelli systems.

\begin{thm}
\label{t:local_min}
For every energy value $e\in(e_0(L),e_1(L,\sigma))$, the Lagrangian system of $(L,\sigma)$ possesses a non-self-intersecting periodic orbit $(\gamma_e,p_e)$ with energy $e$ such that every element in $\pi^{-1}(\gamma_e,p_e)$ is a local minimizer of the action functional $A_e$.
\hfill\qed
\end{thm}

\section{Proof of the Main Theorem}
\label{s:proof}
In this section we carry out the proof of Theorem~\ref{t:main}. Since the case where the magnetic 2-form $\sigma$ is exact is covered by \cite{Abbondandolo:2014rb}, we focus on the case where $\sigma$ is not exact, so that
\begin{align}
\label{e:sigma_non_exact}
\int_{S^2}\sigma \neq 0.
\end{align}

\subsection{Minimax procedures}
For each energy value $e\in(e_0(L),e_1(L,\sigma))$, consider the local minimizer $(\gamma_e,p_e)$ of $A_e$ given by Theorem~\ref{t:local_min}, and choose an arbitrary $u_e\in\pi^{-1}(\gamma_e,p_e)$. We fix an arbitrary energy value
\[e_* \in (e_0(L),e_1(L,\sigma))\]
such that, for all $m\in\mathds{N}$, the iterated critical point $[u_{e_*}^m]$ belongs to a critical circle that is isolated in $\mathrm{crit}(A_{e_*})$ (if there is no energy value $e_*$ with such a property, there are infinitely many periodic orbits on every energy level in the range $(e_0(L),e_1(L,\sigma))$). The critical points $[u_{e_*}^m]$ are still local minimizers of $A_{e_*}$, as they are iterates of a local minimizer, see \cite[Lemma~3.1]{Abbondandolo:2015lt} and Remark~\ref{r:local}.

Given any subset $Y\subset\widetilde{\M}$, for each $m\in\mathds{N}$ we will write
\begin{align*}
Y^m
:=
\widetilde\psi^m(Y)
=
\big\{[y^m]\ \big|\ [y]\in Y\big\}.
\end{align*}
The Palais-Smale condition holds locally for the free-period action functional of Tonelli Lagrangians, see \cite[Prop.~3.12]{Contreras:2006yo} or \cite[Lemma~5.3]{Abbondandolo:2013is}. This, together with Remark~\ref{r:local}, implies that the functional $A_{e_*}$ satisfies the Palais-Smale condition locally as well. Therefore, a sufficiently small bounded open neighborhood $\mathcal{W}$ of the critical circle of $[u_{e_*}]$ does not contain other critical circles of $A_{e_*}$ and satisfies
\begin{align*}
 &\inf_{\partial \mathcal{W}} A_{e_*} > A_{e_*}([u_{e_*}]),\\
&\mathcal{W}^{m_0}\cap Z^{n_1}(\mathcal{W}^{m_1}) =\varnothing\ \mbox{ whenever }\ (m_0,0)\neq(m_1,n_1).
\end{align*}
For any $e\in(e_0(L),e_1(L,\sigma))$, we denote by $M_e$ the closure of the set of local minimizers of $A_e|_{\mathcal{W}}$. For all $m_0,m_1\in\mathds{N}$ and $n_0,n_1\in\mathds{Z}$ such that $(m_0,n_0)\neq(m_1,n_1)$, we denote by
\[\mathcal{P}_e(m_0,n_0,m_1,n_1)\]
the family of continuous paths $\Theta:[0,1]\to\widetilde{\M}$ such that $\Theta(0)\in Z^{n_0}(M_e^{m_0})$ and $\Theta(1)\in Z^{n_1}(M_e^{m_1})$. We define the corresponding minmax value
\begin{align*}
c_e(m_0,n_0,m_1,n_1):= \inf\big\{ \max A_e\circ\Theta\ \big|\ \Theta\in\mathcal{P}_e(m_0,n_0,m_1,n_1) \big\}.
\end{align*}

\begin{lem}\label{l:monotonicity}
There is an open neighborhood $I\subset(e_0(L),e_1(L,\sigma))$ of $e_*$ such that
\begin{itemize}
\item[(i)] $M_e$ is a non-empty compact set for all $e\in I$,
\item[(ii)] for each $e,e'\in I$, we have $\max A_{e'}|_{M_{e'}}<\inf A_e|_{\partial \mathcal{W}}$,
\item[(iii)] for each $m_0,m_1\in\mathds{N}$ and $n_0,n_1\in\mathds{Z}$, the function  $e\mapsto c_e(m_0,n_0,m_1,n_1)$ is well defined and monotone increasing in $I$.
\end{itemize}
\end{lem}
\begin{proof}
The proof is entirely analogous to the arguments in \cite[Lemmas~3.1--3.3]{Abbondandolo:2014rb} and it will be omitted.
\end{proof}

\subsection{The valley of short curves with low period}\label{ss:valley}

We equip our sphere $S^2$ with an arbitrary Riemannian metric $g$, and $\mathcal{M}$ with the Riemannian metric
\begin{equation}
\label{e:Riemannian_metric}
\begin{split}
\langle (\xi,r),(\eta,s) \rangle
=
\int_{\mathds{T}} \Big(g(\xi,\eta) + g(D_t\xi,D_t\eta) \Big)\,\mathrm{d} t
+
rs,
\\
\forall (\xi,r),(\eta,s)\in\mathrm{T}_{(\gamma,p)}\mathcal{M},
\end{split}
\end{equation}
where $D_t$ denotes the covariant derivative associated to $g$.
The space $\mathcal{M}$ is not complete with respect to the Riemannian metric \eqref{e:Riemannian_metric}, nor is its universal cover equipped with the pulled-back Riemannian metric. Indeed, there are Cauchy sequences $\{(\gamma_n,p_n)\ |\ n\in\mathds{N}\}\subset\mathcal{M}$ such that $p_n\to0$. However, it turns out that this does not pose any problem while applying arguments from non-linear analysis to the functional $A_e$. Indeed, the functional $A_e$ has a ``valley'' near the non-complete ends of $\mathcal{M}$, as we will review now (see \cite[Section~3]{Contreras:2006yo} and \cite[Section~3]{Asselle:2014hc} for analogous arguments in slightly different settings).

We write $\|\dot\gamma\|_{L^2}$ for the $L^2$-norm of the derivative of any curve $\gamma\in W^{1,2}(\mathds{T};S^2)$ measured with respect to $g$, i.e.
\begin{align*}
 \|\dot\gamma\|_{L^2}^2 = \int_{\mathds{T}} g(\dot\gamma(t),\dot\gamma(t))\,\mathrm{d} t.
\end{align*}
We introduce the open subsets
\begin{align}
\label{e:U_delta}
\mathcal{U}_\tau:=
\big\{
(\gamma,p)\in\mathcal{M}\ \big|\ \|\dot\gamma\|_{L^2}^2<\tau\,p,\ \ p<\tau
\big\},\qquad\tau>0.
\end{align}
If $\tau$ is small enough, $\mathcal{U}_\tau$ is connected and evenly covered by $\pi:\widetilde{\M}\to\mathcal{M}$. Namely, there exists a connected component $\mathcal{V}_\tau\subset\pi^{-1}(\mathcal{U}_\tau)$ such that $\pi^{-1}(\mathcal{U}_\tau)$ can be written as a disjoint union
\begin{align*}
\pi^{-1}(\mathcal{U}_\tau)
=
\bigsqcup_{n\in\mathds{Z}}
Z^n(\mathcal{V}_\tau).
\end{align*}
We choose such a connected component $\mathcal{V}_\tau$ so that, for all $[u]=[(\gamma,p)]\in\mathcal{V}_\tau$ with $\gamma(1)$  stationary curve at some point $q\in S^2$, we have
$$A_e([u])=p(1)\big(L(q,0)+e\big).$$

\begin{lem}\label{l:constant_loops}
For all $\tau>0$ sufficiently small, we have
\begin{align*}
\inf A_e|_{\mathcal{V}_\tau}=0,
\qquad
\inf A_e|_{\partial\mathcal{V}_\tau}>0.
\end{align*}
Moreover
\begin{equation*}
\lim_{\tau\to 0^+}\big( \sup A_e|_{\mathcal{V}_\tau} \big)=0 .
\end{equation*}
\end{lem}

\begin{proof}
We cover the sphere with two open balls $D_1,D_2\subset S^2$, and choose a primitive $\theta_i$ of $\sigma$ on $D_i$. Let $\tau>0$ be sufficiently small so that for any $\gamma\in W^{1,2}(\mathds{T};S^2)$ with length less than $\tau$ there exists $\iota(\gamma)\in\{1,2\}$ such that $\gamma$ is entirely contained in $D_{\iota(\gamma)}$. The restriction of the functional $A_e$ to $\mathcal{V}_{\tau}$ takes the following form: for each $[u]\in\mathcal{V}_{\tau}$ with $(\gamma,p):=\pi([u])$, we have
\begin{align*}
A_e([u])
=
p \int_{\mathds{T}} L(\gamma(t),\dot\gamma(t)/p)\,\mathrm{d} t + p\,e + \int_\gamma \theta_{\iota(\gamma)}.
\end{align*}
Since we are assuming that the restriction of the Tonelli Lagrangian $L$ to any fiber of $\mathrm{T} M$ is a polynomial of degree 2 outside a compact set, there exist  constants $0<h_1<h_2$ such that, for all $(q,v)\in\mathrm{T} M$, we have
\begin{equation}\label{e:lower_bound_L}
\begin{split}
L(q,v) &\geq L(q,0) + \partial_vL(q,0)v + h_1\,g_q(v,v)\\
& \geq -E(q,0) + \partial_vL(q,0)v + h_1\,g_q(v,v)\\
& \geq -e_0(L)  + \partial_vL(q,0)v + h_1\,g_q(v,v),
\end{split}
\end{equation}
and
\begin{align}\label{e:upper_bound_L}
L(q,v) &\leq h_2 \big(g_q(v,v)+1\big).
\end{align}
We denote by $\lambda$ the 1-form on $S^2$ given by $\partial_vL(\cdot,0)$. The lower bound~\eqref{e:lower_bound_L} implies that, for all $[u]\in\mathcal{V}_\tau$ with $(\gamma,p):=\pi([u])$, we have
\begin{align*}
A_e([u]) & \geq h_1 \frac{\|\dot\gamma\|_{L^2}^2}{p} + \underbrace{\big(e-e_0(L)\big)}_{>0}\,p - \left| \int_{\gamma}(\lambda + \theta_{\iota(\gamma)}) \right|\\
& \geq  h_1 \frac{\|\dot\gamma\|_{L^2}^2}{p} + \big(e-e_0(L)\big)\,p - \frac14 \|\mathrm{d} \lambda + \underbrace{\mathrm{d}\theta_{\iota(\gamma)}}_\sigma\|_{L^\infty}\,\|\dot\gamma\|_{L^2}^2
\end{align*}
where the latter inequality follows from \cite[Lemma~7.1]{Abbondandolo:2013is}. This readily implies that $A_e>0$ on $\mathcal{V}_\tau$ provided
\begin{align*}
\frac{h_1}{\tau} > \frac14  \|\mathrm{d} \lambda + \sigma\|_{L^\infty}.
\end{align*}
Assume now that $[u]\in\partial\mathcal{V}_\tau$. If $p=\tau$, we have
\begin{align*}
A_e([u]) > (e-e_0(L))\,\tau >0.
\end{align*}
If  $p<\tau$, then $\|\dot\gamma\|_{L^2}^2=p\,\tau$, and therefore
\begin{align*}
A_e([u]) & \geq  h_1 \tau  - \frac14 \|\mathrm{d} \lambda + \sigma\|_{L^\infty}\,\tau^2 >0.
\end{align*}
Overall, this proves that $\inf A_e|_{\partial\mathcal{V}_\tau}>0$.

Inequality~\eqref{e:upper_bound_L} implies that, for all $[u]\in\mathcal{V}_\tau$ with $(\gamma,p):=\pi([u])$, we have
\begin{align*}
A_e([u])
& \leq
h_2 \frac{\|\dot\gamma\|_{L^2}^2}{p} + h_2\,p + e\,p + \int_{\gamma} \theta_{\iota(\gamma)}\\
& \leq
h_2 \frac{\|\dot\gamma\|_{L^2}^2}{p} + h_2\,p + e\,p + \frac14 \|\sigma\|_{L^\infty}\,\|\dot\gamma\|_{L^2}^2\\
& \leq h_2\,\tau + h_2\,\tau + e\,\tau + \frac14  \|\sigma\|_{L^\infty} \,\tau^2,
\end{align*}
where, as before, the second inequality follows from \cite[Lemma~7.1]{Abbondandolo:2013is}. This readily implies that $\sup A_e|_{\mathcal{V}_\tau} \to 0$ as $\tau\to0^+$, which, together with the fact that $A_e>0$ on $\mathcal{V}_\tau$, also implies that $\inf A_e|_{\mathcal{V}_\tau} = 0$.
\end{proof}

\subsection{Essential families}\label{ss:essential_families}

Let us fix an energy value $e\in I$. We say that a union of critical circles
\[
\mathcal{E} \subset \mathrm{crit}  A_e\cap \{ A_e=c_e(m_0,n_0,m_1,n_1)\}
\]
is an \textbf{essential family} for $\mathcal{P}_e(m_0,n_0,m_1,n_1)$ when for every neighborhood $\mathcal{U}$ of $\mathcal{E} $ there exists a path $\Theta\in\mathcal{P}_e(m_0,n_0,m_1,n_1)$ whose image $\Theta([0,1])$ is contained in the union $\mathcal{U}\cup\{A_e<c_e(m_0,n_0,m_1,n_1)\}$.

We denote by $I_\mathrm{discr}$ the subset of those $e\in(e_0(L),e_1(L,\sigma))$ such that the set of critical points $\mathrm{crit}(A_e)$ is a union of isolated critical circles (that is, the periodic orbits with energy $e$ are isolated). Notice that  every energy level $e\in(e_0(L),e_1(L,\sigma))\setminus I_\mathrm{discr}$ contains infinitely many periodic orbits. The existence of essential families can be guaranteed on generic energy levels in $I_\mathrm{discr}$. The precise statement is the following.

\begin{lem}\label{l:Palais_Smale}
There is a subset $I'\subseteq I$ of full Lebesgue measure such that, for all $e\in I'\capI_\mathrm{discr}$, $m_0,m_1\in\mathds{N}$, and $n_0,n_1\in\mathds{Z}$ with $(m_0,n_0)\neq(m_1,n_1)$, the space of paths $\mathcal{P}_e(m_0,n_0,m_1,n_1)$ admits an essential family.
\end{lem}
\begin{proof}
The proof goes along the lines of the one of \cite[Lemma~3.5]{Abbondandolo:2014rb}, but the fact that we are working on the universal cover of $\mathcal{M}$ with the functional $A_e$ requires some variations of the original argument, and therefore we provide full details for the reader's convenience.

For all $m_0,m_1\in\mathds{N}$ and $n_0,n_1\in\mathds{Z}$ such that $(m_0,n_0)\neq(m_1,n_1)$, we denote by $I(m_0,n_0,m_1,n_1)$ the subset of those $e'\in I$ such that the function
\begin{align}
\label{e:minmax_function}
e\mapsto c_e(m_0,n_0,m_1,n_1)
\end{align}
is differentiable at $e'$. By Lemma~\ref{l:monotonicity}(iii), the function~\eqref{e:minmax_function} is monotone increasing in $e$, and therefore $I(m_0,n_0,m_1,n_1)$ is a full measure subset of $I$. We define the subset $I'$ of the statement as
\begin{align*}
I':=  \!\!\!\!\!\!\!\!\! \bigcap_{(m_0,n_0)\neq(m_1,n_1)} \!\!\!\!\!\!\!\!\! I(m_0,n_0,m_1,n_1).
\end{align*}
Being a countable intersection of full Lebesgue measure subsets of $I$, the subset $I'\subset I$ has full Lebesgue measure as well.

Now, we fix $e\in I'\capI_\mathrm{discr}$ and two distinct $(m_0,n_0),(m_1,n_1)\in\mathds{N}\times\mathds{Z}$. In order to simplify the notation, we will just write $c_e$ and $\mathcal{P}_e$ for $c_e(m_0,n_0,m_1,n_1)$ and $\mathcal{P}_e(m_0,n_0,m_1,n_1)$ respectively. We choose an arbitrary strictly decreasing sequence $\{e_\alpha\ |\ \alpha\in\mathds{N}\}\subset I$ such that $e_\alpha\to e$ as $\alpha\to\infty$, and we set $\epsilon_\alpha:=e_\alpha-e$. By definition of $I'$, there exists  $k_0=k_0(e)>0$ such that
\begin{align*}
|c_{e_\alpha}-c_e| \leq k_0\epsilon_\alpha,\qquad
\forall \alpha\in\mathds{N}.
\end{align*}
For all $[u]=[(\gamma,p)]\in\widetilde{\M}$ such that $A_e([u])\geq c_e-\epsilon_\alpha$ and $A_{e_\alpha}([u])\leq c_{e_\alpha}+\epsilon_\alpha$, the period $p(1)$ of the curve $u(1)\in\mathcal{M}$ can be bounded as
\begin{align*}
p(1)
=
\frac{A_{e_\alpha}([u])-A_{e}([u])}{\epsilon_\alpha}
\leq
\frac{c_{e_\alpha}+\epsilon_\alpha-c_{e}+\epsilon_\alpha}{\epsilon_\alpha}
\leq
k_0+2=:k_1,
\end{align*}
while the action $A_e([u])$ can be bounded as
\begin{align*}
A_e([u])\leq A_{e_\alpha}([u])\leq c_{e_\alpha} + \epsilon_\alpha
\leq c_e + (k_0+1)\epsilon_\alpha
\leq c_e + k_1\epsilon_\alpha.
\end{align*}
We introduce the subspaces
\begin{align*}
\mathcal{X}_r :=
\Big\{
[u]=[(\gamma,p)]\in\widetilde{\M}\ \Big|\ p(1)\leq r
\Big\},
\qquad
r>0.
\end{align*}
By the definition of the minmax value $c_{e_\alpha}$ and by the estimates that we have just provided, for each $\alpha\in\mathds{N}$ there exists a path $\Theta_\alpha\in\mathcal{P}_{e_\alpha}$ such that
\begin{align*}
\Theta_\alpha([0,1])
\subset
\{A_e \leq c_e-\epsilon_\alpha\} \cup \big( \mathcal{X}_{k_1} \cap \{A_e\leq c_e + k_1\epsilon_\alpha\} \big).
\end{align*}
We recall that, by the definition of the spaces of paths $\mathcal{P}_{e_\alpha}$, we have that
\[\Theta_\alpha(i)\in Z^{n_i}(M_{e_\alpha}^{m_i}) \subset Z^{n_i}(\mathcal{W}^{m_i}),\qquad i=0,1.\]
Lemma~\ref{l:monotonicity}(ii) readily implies that we can attach two suitable tails to the path $\Theta_\alpha$: we can find two continuous paths
\begin{align*}
\Phi_\alpha: & [0,1]\to Z^{n_0}(\mathcal{W}^{m_0})\cap\{A_e\leq A_e(\Theta_\alpha(0))\},\\
\Psi_\alpha: & [0,1]\to Z^{n_1}(\mathcal{W}^{m_1})\cap\{A_e\leq A_e(\Theta_\alpha(1))\},
\end{align*}
such that $\Phi_\alpha(0)\in Z^{n_0}(M_{e}^{m_0})$, $\Phi_\alpha(1)=\Theta_\alpha(0)$, $\Psi_\alpha(0)=\Theta_\alpha(1)$, and $\Psi_\alpha(1)\in Z^{n_1}(M_{e}^{m_1})$; see \cite[Lemma~3.2]{Abbondandolo:2014rb} for a proof of this elementary fact. Since the open set $\mathcal{W}$ is bounded, there exists $k_2>k_1$ large enough such that
\begin{align*}
Z^{n_0}(\mathcal{W}^{m_0}) \cup Z^{n_1}(\mathcal{W}^{m_1})\subset \mathcal{X}_{k_2}.
\end{align*}
We define the continuous path
\begin{align*}
&\Upsilon_\alpha:[0,1] \to \{A_e \leq c_e-\epsilon_\alpha\} \cup \big( \mathcal{X}_{k_2} \cap \{A_e \leq c_e+k_1\epsilon_\alpha\} \big),\\
&\Upsilon_\alpha(s):=
\left\{
  \begin{array}{lll}
    \Phi_\alpha(3s), &  & s\in[0,1/3], \vspace{5pt}\\
    \Theta_\alpha(3(s-1/3)), &  & s\in[1/3,2/3], \vspace{5pt}\\
    \Psi_\alpha(3(s-2/3)), &  & s\in[2/3,1]. \\
  \end{array}
\right.
\end{align*}
Notice that $\Upsilon_\alpha\in\mathcal{P}_e$, and $\max A_e\circ\Upsilon_\alpha\to c_e$ as $\alpha\to\infty$.

We claim that $\mathrm{crit}(A_e)\cap A_e^{-1}(c_e)\cap\mathcal{X}_{k_2+2}$ is an essential family for $\mathcal{P}_e$. Let $\mathcal{U}\subset\widetilde{\M}$ be an arbitrary open set such that
\begin{align*}
\mathcal{U}\cap\mathrm{crit}(A_e) = \mathrm{crit}(A_e)\cap A_e^{-1}(c_e)\cap\mathcal{X}_{k_2+2}.
\end{align*}
Our goal for the remaining of the proof is to deform one of our paths $\Upsilon_\alpha$, away from its endpoints, so that the modified path will have image inside $\{A_e<c_e\}\cup\mathcal{U}$. Notice that, since $e\inI_\mathrm{discr}$, if $\mu>0$ is small enough we have
\begin{align}\label{e:critical_points_in_U}
\mathcal{U}\cap\mathrm{crit}(A_e) = \mathrm{crit}(A_e)\cap A_e^{-1}[c_e-\mu,c_e+\mu]\cap\mathcal{X}_{k_2+2},
\end{align}
and $\mathcal{U}$ contains at most finitely many critical circles of $A_e$. In particular, we can find a smaller open neighborhood $\mathcal{U}'\subset\mathcal{U}$ of $\mathcal{U}\cap\mathrm{crit}(A_e)$ and some $\ell>0$ such that every smooth path $\Theta:[0,1]\to\overline\mathcal{U}$ with $\Theta(0)\in\mathcal{U}'$ and $\Theta(1)\in\partial\mathcal{U}$ has length at least $\ell$. Here, the length is the one measured with respect to the pull-back of the Riemannian metric \eqref{e:Riemannian_metric} to the universal cover $\widetilde{\M}$.

Consider the open subsets $\mathcal{U}_\tau\subset\mathcal{M}$ introduced in~\eqref{e:U_delta}, and the selected connected components of their preimage $\mathcal{V}_\tau\subset\pi^{-1}(\mathcal{U}_\tau)$.
Since $e\inI_\mathrm{discr}$, the set $M_e$ is the union of finitely many critical circles of $A_e$. In particular, there exists $\tau_2>0$ small enough such that
\begin{align*}
 \{\Upsilon_\alpha(0),\Upsilon_\alpha(1)\ |\ \alpha\in\mathds{N}\} \cap \pi^{-1}(\mathcal{U}_{\tau_2}) =\varnothing.
\end{align*}
If needed, we reduce $\tau_2>0$ so that the open subset $\mathcal{U}_{\tau_2}$ is connected and evenly covered by $\pi:\widetilde{\M}\to\mathcal{M}$. By Lemma~\ref{l:constant_loops}, there exist $\delta>0$ and $0<\tau_1<\tau_2$ such that, for all $n\in\mathds{N}$,
\begin{align}\label{e:estimates_near_stationary_curves_outer}
\inf A_e|_{\partial(Z^n(\mathcal{V}_{\tau_2}))}-\sup A_e|_{Z^n(\mathcal{V}_{\tau_1})}\geq\delta.
\end{align}
Finally, we fix an index $\alpha\in\mathds{N}$ large enough so that
\begin{align}\label{e:bound_epsilon_alpha}
k_1\epsilon_\alpha<\min\big\{\mu,\delta\big\}.
\end{align}

In the following, we will denote by $\|\cdot\|$ the Riemannian norm induced by the Riemannian metric~\eqref{e:Riemannian_metric}. With a slight abuse of notation, we will denote by $\|\cdot\|$ also the Riemannian norm that is pulled-back to the universal cover $\widetilde{\M}$.
Fix $\tau_0\in(0,\tau_1)$ and introduce a vector field on $\widetilde{\M}$ of the form $V:=f\,\nabla A_e$, for some suitable smooth function $f:\widetilde{\M}\to[-1,0]$, such that
\begin{itemize}
\item[(i)] $\|V([u])\|\leq2$ for all $[u]\in\widetilde{\M}$,

\item[(ii)] $\mathrm{supp}(V)\subset A_e^{-1}[c_e-\epsilon_{\alpha-1},c_e + k_1\epsilon_{\alpha-1}]\setminus \pi^{-1}(\mathcal{U}_{\tau_0})$,

\item[(iii)] $\mathrm{d} A_e([u])V([u])\leq - \min\big\{\|\nabla A_e([u])\|^2,1\big\}$ for all $[u]\in \widetilde{\M}\setminus\pi^{-1}(\mathcal{U}_{\tau_1})$ such that $A_e([u])\in[c_e-\epsilon_{\alpha},c_e + k_1\epsilon_{\alpha}]$.
\end{itemize}
We denote by $\phi_t:\widetilde{\M}\to\widetilde{\M}$ the flow of $V$. This flow is complete. Indeed, since the vector field $V$ is uniformly bounded, the flow lines that may not be defined for all positive time are those that enter all sets $\mathcal{X}_r$, for $r>0$ arbitrarily small. Since $V$ is non-negatively  proportional to $-\nabla A_e$, its flow lines are non-negative reparametrizations of those of $-\nabla A_e$. Finally, if a flow line of $-\nabla A_e$ is not defined for all positive times, then it must enter the set $\pi^{-1}(\mathcal{U}_{\tau_0})$ (see \cite[Proposition~3.1(2)]{Asselle:2014hc} for a proof of this fact), but this latter set is outside the support of $V$. Actually, since $\|V\|\leq 2$, we have
\begin{align*}
\phi_1(\mathcal{X}_{k_2})\subset\mathcal{X}_{k_2+2}.
\end{align*}

The free-period action form $\eta_e$ satisfies a generalized Palais-Smale condition on subsets of $\mathcal{M}$ where the period is bounded from above and bounded away from zero, see \cite[Theorem~2.1(2)]{Asselle:2014hc}. Moreover, for each sequence $\{(\gamma_n,p_n)\ |\ n\in\mathds{N}\}\subset\mathcal{M}$ such that $p_n\to0$ and $\|\eta_e(\gamma_n,p_n)\|\to0$ as $n\to\infty$, we have $\|\dot\gamma_n\|^2_{L^2}/p_n\to0$ as $n\to\infty$,
see \cite[Theorem~2.1(1)]{Asselle:2014hc}. In particular, $(\gamma_n,p_n)$ belongs to $\mathcal{U}_{\tau_0}$ for $n$ large enough.  This, together with~\eqref{e:critical_points_in_U}, implies that  there exists a constant $\nu\in(0,1)$ such that
\begin{align}\label{e:nabla_A_e_bounded_away_from_zero}
\| \nabla A_e([u]) \| \geq \nu,
\quad
\forall [u]\in A_e^{-1}[c_e-\mu,c_e+\mu]\cap\mathcal{X}_{k_2+2} \setminus(\pi^{-1}(\mathcal{U}_{\tau_0})\cup\mathcal{U}').
\end{align}

We fix an index $\beta\geq\alpha$ large enough so that $k_1\epsilon_\beta<\min\big\{ \ell\nu,\nu^2 \big\}$, which together with~\eqref{e:bound_epsilon_alpha} implies
\begin{align}\label{e:bound_epsilon_beta}
k_1\epsilon_\beta < \min\big\{\mu,\delta,\ell\nu,\nu^2  \big\}.
\end{align}
The composition $\phi_1\circ\Upsilon_{\beta}$ belongs to $\mathcal{P}_{e}$. We claim that its image $\phi_1\circ\Upsilon_{\beta}([0,1])$ is contained in $\{A_e<c_e\}\cup\mathcal{U}$, which sets our goal for the proof. First of all, since $A_e$ does not increase along the flow lines of $\phi_t$, we have
\begin{align}
\label{e:upper_bound_path}
A_e(\phi_t\circ\Upsilon_{\beta})\leq A_e(\Upsilon_{\beta})\leq c_e+k_1\epsilon_\beta,\qquad\forall t\in[0,1].
\end{align}
There are three possible cases to consider:
\begin{itemize}
\item If $\phi_t\circ\Upsilon_{\beta}(s)\in \mathcal{U}'$ for some $t\in[0,1]$ and $\phi_1\circ\Upsilon_{\beta}(s)\not\in\mathcal{U}$, Equations \eqref{e:nabla_A_e_bounded_away_from_zero}, \eqref{e:bound_epsilon_beta}, and \eqref{e:upper_bound_path} imply that
\begin{align*}
\qquad\qquad A_e(\phi_1\circ\Upsilon_{\beta}(s))
&
=
A_e(\phi_t\circ\Upsilon_{\beta}(s))
+
\int_t^1
\mathrm{d} A_e(\phi_r\circ\Upsilon_{\beta}(s))V(\phi_r\circ\Upsilon_{\beta}(s))\,\mathrm{d} r\\
&
\leq c_e + k_1\epsilon_\beta - \nu \int_t^1 \|V(\phi_r\circ\Upsilon_{\beta}(s))\|\,\mathrm{d} r
\\
& \leq
c_e + k_1\epsilon_\beta - \ell\nu\\
& < c_e.
\end{align*}

\item If $\phi_t\circ\Upsilon_{\beta}(s)\in\pi^{-1}(\mathcal{U}_{\tau_1})$ for some $t\in[0,1]$, then, since $\phi_t\circ\Upsilon_{\beta}(s)\not\in\pi^{-1}(\mathcal{U}_{\tau_2})$, Equations \eqref{e:estimates_near_stationary_curves_outer},  \eqref{e:bound_epsilon_beta}, and \eqref{e:upper_bound_path} imply
\begin{align*}
A_e(\phi_1\circ\Upsilon_{\beta}(s))
\leq
A_e(\phi_t\circ\Upsilon_{\beta}(s))
\leq
c_{e} + k_1\epsilon_{\beta} - \delta
< c_e.
\end{align*}

\item If $\phi_t\circ\Upsilon_{\beta}(s)\not\in \mathcal{U}'\cup \pi^{-1}(\mathcal{U}_{\tau_1})$ for all $t\in[0,1]$, then property~(iii) in the definition of $V$ above, together with Equations~\eqref{e:nabla_A_e_bounded_away_from_zero}, \eqref{e:bound_epsilon_beta}, and \eqref{e:upper_bound_path}, implies
\begin{align*}
\qquad A_e(\phi_1\circ\Upsilon_{\beta}(s))
&
=
A_e(\Upsilon_{\beta}(s))
+
\int_0^1
\mathrm{d} A_e(\phi_r\circ\Upsilon_{\beta}(s))V(\phi_r\circ\Upsilon_{\beta}(s))\,\mathrm{d} r\\
&
=
c_e + k_1\epsilon_{\beta}
-
\int_0^1
\|\mathrm{d} A_e(\phi_r\circ\Upsilon_{\beta}(s))\|^2\,\mathrm{d} r\\
&\leq c_e + k_1\epsilon_{\beta} - \nu^2\\
&<c_e.
\end{align*}
\end{itemize}
Overall, we showed that, for an arbitrary $s\in[0,1]$, if $\phi_1\circ\Upsilon_{\beta}(s)$ is not contained in $\mathcal{U}$, then it is contained in the sublevel set $\{A_e<c_e\}$.
\end{proof}

\begin{lem}\label{l:iterated_non_mountain_pass}
For each $e\in I'\capI_\mathrm{discr}$ and $[v]\in\mathrm{crit}(A_e)$, there exists a constant $m([v])\in\mathds{N}$ with the following property. Consider the critical circle $\mathcal{C}$ of a critical point $Z^n([v^m])$, where $n\in\mathds{Z}$ and $m>m([v])$. If $\mathcal{E}$ is an essential family containing $\mathcal C$, then $\mathcal{E}\setminus\mathcal{C}$ is an essential family for the same space of paths as well.
\end{lem}
\begin{proof}
We set
$a_{m,n}:=A_e(Z^n([v^m]))$, where $m\in\mathds{N}$ and $n\in\mathds{Z}$.
By Theorem~\ref{t:non_mountain}, there exists $m([v])\in\mathds{N}$ such that, for all integers $m>m([v])$, the following statement holds. There exists an (arbitrarily small) open neighborhood $\mathcal{W}$ of the critical circle of $[v^m]$ such that the inclusion induces an injective map between path-connected components
\[ \pi_0\big(\{A_e<a_{m,0}\}\big) \hookrightarrow \pi_0\big(\{A_e<a_{m,0}\}\cup\mathcal{W}\big).
\]
For every $n\in\mathds{Z}$, we denote by $\mathcal{W}_n:=Z^n(\mathcal{W})$ the corresponding neighborhood of the critical circle $\mathcal C$ of $Z^n([v^m])$. Clearly, the inclusion induces an injective map
\begin{align}\label{e:injective_pi_0}
 \pi_0\big(\{A_e<a_{m,n}\}\big) \hookrightarrow \pi_0\big(\{A_e<a_{m,n}\}\cup\mathcal{W}_n\big).
\end{align}
Now, assume that $\mathcal C$ belongs to an essential family $\mathcal{E} $ for $\mathcal{P}_e(m_0,n_0,m_1,n_1)$. In particular $a_{m,n}=c_e(m_0,n_0,m_1,n_1)$.

We require the neighborhood $\mathcal{W}$ to be small enough so that for all neighborhoods $\mathcal{W}'$ of $\mathcal{E}\setminus\mathcal C$ sufficiently small, we have $\overline{\mathcal{W}}_n\cap \overline{\mathcal{W}'}=\varnothing$. The existence of such a disjoint $\overline{\mathcal{W}'}$ is guaranteed by the fact that the set of critical points of $A_e$ comes in isolated critical circles. Since, by Lemma~\ref{l:Palais_Smale}, $\mathcal{P}_e(m_0,n_0,m_1,n_1)$ admits an essential family, there exists a continuous path $\Theta\in\mathcal{P}_e(m_0,n_0,m_1,n_1)$ whose image is contained in the union
\[\mathcal{W}_n\cup\mathcal{W}'\cup\{A_e<c_e(m_0,n_0,m_1,n_1)\}.\]
Notice that
\begin{align}\label{e:endpoints_below}
\max\big\{A_e(\Theta(0)),A_e(\Theta(1))\big\} < c_e(m_0,n_0,m_1,n_1).
\end{align}
Indeed, $\Theta(0)$ and $\Theta(1)$ belong to distinct critical circles that are isolated local minimizers of $A_e$, and this latter functional satisfies the Palais-Smale condition locally.

By~\eqref{e:endpoints_below} and since the map~\eqref{e:injective_pi_0} is injective, there exists another path $\Theta'\in\mathcal{P}_e(m_0,n_0,m_1,n_1)$ whose image is contained in the union
\[\mathcal{W}'\cup\{A_e<c_e(m_0,n_0,m_1,n_1)\}.\]
Therefore, $\mathcal{E}\setminus\mathcal C$ is also an essential family for $\mathcal{P}_e(m_0,n_0,m_1,n_1)$.
\end{proof}

Now, let $I_\mathrm{finite}$ be the (possibly empty) subset of those energy values $e\inI_\mathrm{discr}$ such that there are only finitely many (non-iterated) periodic orbits with energy $e$. In order to prove Theorem~\ref{t:main}, all we need to do is to prove that the intersection $I'\capI_\mathrm{finite}$ is empty. We will show this in Theorem~\ref{t:I'}, after exploring what would happen on energy values in $I'\capI_\mathrm{finite}$.

\begin{lem}\label{cl:finite_essential_family_1}
For each energy level $e\in I'\capI_\mathrm{finite}$ and compact interval $[a_0,a_1]\subset\mathds{R}$, there exists a finite union of critical circles $\mathcal{E}\subset\mathrm{crit}(A_e)$ such that, for all $m_0,m_1\in\mathds{N}$ and $n_0,n_1\in\mathds{Z}$ with $c_e(m_0,n_0,m_1,n_1)\in[a_0,a_1]$, $\mathcal{E}$ contains an essential family for $\mathcal{P}_e(m_0,n_0,m_1,n_1)$.
\end{lem}

\begin{proof}
Let $(\gamma_1,p_1),...,(\gamma_r,p_r)$ be the only non-iterated periodic orbits with energy $e$, where $r$ is some natural number, and choose $[v_i]\in\pi^{-1}(\gamma_i,p_i)$ for all $i=1,...,r$. Consider the constants $m([v_i])\in\mathds{N}$ given by Lemma~\ref{l:iterated_non_mountain_pass}, so that if we remove the critical circle of any $Z^n([v_i^m])$ with $n\in\mathds{Z}$ and $m>m_{\max}$ from an essential family contained in $A_e^{-1}(c_e(m_0,n_0,m_1,n_1))$, the result is still an essential family for the same space of paths. We set
\begin{align*}
m_{\max} := \max\big\{ m([v_1]), ..., m([v_r]) \big\} \in \mathds{N}
\end{align*}
By Equations~\eqref{e:action_shift} and~\eqref{e:sigma_non_exact}, we infer that there exists $n_{\max}\in\mathds{N}$ such that
$A_e(Z^n([v_i^m]))\not\in[a_0,a_1]$ for all $i\in\{1,...,r\}$, $m\in\mathds{N}$, and $n\in\mathds{Z}$ with $m\leq m_{\max}$ and $|n|> n_{\max}$. We claim that the statement of the lemma holds taking
\begin{align*}
\mathcal{E} :=
\Big\{
Z^n([v_i^m])\
\Big|\
i\in\{1,...,r\},\  1\leq m\leq m_{\max},\  |n|\leq n_{\max}
\Big\}.
\end{align*}
Indeed, consider $m_0,m_1\in\mathds{N}$ and $n_0,n_1\in\mathds{Z}$ such that $c_e(m_0,n_0,m_1,n_1)\in[a_0,a_1]$. Let $\mathcal{E}'$ be an essential family for $\mathcal{P}_e(m_0,n_0,m_1,n_1)$, whose existence is guaranteed by Lemma~\ref{l:Palais_Smale}. By Lemma~\ref{l:iterated_non_mountain_pass}, if we remove from $\mathcal{E}'$ all the critical circles of periodic orbits of the form $Z^n([v_i^m])$ for $m>m_{\max}$, the resulting set is still an essential family for $\mathcal{P}_e(m_0,n_0,m_1,n_1)$. Therefore, $\mathcal{E}'\cap\mathcal{E}$ is an essential family for $\mathcal{P}_e(m_0,n_0,m_1,n_1)$.
\end{proof}

Let $m_0\in\mathds{N}$ and $n_0\in\mathds{Z}$. For all $m_1\in\mathds{N}$, and $n_1\in\mathds{Z}$ we know that $c_e(m_0,n_0,m_1,n_1)$ is bounded from below by $\min A_e|_{Z^{n_0}(M_e^{m_0})}$. Hence, the following quantity is a well-defined real number:
\begin{align*}
c_e(m_0,n_0)
:=
\inf
\left\{
c_e(m_0,n_0,m_1,n_1)\  \left|\
  \begin{array}{@{}l@{}}
    m_1\in\mathds{N},\ n_1\in\mathds{Z} \\
    \mbox{with }(m_1,n_1)\neq(m_0,n_0)
  \end{array}
\right.\right\}.
\end{align*}
Notice that the deck transformation $Z^k$ induces a homeomorphism between the spaces of paths $\mathcal{P}_e(m_0,n_0,m_1,n_1)$ and $\mathcal{P}_e(m_0,n_0+k,m_1,n_1+k)$, and we have
\begin{align*}
 c_e(m_0,n_0+k,m_1,n_1+k) = c_e(m_0,n_0,m_1,n_1) + k \int_{S^2}\sigma,\qquad\forall k\in\mathds{Z}.
\end{align*}
This readily implies
\begin{align}
\label{e:shift_c}
 c_e(m_0,n_0+k) = c_e(m_0,n_0) + k \int_{S^2}\sigma,\qquad\forall k\in\mathds{Z}.
\end{align}
The infimum in the definition of $c_e(m_0,n_0)$ is actually attained provided $e\in I'\capI_\mathrm{finite}$.

\begin{lem}\label{cl:finite_essential_family_2}
If $e\in I'\capI_\mathrm{finite}$, for all $(m_0,n_0)\in\mathds{N}\times\mathds{Z}$ there exist $(m_1,n_1)\in\mathds{N}\times\mathds{Z}$ such that $(m_0,n_0)\neq(m_1,n_1)$ and
$c_e(m_0,n_0) = c_e(m_0,n_0,m_1,n_1)$.
\end{lem}
\begin{proof}
Let us fix $(m_0,n_0)\in\mathds{N}\times\mathds{Z}$, and set
\begin{align*}
a_0:= \min A_e|_{Z^{n_0}(M_e^{m_0})} \leq c_e(m_0,n_0,m_0+1,n_0) =:a_1.
\end{align*}
Notice that $c_e(m_0,n_0)\in[a_0,a_1]$. By Lemma~\ref{cl:finite_essential_family_1}, there exists a finite union of critical circles $\mathcal{E}\subset\mathrm{crit}(A_e)$ such that, whenever $c_e(m_0,n_0,m_1,n_1)\in[a_0,a_1]$, $\mathcal{E}$ contains an essential family for $\mathcal{P}_e(m_0,n_0,m_1,n_1)$. We introduce the finite set of critical values
\begin{align*}
F:= \big\{A_e([w])\ \big|\ [w]\in\mathcal{E}\big\}.
\end{align*}
The value $c_e(m_0,n_0)$ is the infimum of those $c_e(m_0,n_0,m_1,n_1)$ belonging to the finite set $F$, and therefore it is a minimum.
\end{proof}

\subsection{The main multiplicity result}

Theorem~\ref{t:main} is an immediate consequence of the following more precise statement.

\begin{thm}\label{t:I'}
The set $I'\cap I_{\mathrm{finite}}$ is empty. Namely, for all energy values $e\in I'$, there are infinitely many periodic orbits with energy $e$.
\end{thm}

In the proof of Theorem~\ref{t:I'}, we will need the following abstract lemma established in \cite[Lemma~2.5]{Abbondandolo:2014rb} for the free-period action functional. Being a local statement, such a lemma holds for the functional $A_e$ as well (see Remark~\ref{r:local}).
\begin{lem}
\label{l:local_homology_finite_rank}
Every isolated critical circle $\mathcal C\subset\mathrm{crit}(A_e)\cap A_e^{-1}(c)$ has an arbitrarily small open neighborhood $\mathcal U$ such that the intersection $\mathcal U\cap\{A_e<c\}$ has only finitely many connected components.
\hfill\qed
\end{lem}

\begin{proof}[Proof of Theorem~\ref{t:I'}]
We assume by contradiction that there exists $e\in I'\cap I_{\mathrm{finite}}$. We set
\begin{align*}
a_0:= 0 < \left|\int_{S^2}\sigma\right| =:a_1.
\end{align*}
Lemma~\ref{cl:finite_essential_family_1} provides a finite union of critical circles
\[\mathcal{E} = \mathcal C_1\cup...\cup \mathcal C_s \subset \mathrm{crit}(A_e)\] such that, whenever $c_e(m_0,n_0,m_1,n_1)\in[a_0,a_1]$,  $\mathcal{E}$ contains an essential family for $\mathcal{P}_e(m_0,n_0,m_1,n_1)$. By Equation~\eqref{e:shift_c} and Lemma~\ref{cl:finite_essential_family_2}, for each $m\in\mathds{N}$ there exist $n_m\in\mathds{Z}$ and $(m'_m,n_m')\in\mathds{N}\times \mathds{Z}$ such that
\begin{align*}
a_0 \leq c_e(m,n_m) = c_e(m,n_m,m_m',n_m') < a_1.
\end{align*}
In particular, $\mathcal{E}$ contains an essential family for $\mathcal{P}_e(m,n_m,m_m',n_m')$.
For each $i=1,...,s$, we consider an open neighborhood $\mathcal U_{i}$ of the critical circle $\mathcal C_i$ given by Lemma~\ref{l:local_homology_finite_rank}. We define
\[
\mathcal{F}:= \bigcup_{i=1,...,s}\Big\{ \mathcal V\ \Big|\ \mathcal V\ \mbox{is a connected component of } \mathcal U_i\cap\{A_e<A_e(\mathcal C_i)\} \Big\}.
\]
Notice that $\mathcal{F}$ has finite cardinality according to Lemma~\ref{l:local_homology_finite_rank}. For each $m\in\mathds{N}$, there exists $\mathcal V_m\in \mathcal{F}$ with the following property:
there exists a path $\Theta_m\in\mathcal{P}_e(m,n_m,m'_m,n'_m)$ and $s_m\in[0,1]$ such that the restriction $\Theta_m|_{[0,s_m]}$ is contained in the sublevel set $\{A_e<c_e(m,n_m,m'_m,n'_m)\}$, and $\Theta_m(s_m)\in \mathcal V_m$. Since $\mathcal{F}$ is finite, by the pigeonhole principle there exist distinct $m_1, m_2\in\mathds{N}$ such that $\mathcal V_{m_1}= \mathcal V_{m_2}$. In particular, $c_e(m_1,n_{m_1},m'_{m_1},n'_{m_1})=c_e(m_2,n_{m_2},m'_{m_2},n'_{m_2})$.

Consider the path $\Theta:[0,1]\to \widetilde{\M}$ obtained by concatenation of three paths: the restricted path $\Theta_{m_1}|_{[0,s_{m_1}]}$, some path connecting $\Theta_{m_1}(s_{m_1})$ with $\Theta_{m_2}(s_{m_2})$ within $\mathcal V_{m_1}$, and the restricted path $\Theta_{m_2}|_{[0,s_{m_2}]}$ traversed in the opposite direction. By construction, $\Theta\in\mathcal{P}_e(m_1,n_{m_1},m_2,n_{m_2})$. However,
\begin{align*}
\max A_e\circ\Theta < c_e(m_1,n_{m_1},m'_{m_1},n'_{m_1}) = c_e(m_1,n_{m_1}) \leq c_e(m_1,n_{m_1},m_2,n_{m_2}),
\end{align*}
which contradicts the definition of $c_e(m_1,n_{m_1},m_2,n_{m_2})$.
\end{proof}

redpajama_set_name: RedPajamaArXiv


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Salt Lake Sports Recovery's therapy is a highly specific process leading to the diagnosis and treatment of musculoskeletal injuries through non-invasive methods. Salt Lake Sports Recovery draws upon a myriad of established techniques that have been processed into a formulaic method to assess and correct pain and dysfunction. The therapy combines manual and active therapy to promote healing of muscles, tendons, ligaments, fascia and joints.
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text:
Jack Healey Commentary November 6th, 2018
Winning the World Series is the main thing, but as long as they have individual awards you might as well talk about them. That being said why isn't JD Martinez at least a finalist in the AL MVP race? I get it about DH and all that, but MVP is an offensive award and always has been. I've pointed this out before, but Brooks Robinson is the greatest defensive 3rd basemen of all-time so why did his only MVP award which was in 1964 come in his best offensive year? Why didn't he win the award in any of the other years? Because it's an offensive award that's why. I think the ultimate winner will be Mookie Betts so its a moot point about JD Martinez, but come on he deserves to be a finalist.
The more I watch Tom Brady the more I think that he is grossly underrated.
I went 9-4 in my NFL picks in week #9 and now stand at 81-46.
I was a perfect 6-0 in the high school football semi's and now stand at at 103-36.
The three state championship football games will be at Rutland's Alumni Field this Saturday November 10th.
At 11 AM it's Division 3 between Poultney and Woodstock
At 2PM it's Division 2 between Fair Haven and Burr and Burton
At 5PM it's Division 1 between Rutland and Mt. Mansfield
In the Division 3 state title game Poultney at [8-2] vs. Woodstock at [10-0]. Poultney is looking for its 2nd state title and first since defeating Windsor in 2007. Poultney is 1-1 in Division 3 state title games and 0-1 in Division 2 state title games. Poultney lost to South Burlington 7-6 in the first Division 2 title game in 1970
Woodstock is making its 2nd straight appearance in the title game and is looking for its 11th state championship and first since 2013.Woodstock is 10-6 in state title games. Woodstock defeated Poultney 50-13 in the regular season game in week #6.


See Gerald Frug and David Barron, City Bound: How States Stifle Urban Innovation, p. 141, (2008). Graph is user image of publicly available population data from Dallas Commons digital site.
intertwined. Dallas can craft a portfolio of priorities that will propel it into a more prosperous next century.
See Marjorie Valbrun, Rethinking Criminal JusticeWhy Black Prosecutors Matter, (www.newamericanmedia.org), (November 19, 2010).
Charles Hynes, The Evolving Prosecutor: Broadening The Vision, Expanding The Role, 24 Fall Crim. Just 1, 2 (2009). M. Elaine Nugent, The Changing Nature of Prosecution: Community Prosecution vs. Traditional Prosecution Approaches , (February 2004).
See The Changing Nature at p. 33.
people look at an urban area and determine it is dilapidated. They fix it up. It becomes dilapidated again. We have seen the vicious cycle play out in many communities. By engaging the community in the mission to alter appearance and perception, the cycle might be avoided altogether. Dallas Mayor Mike Rawlings understands the importance of appearance and perception. In early 2012, he rolled out an initiative (entitled Grow South) that focuses on improving the most dilapidated and underdeveloped areas of the city. A central tenet of his plan is instituting a culture of clean in the City of Dallas. His commitment to this topic is obvious because many of his public remarks over the last two months have trumpeted the need for a cleaner image. Dallas had the makings of the right idea in 2010. During that time, the Dallas-Forth Worth area was preparing to host Super Bowl XLV. In order to present itself as a modern global city on the rise, not one covered in spray paint,10 Dallas officials tabbed a prominent local lawyer as its graffiti czar.11 The graffiti czar was charged with leading the City of Dallas Graffiti Abatement Program. Most importantly, the czar would coordinate the efforts of local businesses to pick a day when they would sponsor the clean up of the areas near their respective businesses. The brilliance of this idea is that it engages private businesses in the project of cleaning up public spaces. The businesses do this because they recognize that their individual fortunes are tied to a larger communal fortune. Even still, the city must go one step further.
See Dallas Morning News City Hall Blog (quoting then-Mayor Tom Leppert in December 2010): http://cityhallblog.dallasnews.com/2010/12/dallas-attorney-john-barr-name.html/ . 11 See id.
Everyday citizens of Dallas must be engaged in this massive facelift as well. Just as businesses recognize the impact nearby vandalism has on its bottom line, so too must citizens realize that local eyesores (litter, vandalism, graffiti, etc.) debase their own living experiences as well. This is a particularly tough sell because it requires widespread buy-in to counteract the status quo. The hearts and minds of residents must fundamentally change to truly eradicate these problems. The city can work with communities to get people invested in the physical appearance of their neighborhoods in a contagious way. In 2010, Dallas held a total of 25 community clean up events ranging from graffiti wipe out days to mowing vacant lots. Again, this is the right idea. The next level of effectiveness must engage residents more fully, however. Dallas launched a new website (http://www.dallascommunityoutreach.com/) aimed at bringing more residents into the fold. The city must build on the momentum it has created by empowering residents to plan their own neighborhood-based events. Rather than planning scores of events and telling people to come, the city can provide resident-leaders with resources and allow them to plan the events themselves. The picture below is an example of one such event.
As seen in a 2011 study of Dallas Public Schools, found at: www.studentmotivation.org/dallasisd.
not do. Much has been written about the causes of and remedies for poor student performance. A recurrent theme in nearly all the literature is parental involvement. Parents taking on an active role in the educational environment is a crucial nexus to the central argument of this paper.
From the Dallas Independent School Districts website: www.dallasisd.org/page/7797.
See Judith Martinez, Parental Involvement: Key to Student Achievement, (Feb. 23, 2004).
See J. Berry, K. Portney, and K. Thomson, The Rebirth of Urban Democracy, pp. 10-14, 294-296 (1993). Id. at 13.
Id. at 294. Id. at 294-95. 19 See id. 20 Id. at 295. 21 Id.
Dallas (depicted in the map below), designed according to historical, natural, and economic boundaries. Each team is challenged to develop strategic plans for its dedicated area based on the following topics: economic development policy, small business, and finance/funding sources.22 This approach has allowed for a narrow tailoring of solutions, by planning area, which has begun to make a meaningful impact in Dallas.
As laudable as this innovation has been, there is still much to be done. Refracting this effort through the lens offered by The Rebirth of Urban Democracy, it is lacking in three ways: 1) these groups still do not have anything more than planning power; 2) there is no evidence city hall administrators must work with these groups; and 3) this effort is primarily based in Southern Dallas, not the other affluent areas of Dallas. Mayor Rawlings has indicated that he would like to add at least 30 neighborhood associations throughout Dallas in the next year. The effectiveness of these organizations will depend on how much they activate the energy and effort of Dallas residents.
See Southern Dallas Task Forces Interim Team Report, p. 11 (2009).
City of Dallas, forwardDallas! Comprehensive Plan: Vision, (June 2006).
Survey Respondents' Top Priorities for the City of Dallas.
Based on his remarks to our class on 3/29/12 in Austin North.
part, even if small, to revitalize Jerusalem. Everyone was included, and in just 52 days, their collaborative effort had rebuilt the citys wall and gates.27 Revitalizing Americas urban cities will require a special type of leadership. Some will want to be Moses-types that deliver a city to the Promised Land with a set of instructions. Others will want to be David-types that conquer Goliath-sized problems with personal courage and faith. However, the project of urban revitalization will require modern-day Nehemiahs. These leaders will change cities by galvanizing the people who live in them. They will encourage citizens to pick up the tools and talents they possess to make whatever contribution they can. Piece-by-piece. Person-by-person. Everyone can collaborate to rebuild the Jerusalems of America. To achieve this dream, cities need leaders and policies that transform everyday citizens into community stakeholders. Grand plans are simply not enough. Neither are grand ambitions. These concepts are perhaps necessary but not sufficient. Meaningful citizen participation is the silent bridge from the decaying city to the thriving metropolis. By engaging community stakeholders in the project of managing the civic portfolio of priorities, Dallas and its peer cities can be genuinely revitalized.
See Book of Nehemiah, Chapter 6,Verse 15.
Include parents in school decisions, developing parent leaders and representatives.
Sample Practices Active PTA/PTO or other parent organizations, advisory councils, or committees (e.g., curriculum, safety, personnel) for parent leadership and participation. Independent advocacy groups to lobby and work for school reform and improvements. District-level councils and committees for family and community involvement. Information on school or local elections for school representatives. Networks to link all families with parent representatives. Challenges Include parent leaders from all racial, ethnic, socioeconomic, and other groups in the school. Offer training to enable leaders to serve as representatives of other families, with input from and return of information to all parents. Include students (along with parents) in decision-making groups. Redefinitions "Decision making" to mean a process of partnership, of shared views and actions toward shared goals, not just a power struggle between conflicting ideas. Parent "leader" to mean a real representative, with opportunities and support to hear from and communicate with other families.
Identify and integrate resources and services from the community to strengthen school programs, family practices, and student learning and development.
As articulated by Judith Martinez. Parental Involvement: Key to Student Achievement. Feb. 23, 2004. Citing Epsteins Framework of Six Types of Involvement at pp. 4-6.
talents to support students, families, and schools. "Community" means all who are interested in and affected by the quality of education, not just those with children in the schools.


The Wekiva Parkway (SR 429) is a $1.6B project that connects to SR 417, completing the beltway around Central Florida, while helping to protect the natural resources surrounding the Wekiva River. The 25-mile tolled expressway provides travel alternatives and relieve US 441, SR 46 and other area roads of traffic congestion resulting from growth and travel between Orange, Lake and Seminole Counties. This $56.2M phase of the project includes two adjacent precast, post-tensioned concrete U-beams spanning over a connector road. Each bridge has a total length of 469' (137', 195', and 137'). The bridge was erected segment-by-segment using ground-based cranes with prefabricated shoring towers for temporary support at beam ends.
FINLEY provided construction engineering including construction analysis, erection manual, geometry control manual, mass concrete plan, temporary works, erection elevation, on-site-review and technical support during construction.


text:
Q: Code for Ant not Running in Maven This is probably the most strange thing I have ever seen in Java: The following code (that uses no external libraries) runs flawlessly using Ant (on Netbeans, on a Windows-7 Machine) :
public static void main( String[] args ) throws IOException, InterruptedException
{
  String prog = "\"C:\\Program Files\\R\\R-2.14.2\\bin\\Rscript.exe\"";
  String[] progParams = new String[]{"C:\\Users\\pantelis\\Desktop\\SB\\MC_Sarfraz_ver5.R"};
  Process p = Runtime.getRuntime().exec(prog+" "+progParams[0]);
  p.waitFor();
  System.out.println(p.exitValue()==0);
  System.out.println("Done!!!");
}

and it returns the output message:
true
Done!!!

meaning that it completed smoothly. When I'm trying to run the very same code using Maven, I get no exceptions but the code finishes with:
false
Done!

and the process is never executed. This is my POM file:
<project xmlns="http://maven.apache.org/POM/4.0.0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
 xsi:schemaLocation="http://maven.apache.org/POM/4.0.0 http://maven.apache.org/xsd/maven-4.0.0.xsd">
  <modelVersion>4.0.0</modelVersion>
  <groupId>com.mycompany</groupId>
  <artifactId>MVNTest</artifactId>
  <version>1.0-SNAPSHOT</version>
  <packaging>jar</packaging>
  <name>MVNTest</name>
  <url>http://maven.apache.org</url>
  <properties>
    <project.build.sourceEncoding>UTF-8</project.build.sourceEncoding>
  </properties>
  <dependencies>
    <dependency>
      <groupId>junit</groupId>
      <artifactId>junit</artifactId>
      <version>3.8.1</version>
      <scope>test</scope>
    </dependency>
  </dependencies>
</project>

Update 1: I used the following code to print any error messages generated by the process:
String line;
BufferedReader input = new BufferedReader(new InputStreamReader(p.getErrorStream()));
while ((line = input.readLine()) != null) {
  System.out.println(line);
}

Here is the output:
Error in library(RDCOMClient) : there is no package called 'RDCOMClient'
Execution halted

This error message appears only when I run using Maven.
Update 2: When I run using Maven, I use JDK7. Here is the corresponding command:
cd C:\Users\pantelis\Documents\NetBeansProjects\MVNTest;
"JAVA_HOME=C:\\Program Files\\Java\\jdk1.7.0_04"
"\"C:\\Program Files\\NetBeans 7.1.2\\java\\maven\\bin\\mvn.bat\""
"-Dexec.args=-classpath %classpath com.mycompany.mvntest.App"
"-Dexec.executable=C:\\Program Files\\Java\\jdk1.7.0_04\\bin\\java.exe"
-Dexec.classpathScope=runtime process-classes
org.codehaus.mojo:exec-maven-plugin:1.2:exec

Update 3: These are the environmental variables that I get when I run with Ant:
USERPROFILE -> C:\Users\pantelis
ProgramData -> C:\ProgramData
PATHEXT -> .COM;.EXE;.BAT;.CMD;.VBS;.VBE;.JS;.JSE;.WSF;.WSH;.MSC
Classpath -> C:\LINGO9\Lingo9_0.jar
ProgramFiles(x86) -> C:\Program Files (x86)
TEMP -> C:\Users\pantelis\AppData\Local\Temp
SystemDrive -> C:
ProgramFiles -> C:\Program Files
Path -> C:\Program Files (x86)\MiKTeX 2.9\miktex\bin;C:\Windows\system32;C:\Windows;C:\Windows\System32\Wbem;C:\Windows\System32\WindowsPowerShell\v1.0\;C:\Program Files\WIDCOMM\Bluetooth Software\;C:\Program Files\WIDCOMM\Bluetooth Software\syswow64;\;C:\Program Files (x86)\Sony\VAIO Startup Setting Tool;C:\Program Files (x86)\ATI Technologies\ATI.ACE\Core-Static;C:\LINGO9\;C:\Program Files (x86)\OpenVPN\bin
HOMEDRIVE -> C:
PROCESSOR_REVISION -> 2505
USERDOMAIN -> pantelis-VAIO
ALLUSERSPROFILE -> C:\ProgramData
ProgramW6432 -> C:\Program Files
PROCESSOR_IDENTIFIER -> Intel64 Family 6 Model 37 Stepping 5, GenuineIntel
SESSIONNAME -> Console
TMP -> C:\Users\pantelis\AppData\Local\Temp
CommonProgramFiles -> C:\Program Files\Common Files
=:: -> ::\
LOGONSERVER -> \\PANTELIS-VAIO
PROCESSOR_ARCHITECTURE -> AMD64
FP_NO_HOST_CHECK -> NO
OS -> Windows_NT
HOMEPATH -> \Users\pantelis
PROCESSOR_LEVEL -> 6
CommonProgramW6432 -> C:\Program Files\Common Files
configsetroot -> C:\Windows\ConfigSetRoot
LOCALAPPDATA -> C:\Users\pantelis\AppData\Local
COMPUTERNAME -> PANTELIS-VAIO
windir -> C:\Windows
SystemRoot -> C:\Windows
NUMBER_OF_PROCESSORS -> 4
USERNAME -> pantelis
PUBLIC -> C:\Users\Public
PSModulePath -> C:\Windows\system32\WindowsPowerShell\v1.0\Modules\
CommonProgramFiles(x86) -> C:\Program Files (x86)\Common Files
ComSpec -> C:\Windows\system32\cmd.exe
APPDATA -> C:\Users\pantelis\AppData\Roaming

And here are the environmental variables when I run with Maven:
USERPROFILE -> C:\Users\pantelis
JAVA_HOME -> C:\Program Files\Java\jdk1.7.0_04
PROGRAMDATA -> C:\ProgramData
MAVEN_CMD_LINE_ARGS -> "-Dexec.args=-classpath %classpath com.mycompany.mvntest.App" "-Dexec.executable=C:\Program Files\Java\jdk1.7.0_04\bin\java.exe" -Dexec.classpathScope=runtime process-classes org.codehaus.mojo:exec-maven-plugin:1.2:exec
CONFIGSETROOT -> C:\Windows\ConfigSetRoot
COMMONPROGRAMFILES -> C:\Program Files\Common Files
PROCESSOR_REVISION -> 2505
USERDOMAIN -> pantelis-VAIO
ALLUSERSPROFILE -> C:\ProgramData
PROGRAMW6432 -> C:\Program Files
SESSIONNAME -> Console
TMP -> C:\Users\pantelis\AppData\Local\Temp
PSMODULEPATH -> C:\Windows\system32\WindowsPowerShell\v1.0\Modules\
NB_EXEC_MAVEN_PROCESS_UUID -> b5daa24f-51dc-48ac-ad6d-820dc245b80a
M2_HOME -> C:\Program Files\NetBeans 7.1.2\java\maven\bin\..
LOGONSERVER -> \\PANTELIS-VAIO
=:: -> ::\
PATH -> C:\Program Files (x86)\MiKTeX 2.9\miktex\bin;C:\Windows\system32;C:\Windows;C:\Windows\System32\Wbem;C:\Windows\System32\WindowsPowerShell\v1.0\;C:\Program Files\WIDCOMM\Bluetooth Software\;C:\Program Files\WIDCOMM\Bluetooth Software\syswow64;\;C:\Program Files (x86)\Sony\VAIO Startup Setting Tool;C:\Program Files (x86)\ATI Technologies\ATI.ACE\Core-Static;C:\LINGO9\;C:\Program Files (x86)\OpenVPN\bin
PROMPT -> $P$G
COMMONPROGRAMW6432 -> C:\Program Files\Common Files
PROCESSOR_LEVEL -> 6
LOCALAPPDATA -> C:\Users\pantelis\AppData\Local
COMPUTERNAME -> PANTELIS-VAIO
CLASSWORLDS_LAUNCHER -> org.codehaus.plexus.classworlds.launcher.Launcher
USERNAME -> pantelis
APPDATA -> C:\Users\pantelis\AppData\Roaming
WINDIR -> C:\Windows
PATHEXT -> .COM;.EXE;.BAT;.CMD;.VBS;.VBE;.JS;.JSE;.WSF;.WSH;.MSC
PROGRAMFILES(X86) -> C:\Program Files (x86)
TEMP -> C:\Users\pantelis\AppData\Local\Temp
HOMEDRIVE -> C:
SYSTEMDRIVE -> C:
=C: -> C:\Users\pantelis\Documents\NetBeansProjects\MVNTest
ERROR_CODE -> 0
COMMONPROGRAMFILES(X86) -> C:\Program Files (x86)\Common Files
CLASSWORLDS_JAR -> "C:\Program Files\NetBeans 7.1.2\java\maven\bin\..\boot\plexus-classworlds-2.4.jar"
PROCESSOR_IDENTIFIER -> Intel64 Family 6 Model 37 Stepping 5, GenuineIntel
MAVEN_JAVA_EXE -> "C:\Program Files\Java\jdk1.7.0_04\bin\java.exe"
CLASSPATH -> C:\LINGO9\Lingo9_0.jar
PROCESSOR_ARCHITECTURE -> AMD64
FP_NO_HOST_CHECK -> NO
OS -> Windows_NT
HOMEPATH -> \Users\pantelis
COMSPEC -> C:\Windows\system32\cmd.exe
PROGRAMFILES -> C:\Program Files
HOME -> C:\Users\pantelis
NUMBER_OF_PROCESSORS -> 4
PUBLIC -> C:\Users\Public
SYSTEMROOT -> C:\Windows

A: After the update, based on the error you get when running from maven, it seems some (class)path is not set up correctly: RDCOMClient seems to be related to R, so it is the setup that is used to run Rscript.exe that is the problem.
Print out the environmental variables when you are running via ant and also under maven and make a comparison to determine the source of the missing library error
Update from comments:
The problem might be related to how the project is executed from maven: exec:exec starts a new JVM, whereas exec:java runs the project in the same JVM that is running maven.
Update from comments:
Apparently, the "Compile on Save" settings in NetBeans change the maven goal from exec:exec to exec:java.  Alternatively, you can configure the pom.xml manually (see the exec maven plugin page that has examples and descriptions for the two goals)

A: Does this happen with all R packages ? Or only RDCOMClient ?
I'll be very surprised if this is a Maven/Ant issue.
Reading the error stream is unnecessary. R outputs the error in a file or you can use sink("filename") to write R output to a file.
Please treat as comment.

A: I think, you need to specify your prog and params inside an array, and pass that array to exec.
public static void main( String[] args ) throws IOException, InterruptedException
{
  String prog = "\"C:\\Program Files\\R\\R-2.14.2\\bin\\Rscript.exe\"";
  String[] progParams = new String[] {
     prog,
     "C:\\Users\\pantelis\\Desktop\\SB\\MC_Sarfraz_ver5.R"
  };
  Process p = Runtime.getRuntime().exec(progParams);
  p.waitFor();
  System.out.println(p.exitValue()==0);
  System.out.println("Done!!!");
}

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Hope you have a Social Thanksgiving!
Wishing you a bountiful Thanksgiving with family and friends!
Know any Boomers that don't like social media?
This Forbes article by William Arruda explains what will happen if you don't adopt social media into your practices like the Millenials have done. Take a look and share with a friend that might need it! Read more here.
Does your customer service help or hurt your business' reputation?


Laid-back living, intricate meandering streets and acres of sandy white beach – holidays to Ca'n Picafort are the perfect way to unwind. Located on the northeast coast of Majorca, on the Eastern end of the Bay of Alcudia, Ca'n Picafort is a quiet and relaxing beach resort which presents the perfect alternative to the bright lights and raucous nightlife of many of Spain's busier resorts.
Ideal for those looking for romantic all inclusive getaways, while still offering some great facilities for families, Ca'n Picafort holidays are filled with enough sunshine, sea and sand to delight even the most discerning holidaymaker. Revel in the sight of the majestic Tramuntana Mountains, enjoy a fun-filled trip to the nearby water park with the entire family or indulge your sense of adventure with a myriad of action-packed activities throughout the Santa Margalida region – holidays to Ca'n Picafort offer countless opportunities and plenty of things to do for those with little ones in tow.
Famous for its sandy beaches and family friendly atmosphere, holidays in C'an Picafort are a popular choice for family holidays.
With its palm-lined streets and stretches of white sands, Ca'n Picafort is quiet resort and a sight to behold, establishing itself as one of the prettiest locations in the Balearic Islands for a 2018 / 2019 holiday. You'd mistaken if you thought it was all about lazy lounging in Ca'n Picafort. Its main strip, the Paseo Colon, is home to a stretch of restaurants, shops and bars running right down to the seafront. There's a heavy German influence, as the resort tends to balance its focus between German and British tourists. If you follow the promenade along the seafront, dipping in and out of the shops, you'll reach the modern part of town, Son Baulo.
Proof that good things come in small packages, Ca'n Picafort is easy to get around by foot, with its idyllic surroundings serving as a beautiful backdrop to long afternoon walks in the sunshine. Venture just a little out of town and you'll find the beautiful oyster-shaped bay of Puerto Pollensa. With its cobbled promenade and quaint marina, Puerto Pollensa presents the ideal opportunity for a day trip, with a bustling weekly market taking place every Wednesday.
Just 20 minutes drive north up the coast and you'll reach Alcudia, altogether more go-go-go than Ca'n Picafort. With a delightful marina, narrow cobbled streets and a magical old town lined with tradtional Balearics eateries and the beautiful San Jaume Church, Alcudia is another top day trip destination.
For a range of mouth-watering restaurants as well as some great shops and bars on your holidays in C'an Picafort, head down to Paseo Colon. Here, you'll find many quaint eateries serving up the tastiest seafood dishes as well as some laid-back beach bars for sinking cool beers and sampling many a jug of Sangria.
Along the seafront you'll find the typical affair of seafood restaurants and Balearics tapas and you can follow the promenade to the Son Baulo area for more international restaurants and brands you might recognise.
From maze fun parks to go-karting and the most notable waterpark in Majorca, there is an array of activities and attractions to keep the children amused. Our top pick of a family hotel in Ca'n Picafort is the Exagon Park, a 4 star hotel with many kid's facilities and excellent access to the beach.
Alcudia Waterpark can be reached from Ca'n Picafort along a main road. It's a simple journey and doesn't take long. You can then follow that road along to Alcudia. You'll also find typical holiday activites like karting, buggy riding and outdoor excursions such as cave exploring and off-road tours, all available from tour operators in town. If in doubt, ask at the desk of your hotel.
There are lots of activities to take up in the Santa Margalida region and along the coastline between Ca'n Picafort, Playa de Muro and Alcudia. Rock climb, 4 wheel off-road tours and cliff-diving are just a few. For a relaxing stroll and the chance to spot some majestic wildlife, explore the Parque Natural s'Albufera de Mallorca, home to some rare birds.
All roads lead to the beach on your holidays in C'an Picafort, and if you continue down the road, towards the Son Baulo area of the resort, you'll find two of the bigger beaches. These gentle waters are perfect for little ones, while adventurous types might like to sample some of the great wealth of watersports on offer.
If you ever tire of the beaches in the resort on your Ca'n Picafort holidays, you'll find some fantastic attractions and more beaches in the nearby town of Alcudia or Puerto Pollensa. A bus route runs into Playa de Muro then up to Alcudia, although you can also get a taxi if you don't feel like tackling public transport with all your beach gear.
Ca'n Picafort is not typically known for its nightlife, which is its biggest draw for most tourists to this area. There are numerous bars, however, and in peak season these are busy and open late. There's also two nightclubs, but they're not enormous. If you're looking for nightlife, head up the coast to Alcudia, which is a larger resort and home to many more places to let loose.
Ca'n Picafort is a sweet destination for a romantic getaway because of its accessibility to some truly beautiful areas of Majorca. Booking flights and an all inclusive hotel are also affordable, making Ca'n Picafort a quick and easy cheap holiday to treat your partner with.
Ca'n Picafort, before the tourism boom of the 1970s, was just a small fishing village. Once the hotels moved in, much of the tradition of the village evaporated, although there's still a small functioning harbour from which most of the pleasure trips depart.
However, the Santa Margalida region is home to over a hundred archaelogical sites, including the fascinating and popular necropolis at Son Real, which was acquired as a protected site by the Balearics government in 2002. Worth a visit, if you're into history.
If you'd rather immerse yourself in the culture of a medieval town, simply take a stroll through Alcudia's ancient streets, catching a glimpse at the small remnants and ruins left over from Alcudia's past.