Agency 323 is a full-service interactive agency. We are one of the leading interactive marketing and digital development companies in Texas with expertise in technology, strategy, application development and design services. We serve a variety of clients across a number of industries—from Fortune 500 companies to small businesses—giving us unique perspectives into the technologies and opportunities that can transform businesses in today's continually evolving market.
Agency 323 was founded in 2008 by our Managing Partner, Charles Lauller. Our initial vision was to help educate and empower organizations with the proper tool set (technology) and mindset (sociology) to execute best practice interactive campaigns that add tangible and measurable value to achieving their business goals. Today, that vision remains true, as interactive has become a vital strategic foundation to virtually every business.
Whether the project is large or small, Agency 323's approach in finding the right interactive solution is about asking the right questions at the right time; adhering to proven processes that focus and guide our teams; executing on the basis of solid, informed rationale; and measuring the results for continuous improvement.
We believe wholeheartedly that effective interactive experiences engage their target audience and easily connect people with information, services and products. We believe strengthening the relationship a business shares with its customers, employees, and the community is the key to building great interactive solutions and value for the brand.
As managing partner, Charles' role is to collaboratively build and execute strategies for organizations who seek to engage their target audience online. As an integrated agency, Agency 323 focuses on four specific areas: Analysis & Strategic Planning, Listening & Discovery, Engagement & Execution, and Monetization and Measurement. We seamlessly integrated and extend your existing marketing channels into the new world of the web.
In the early stages of his career, Charles held various sales and sales management positions for high tech companies, including pivotal roles in sales and sales management at Print, Inc, where he helped develop document management strategies and solutions for dozens of the top law firms in the world including Jones Day, Pillsbury Winthrop Shaw Pittman, Orrick Herrington Sutcliffe, and Morrison & Foerster. He helped propel the start-up to $50MM in 5 years.
During his career he has consulted for many of Silicon Valley's top tech companies including Salesforce.com, Network Appliance, McAfee, and Juniper Networks to provide more efficient and cost effective solutions for mission critical business processes.
Angela embodies, "Healthcare Enthusiast: Competitive Champion!" As a recent graduate of the MHA program, Angela further enhanced her non-clinical skills of the current health care environment in a managerial/leadership capacity. She achieved highest honors in completion of curriculum involving the latest theories and practice applications impacting health care globally.
Dave Russell brings a forward thinking digital media perspective from 17+ years of experience to Agency 323. Dave has walked the edge of innovation at some of the best and brightest agency in America: GSD&M, Wieden & Kennedy, EuroRSCG and TM Advertising.
Dave has spent his career working on an impressive roster of clients including Chili's, Microsoft, Intel, Nike, BMW, MasterCard, American Airlines and Nationwide Insurance. Alongside each of these brands he pushed the digital space forward, writing and re-writing the rules. Dave has been at the epicenter of the evolving media space. He is a sought after panelist, consultant and advisory board member at digital conferences and digital media startups.
He is a graduate of Brigham Young University with a B.S. in Business Management. He is married and has four great children. And yes, he knows his way around a golf course.
Jason Stebbing offers a solutions, conversion driven approach to creative design and functional implementation. With 11+ years of marketing, graphic design, art direction and web centric project experience, he adds a unique approach to solving creative challenges. Jason's experience spans both large and small business engagements from enterprise level implementation support to turnkey delivery of boutique web presence.
Jason's career has evolved from early experience with simple front-end coding to dynamic, user interface design to drive client objectives and achieve conversion goals. His love for high value design fuels his daily routine, and delivers high client satisfaction.
He holds a bachelor's degree from the University of Utah, and earned a MBA at Xavier University. Jason spends his personal time with his wife, 4 children and on his bike, on the trail or at the slopes (whenever possible).
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Wednesday, March 13, marked the end of the floor cutoff date – meaning it was the final day for us to pass House bills over to the Senate. This year, cutoff was one of the more interesting days on the House floor we have had since session started.
House Democrats were adamant in getting a universal background check bill passed for firearms purchases—even if the sale was between family members or friends. Despite having a majority in the House, they didn't have the votes to pass the bill. Unfortunately, House Democratic leadership made it very apparent we weren't voting on anything else until they had enough votes to pass it. Gov. Jay Inslee and even Vice President Joe Biden were calling members trying to use their influence on members from both sides of the aisle. The arm twisting didn't work. After almost nine hours Democrats informed us we were adjourning for the night. Of course the silver-lining is a number of bad bills, particularly legislation that was passed out of the labor committee, ended up dying. Yes, bad gun legislation killed poor labor bills that criminalize work.
House Bill 1440 would have made it almost impossible to be an independent contractor in Washington. The proposed bill states any independent contractor hired by an employer for a job is an employee of that company and the burden of proof is on the employer to prove otherwise. Basically, employers are presumed guilty and must prove their innocence to the Department of Labor and Industries in order to avoid additional premiums, penalties and maybe even an audit. This is a great example of why I am saying Olympia is criminalizing work.
House Bill 1023 would require apprenticeship utilization on all public works projects. This bill would have given an advantage to Seattle-based businesses over businesses in Eastern Washington.
House Bill 1025 would extend prevailing wage requirements. Another huge cost increase to businesses and taxpayers during a time of economic uncertainty. Prevailing wage increases the cost of public works projects by nearly 40 percent. We can't afford this in eastern Washington as we are finding out with our school construction projects.
Despite our opposition, and President Obama's acknowledgement this policy is bad for small businesses, majority Democrats in Olympia passed the bill on a party-line vote, 51 to 47, with four Democrats voting no. Click "Manweller on HB 1473" to watch my floor speech.
There are a number of bills Rep. Judy Warnick and I have either prime-sponsored or co-sponsored that are important to our district, and have passed the House unanimously or close to it. Below is a summary of some of those bills making their way through the Senate.
House Bill 1068 – is my bill that would allow, but not require, the Kittitas County TV Reception Improvement District to provide an excise tax exemption to television owners receiving satellite television service. It passed the House of Representatives and is moving in the Senate. I introduced this bill after it was brought to my attention TV reception districts were able to charge excise tax for television service that was not being provided. The bill would allow the board of a TV reception district to exempt a television owner from the excise tax if the owner subscribes to satellite television service.
House Bill 1416 – would make numerous changes to Local Improvement Districts (LIDs) that have been created by irrigation districts. This piece of legislation would modernize irrigation district statutes that have been in operation since 1929. There are 400,000 acres of irrigatable land in Irrigation District 13 and 70,000 acres in the Odessa sub area, so it is critical we update our irrigation and water laws.
House Bill 1558 – would extend a tax exemption on bee food, and provide an additional exemption from the sales tax with a certificate. I don't need to tell you the importance of beekeeping to our district and agriculture in general.
House Bill 1419 – would expand the membership of the Washington State Horse Park Authority. The board informed us they needed more people to help manage and promote the horse park so they can continue to attract visitors from around the Northwest, which helps with the numerous other recreational activities in our region.
House Bill 1552 – would address scrap metal theft, a huge problem in our area. It would offer some relief to all the farmers and ranchers in Eastern Washington who have been suffering from metal theft. There is nothing more frustrating that getting up in the morning and seeing that a drug addict has striped your fences, stolen irrigation pipes, and vandalized agricultural equipment.
If you would like more information on these issues or anything else before the Legislature pleas do not hesitate to contact me.


Conquerors are ex-prisoners and forced conscripts who have risen to the rank of elite soldier. Originally pressed into military service as sword fodder, those who are skilled enough to survive are promoted out of the rank and file and given heavy infantry training. They are heavily armored warriors, relying on a defensive style and their massive flails to whittle away at their opponent's stamina until they fall under the Conquerors' massive onslaught. Climbing from the lowest ranked to this prestigious role in the Knight's forces, Conquerors have earned the respect of their peers through determination and feats of arms.
Conquerors are heavily armored fighters armed with massive flails. They destabilize and overcome opponents through multiple defensive abilities and infinite attacks.
Watch new For Honor gameplay, and meet the Conqueror of the Knight faction.
The flail is one of the most vicious weapons used by the heavy infantry of the Iron legion. They are made of heavy iron chains linking a stout wooden handle to a massive steel-spiked ball. The flail inflicts massive wounds through concussion as it crushes both armor and bones.
Customizable armor. Take a look at each of the three customizable sections of the Conquerors' armor.
We are very happy to give our Knights a chance to embody the legendary Conquerors! Check out the 360 views of the Conquerors and close ups to armor parts and weapon.


The 205 Smokestop is our most popular, specialist tobacco smoke removing air purifier providing floor to ceiling air purification for small to medium rooms rooms up to 26m². With almost a full kilo of carbon in its filter and the highest CADR ( Clean air delivery rate ) in its class, the 205 is around 100 times more effective at removing tobacco smoke odour than many much higher priced machines found on the High Street. A really solid choice if you want genuine ability, coupled with virtually silent operation. Great for living room or bedroom, or both!
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The HMFC Medical Director, Dr Stephen Mainini, is a Pulmonary Specialist. With Dr. Mainini's help as a volunteer, HMFC schedules patients with respiratory issues, sleep apnea, and other lung ailments to see him. The next Respiratory Clinic is scheduled for August 30, 2018.
To make an appointment and be included in this clinic complete the request for an appointment below and click submit.


text:
\section{Introduction}
Given a dynamical system $f:M\to M$ acting on an arbitrary  metric space $(M,d)$, a \emph{$\delta$-pseudotrajectory} for $f$ is any sequence of points $(y_n)_{n\in \mathbb{Z}}\subset M$ satisfying
\begin{displaymath}
d(y_{n+1},f(y_n))\le \delta \text{ for every } n\in\mathbb{Z}.
\end{displaymath}
An important question that naturally arises  is to understand whether a pseudotrajectory can be approximated by a real trajectory for $f$. More precisely,   we ask if there exists a point $x\in M$ such that $d(y_n,f^n(x))$ is small for every $n\in \mathbb{Z}$. If this happens,  we say that $x$ \emph{shadows} the pseudotrajectory $(y_n)_{n\in \mathbb{Z}}$. Furthermore, if  every pseudotrajectory has a shadowing point we say that the system $(f,M)$ has the \emph{shadowing property} (see~\cite{Pal00, Pil99} for more precise definitions).

We emphasize that there is a clear motivation for studying shadowing properties. For instance, a pseudotrajectory can be viewed as a result of measurements in a real system (that are usually subjected to round-off error or noise). Then, to study the  shadowing problem is to try to understand the relationship between the real trajectories and the approximate trajectories obtained by such measurements. If the system has the shadowing property we  have  that the numerically obtained trajectories actually reflect the real behavior of trajectories of $f$. From a more theoretical perspective, the shadowing property has proved to be very fruitful when dealing with problems concerned with topological stability and construction of symbolic dynamics \cite{Bow75}.

It is well known that many classes of dynamical systems exhibit  shadowing property.  Indeed, Bowen~\cite{Bow75} and Anosov~\cite{Ano70} proved that \emph{uniformly hyperbolic} dynamical systems  with both discrete and continuous time have  shadowing property. Furthermore, Katok~\cite{Katok} proved that \emph{nonuniformly hyperbolic} dynamical systems also
posses certain type of shadowing property.  More recently, Pilyugin~\cite{Pil97, Pil99} proved that structurally stable systems exhibit shadowing property.  Finally, we recall that it has been noticed that partially hyperbolic dynamical systems have no shadowing property~\cite{BDT}.

The original proofs that uniformly hyperbolic dynamical systems have shadowing property (due to Bowen and Anosov)  rely on the existence of invariant stable and unstable manifolds. Later, Palmer~\cite{Palmer2} and independently Mayer and Sell~\cite{MS} gave quite  ingenious and very  simple analytic proofs of shadowing lemma that don't use the invariant manifold theory.
Their approach also inspired versions of the shadowing lemma that deal with maps on Banach spaces (see~\cite{CLP, Henry}). In addition, analytic proof of the shadowing lemma for nonuniformly hyperbolic dynamics was given in~\cite{DS}.
 Finally, in the recent paper~\cite{BCDMP} the authors have developed shadowing theory for linear operators on an arbitrary Banach space.
 We refer to~\cite{Pal00, Pil99} for more discussion and further references related to shadowing theory.

We emphasize that all the above mentioned results deal with \emph{autonomous} dynamics and the main objective of the present paper is to develop shadowing theory for \emph{nonautonomous} systems acting on an arbitrary Banach space. More precisely, starting with a linear dynamics
\begin{equation}\label{1021}
x_{m+1}=A_m x_m \quad m\in \mathbb{Z},
\end{equation}
where the sequence $(A_m)_{m\in \mathbb{Z}}$ admits an exponential dichotomy (which represents a version of the notion of hyperbolicity for time-varying dynamics), we prove that  a small \emph{nonlinear} perturbation of~\eqref{1021} has the shadowing property.  In particular, results from~\cite{BCDMP} correspond to a special case when $(A_m)_{m\in \mathbb{Z}}$  in~\eqref{1021} is a constant sequence of operators and when linear dynamics is not perturbed.
Moreover, we propose a unified approach (inspired by~\cite{DD}) that allows us to measure
the error in the notion of a pseudotrajectory as well as its deviation from a trajectory in  a variety of ways that include the concept of  standard shadowing (as well as other concepts of shadowing studied in~\cite{BCDMP} and elsewhere) as a very particular case. We note that our methods are inspired by the above mentioned analytic proofs of the shadowing lemma.

Finally, as a nontrivial application of our results, we give a simple proof of the nonautonomous version of the Grobman-Hartman theorem in the discrete-time setting. We emphasize that the first version of this result goes back to Palmer~\cite{Palmer} who considered finite-dimensional dynamics with continuous time. More recent versions are due to Barreira and Valls~\cite{BarVal-DCDS07,
BV1} (see also~\cite{BDV1}) who deal with discrete-time dynamics on an arbitrary Banach space that admits a nonuniform exponential dichotomy.  Although our version of the nonautonomous Grobman-Hartman theorem works under more restrictive assumptions that those we just mentioned (see Remark~\ref{10:53} for a detailed discussion), our proof differs from theirs and
is inspired by the classical construction of the conjugacy between Anosov diffeomorphism and its small perturbation (see~\cite{KH}).

The paper is organized as follows. In Section~\ref{Prel} we introduce the class of sequence spaces that will be used to present a general framework that will unify various types of shadowing. We also recall the classical notion of an exponential dichotomy. Section~\ref{MR} contains main results of our paper,  while in Section~\ref{A} we present some applications.

\section{Preliminaries}\label{Prel}
\subsection{Banach sequence spaces}
In this subsection we present some basic
definitions and properties from the theory of Banach sequence spaces. The material is taken from~\cite{DD, Sasu} where the reader can also find more details.

Let $\mathcal{S}(\mathbb{Z})$ be the set of all sequences $\mathbf{s}=(s_n)_{n\in \mathbb{Z}}$ of real numbers. We say that a linear subspace $B\subset \mathcal{S}(\mathbb{Z})$ is a \emph{normed sequence space} (over $\mathbb{Z}$) if there exists a norm $\lVert \cdot \rVert_B \colon B \to \mathbb{R}_0^+$ such that if $\mathbf{s}'=(s_n')_{n\in \mathbb{Z}}\in B$ and $\lvert s_n\rvert \le \lvert s_n'\rvert$ for $n\in \mathbb{Z}$, then $\mathbf{s}=(s_n)_{n\in \mathbb{Z}}\in B$ and $\lVert \mathbf{s}\rVert_B \le \lVert \mathbf{s}'\rVert_B$. If in addition $(B, \lVert \cdot \rVert_B)$ is complete, we say that $B$ is a \emph{Banach sequence space}.

Let $B$ be a Banach sequence space over $\mathbb{Z}$. We say that $B$ is \emph{admissible} if:
\begin{enumerate}
\item
$\chi_{\{n\}} \in B$ and $\lVert \chi_{\{n\}}\rVert_B >0$ for $n\in \mathbb{Z}$, where $\chi_A$ denotes the characteristic function of the set $A\subset \mathbb{Z}$;
\item
for each $\mathbf{s}=(s_n)_{n\in \mathbb{Z}}\in B$ and $m\in \mathbb{Z}$, the sequence $\mathbf{s}^m=(s_n^m)_{n\in \mathbb{Z}}$ defined by $s_n^m=s_{n+m}$ belongs to $B$ and  $\lVert \mathbf{s}^m \rVert_B = \lVert \mathbf{s}\rVert_B$.
\end{enumerate}
Note that it follows from the definition that for each admissible Banach space $B$ over $\mathbb{Z}$, we have that $\lVert \chi_{\{n\}}\rVert_B=\lVert \chi_{\{0\}}\rVert_B$ for each $n\in \mathbb{Z}$. Throughout this paper we will assume for the sake of simplicity  that $\lVert \chi_{\{0\}}\rVert_B=1$.

We recall some  explicit examples of admissible  Banach sequence spaces over $\mathbb{Z}$ (see~\cite{DD,Sasu}).

\begin{example}\label{ex1}
The set $l^\infty =\{ \mathbf{s}=(s_n)_{n\in \mathbb{Z}} \in \mathcal{S} (\mathbb{Z}): \sup_{n\in \mathbb{Z}} \lvert s_n \rvert < \infty \}$ is a Banach sequence space when equipped with the norm $\lVert \mathbf{s} \rVert =\sup_{n\in \mathbb{Z}} \lvert s_n \rvert$.
\end{example}

\begin{example}\label{ex2}
The set $c_0=\{ \mathbf{s}=(s_n)_{n\in \mathbb{Z}} \in \mathcal{S} (\mathbb{Z}): \lim_{\lvert n\rvert \to \infty} \lvert s_n\rvert=0\}$ is a Banach sequence space when equipped with the norm $\lVert \cdot \rVert$ from the previous example.
\end{example}

\begin{example}\label{ex3}
For each $p\in [1, \infty )$, the set \[ l^p= \bigg{\{} \mathbf{s}=(s_n)_{n\in \mathbb{Z}} \in \mathcal{S}(\mathbb{Z}): \sum_{n\in \mathbb{Z}} \lvert s_n \rvert^p <\infty \bigg{\}} \] is a Banach sequence space when equipped with the norm \[\lVert \mathbf{s} \rVert= \bigg{(}\sum_{n\in \mathbb{Z}} \lvert s_n \rvert^p \bigg{)}^{1/p}.\]
\end{example}

\begin{example}[Orlicz sequence spaces]
Let $\phi \colon (0, +\infty) \to (0, +\infty]$ be a nondecreasing nonconstant left-continuous function. We set $\psi(t)=\int_0^t \phi(s) \, ds$  for $t\ge 0$. Moreover, for each $\mathbf s=(s_n)_{n\in \mathbb{Z}} \in \mathcal S(\mathbb{Z})$, let $M_\phi (\mathbf s)=\sum_{n\in \mathbb{Z}} \psi(\lvert s_n \rvert)$. Then
\[
B=\bigl\{ \mathbf s \in \mathcal S(\mathbb{Z}) : M_\phi (c \mathbf s)<+\infty \ \text{for some} \ c>0 \bigr\}
\]
is a Banach sequence space when equipped with the norm
\[
\lVert \mathbf s \rVert=\inf \bigl\{ c>0 : M_\phi ( \mathbf s / c ) \le 1 \bigr\}.
\]
\end{example}
\subsection{Important construction}
Let us now introduce sequence spaces that will play important role in our arguments.
Let $X$ be an arbitrary Banach space and $B$ any Banach sequence space over $\mathbb{Z}$ with norm $\lVert \cdot \rVert_B$. Set
\[
X_B:=\bigg{\{} \mathbf x=(x_n)_{n\in \mathbb{Z}} \subset X: (\lVert x_n\rVert)_{n\in \mathbb{Z}}\in B \bigg{\}}.
\]
Finally, for $\mathbf x=(x_n)_{n\in \mathbb{Z}} \in X_B$ we define
\begin{equation}\label{nn}
\lVert \mathbf x\rVert_B:=\lVert (\lVert x_n\rVert)_{n\in \mathbb{Z}}\rVert_B.
\end{equation}
\begin{remark}
We emphasize that in~\eqref{nn} we slightly abuse the notation since norms on  $B$ and $X_B$ are denoted in the same way. However, this will cause no confusion since in the rest of the paper we will deal with spaces $X_B$.
\end{remark}
\begin{example}
Let $B=l^\infty$ (see Example~\ref{ex1}). Then,
\[
X_B=\bigg{\{} \mathbf x=(x_n)_{n\in \mathbb{Z}} \subset X: \sup_{n\in \mathbb{Z}} \lVert x_n\rVert<\infty \bigg{\}}.
\]
\end{example}
The proof of the following result is straightforward (see~\cite{DD, Sasu}).
\begin{proposition}
$(X_B, \lVert \cdot \rVert_B)$ is a Banach space.
\end{proposition}
\subsection{Exponential dichotomies}\label{sec: exp dichot}
We now recall the notion of an exponential dichotomy which goes back to the landmark work of Perron~\cite{Per30}.
Let $X=(X, \lVert \cdot \rVert)$ be an arbitrary Banach space and $(A_m)_{m\in \mathbb{Z}}$ a sequence of invertible and bounded linear operators on $X$ such that
\begin{equation}\label{ub}
\sup_{m\in \mathbb{Z}} \lVert A_m\rVert <\infty.
\end{equation}
Set
\[
\EuScript{A}(m,n)=\begin{cases}
A_{m-1}\cdots A_n & \text{if $m>n$,} \\
\text{\rm Id} & \text{if $m=n$,} \\
A_m^{-1}\cdots A_{n-1}^{-1} & \text{if $m<n$.}
\end{cases}
\]
We say that the sequence $(A_m)_{m\in \mathbb{Z}}$ admits an \emph{exponential dichotomy} if:
\begin{enumerate}
\item there exists a sequence $(P_m)_{m\in \mathbb{Z}}$ of projections on $X$ such that
\begin{equation}\label{P}
P_{m+1}A_m=A_mP_m \quad \text{for each $m\in \mathbb{Z}$;}
\end{equation}
\item there exist $C, \lambda >0$ such that
\begin{equation}\label{ED1}
\lVert \EuScript{A}(m,n)P_n\rVert \le Ce^{-\lambda (m-n)} \quad \text{for $m\ge n$}
\end{equation}
and
\begin{equation}\label{ED2}
\lVert \EuScript{A}(m,n)(\text{\rm Id}-P_n)\rVert \le Ce^{-\lambda (n-m)} \quad \text{for $m\le n$.}
\end{equation}
\end{enumerate}
Let $B$ be an admissible Banach sequence space. We define a bounded linear operator $\mathbb A \colon X_B\to X_B$ by

\begin{displaymath}
(\mathbb A  \mathbf x)_n=A_{n-1}x_{n-1}, \quad  \text{for $n\in \mathbb{Z}$ and $\mathbf x=(x_n)_{n\in \mathbb{Z}} \in X_B$.}
\end{displaymath}
It follows easily from~\eqref{ub} that $\mathbb A$ is a well-defined and bounded linear operator on $X_B$.

The following result is only a particular case of the results established by Sasu~\cite{Sasu}.
\begin{theorem}\label{t1}
For a sequence $(A_m)_{m\in \mathbb{Z}}$ of invertible and bounded linear operators on $X$ satisfying~\eqref{ub} and an admissible Banach sequence space $B$, the following statements are equivalent:
\begin{enumerate}
\item $(A_m)_{m\in \mathbb{Z}}$ admits an exponential dichotomy;
\item $\text{\rm Id}-\mathbb A$ is an invertible operator on $X_B$.
\end{enumerate}
\end{theorem}
One can use the previous result to establish the following one (see~\cite{Henry2} for example) which tells us that the notion of an exponential dichotomy is robust under small linear perturbations.

\begin{theorem}\label{rob}
Let $B$ be an admissible Banach sequence space and
assume that $(A_m)_{m\in \mathbb{Z}}$ is a sequence of invertible and bounded operators on $X$ that admits an exponential dichotomy and satisfies~\eqref{ub}. Then, there exists $c>0$  such that for any sequence $(B_m)_{m\in \mathbb{Z}}$ of bounded linear operators on $X$ satisfying
\[
\sup_{m\in \mathbb{Z}} \lVert A_m-B_m \rVert \le c,
\]
we have that  $(B_m)_{m\in \mathbb{Z}}$ also admits an exponential dichotomy. Furthermore, there exists $K>0$ such that $\lVert (\text{\rm Id}-\mathbb B)^{-1}\rVert \le K$, where the operator $\mathbb B \colon X_B \to X_B$ is given by
\[
(\mathbb B  \mathbf x)_n=B_{n-1}x_{n-1}, \quad  \text{for $n\in \mathbb{Z}$ and $\mathbf x=(x_n)_{n\in \mathbb{Z}} \in X_B$.}
\]
\end{theorem}

\begin{remark}\label{4:01}
In fact, by carefully  inspecting  the proof of Theorem~\ref{t1} one can  conclude that we can choose constants $C, \lambda >0$ in the notion of an exponential dichotomy uniformly over all sequences $(B_m)_{m\in \mathbb{Z}}$ satisfying the assumptions of Theorem~\ref{rob}.
\end{remark}
\subsection{A fixed point theorem}
We will also use the following classical consequence of the Banach fixed point theorem (see~\cite{Lanford} for example).
\begin{theorem}\label{FPT}
Let $Z$ be a Banach space and $A$ a differentiable map defined on a neighborhood of $0\in Z$. Furthermore, let $\Gamma$ be a bounded linear operator on $Z$ such that $\text{\rm Id}-\Gamma$ is invertible. Finally, suppose that there exist $\rho>0$ and $\kappa \in (0, 1)$ such that:
\begin{enumerate}
\item for each $z\in Z$ satisfying $\lVert z\rVert \le \rho$,
\[
\lVert (\text{\rm Id}-\Gamma)^{-1}\rVert \cdot \lVert d_zA-\Gamma\rVert \le \kappa;
\]
\item \[
\lVert (\text{\rm Id}-\Gamma)^{-1}\rVert \cdot \lVert A(0)\rVert \le (1-\kappa)\rho.
\]

\end{enumerate}
Then, $A$ has a unique fixed point in $\{z\in Z: \lVert z\rVert \le \rho\}$.
\end{theorem}

\section{Main results}\label{MR}

\subsection{Setup} \label{sec: setup}
Let $B$ be an admissible Banach sequence space,  $X$  a  Banach space and $(A_m)_{m\in \mathbb{Z}}$ a sequence of invertible and bounded linear operators on $X$ that admits an exponential dichotomy and satisfies~\eqref{ub}. Moreover, let $c>0$ be a constant given by Theorem~\ref{rob}.
 Finally, let $f_n \colon X\to X$, $n\in \mathbb{Z}$ be a sequence of differentiable maps such that:
\begin{enumerate}
\item
\begin{equation}\label{fg}
\lVert d_xf_n\rVert \le c\quad \text{for $x\in X$ and $n\in \mathbb{Z}$;}
\end{equation}
\item \begin{equation}\label{UB} \sup_{m\in \mathbb{Z}} \lVert f_m\rVert_{\sup} <\infty,\end{equation}
where $\lVert f_m\rVert_{\sup}:=\sup_{x\in X} \lVert f_m(x)\rVert$;
\item there exist $D>0$ and $r>0$ such that
\begin{equation}\label{der}
\lVert d_xf_n-d_yf_n\rVert \le D\lVert x-y\rVert^r \quad \text{for $x, y\in X$ and $n\in \mathbb{Z}$.}
\end{equation}
\end{enumerate}
We consider a nonautonomous and nonlinear dynamics defined by the equation
\begin{equation}\label{nnd}
x_{n+1}=F_n(x_n),  \quad n\in \mathbb{Z},
\end{equation}
where \[F_n:=A_n+f_n.\]
We now introduce a notion of a pseudotrajectory associated with the system~\eqref{nnd}. Given $\delta >0$, the sequence $(y_n)_{n\in \mathbb{Z}} \subset X$ is said to be an $(\delta, B)$-\emph{pseudotrajectory} for~\eqref{nnd} if
$(y_{n+1}-F_n(y_n))_{n\in \mathbb{Z}}\in X_B$ and
\begin{equation}\label{pseudo}
\lVert (y_{n+1}-F_n(y_n))_{n\in \mathbb{Z}} \rVert_B \le \delta.
\end{equation}
\begin{remark}
When $B=l^\infty$ (see Example~\ref{ex1}), condition~\eqref{pseudo} reduces to
\[
\sup_{n\in \mathbb{Z}}  \lVert y_{n+1}-F_n(y_n) \rVert \le \delta.
\]
The above requirement represents a usual definition of a pseudotrajectory in the context of smooth dynamics (see~\cite{Pal00, Pil99}).
\end{remark}

We say that~\eqref{nnd} has an \emph{$B$-shadowing property} if for every $\varepsilon>0$ there exists $\delta >0$ so that for every $(\delta, B)$-pseudotrajectory $(y_n)_{n\in \mathbb{Z}}$, there exists a sequence $(x_n)_{n\in \mathbb{Z}}$ satisfying \eqref{nnd}  and such that
$(x_n-y_n)_{n\in \mathbb{Z}} \in X_B$ together with
\begin{equation}\label{wv}\lVert (x_n-y_n)_{n\in \mathbb{Z}}\rVert_B \le \varepsilon.\end{equation}
Moreover, if there exists $L>0$ such that $\delta$ can be chosen as $\delta=L\epsilon$, we say that~\eqref{nnd} has the \emph{$B$-Lipschitz shadowing property}.
\begin{remark}
In the case when $B=l^\infty$ the above definition of shadowing is the classical one~\cite{Pal00, Pil99}. On the other hand, if $B=c_0$ (see Example~\ref{ex2}) we speak about \emph{limit shadowing}, while in the case when $B=l^p$ (see Example~\ref{ex3}) we speak about \emph{$l^p$-shadowing}.
Hence, our approach offers a unified treatment of various concepts of shadowing.
\end{remark}

Our main objective is to show that under above assumptions, \eqref{nnd} has the  $B$- Lipschitz shadowing property. We begin with some simple  auxiliary results.
\subsection{Lemmata} \label{sec: lemmata}
\begin{lemma}
Let $\delta >0$ and assume that   $(y_n)_{n\in \mathbb{Z}}$ is  an $(\delta, B)$-pseudotrajectory for~\eqref{nnd}.
Furthermore, we define a map $\mathbf A \colon X_B \to X_B$ given by
\begin{displaymath}
(\mathbf A(\mathbf x))_n=F_{n-1}(y_{n-1}+x_{n-1})-y_n,
\end{displaymath}
for $n\in \mathbb{Z}$ and $\mathbf x=(x_n)_{n\in \mathbb{Z}} \in X_B$. Then, we have that $\mathbf A$ is a well-defined map.
\end{lemma}

\begin{proof}
Observe that
\[
\begin{split}
(\mathbf{A}(\mathbf x))_n &=F_{n-1}(y_{n-1}+x_{n-1})-y_n \\
&=F_{n-1}(y_{n-1}+x_{n-1})-F_{n-1}(y_{n-1})+F_{n-1}(y_{n-1})-y_n \\
&=A_{n-1}x_{n-1}+f_{n-1}(y_{n-1}+x_{n-1})-f_{n-1}(y_{n-1}) \\
&\phantom{=}+F_{n-1}(y_{n-1})-y_n,
\end{split}
\]
for every $n\in \mathbb{Z}$. Then, it follows from~\eqref{fg} that
\[
\lVert (\mathbf{A}(\mathbf x))_n \rVert \le \big{(}\sup_{m\in \mathbb{Z}} \lVert A_m\rVert \big{)}\cdot \lVert x_{n-1}\rVert +c\lVert x_{n-1}\rVert+\lVert F_{n-1}(y_{n-1})-y_n\rVert,
\]
for each $n\in \mathbb{Z}$ and $\mathbf x=(x_n)_{n\in \mathbb{Z}} \in X_B$. In a view of~\eqref{ub} and~\eqref{pseudo}, we  conclude  that $\mathbf{A}(\mathbf x)\in X_B$.
\end{proof}

\begin{lemma}\label{L}
The  map $\mathbf{A} \colon X_B \to X_B$ is differentiable and
\[
(d_{\mathbf x}\mathbf{A}\mathbf{\boldsymbol{\xi}})_n=A_{n-1}\xi_{n-1}+d_{x_{n-1}+y_{n-1}}f_{n-1}\xi_{n-1},
\]
for $\mathbf x=(x_n)_{n\in \mathbb{Z}}$, $\boldsymbol{\xi}=(\xi_n)_{n\in \mathbb{Z}} \in X_B$.
\end{lemma}

\begin{proof}
Let us fix $\mathbf x=(x_n)_{n\in \mathbb{Z}} \in X_B$ and define a linear operator $L\colon X_B \to X_B$ by
\[
(L\boldsymbol{\xi})_n=A_{n-1}\xi_{n-1}+d_{x_{n-1}+y_{n-1}}f_{n-1}\xi_{n-1}, \quad \text{for $\boldsymbol{\xi}=(\xi_n)_{n\in \mathbb{Z}}\in X_B$.}
\]
It follows readily from~\eqref{ub} and~\eqref{fg} that $L$ is a well-defined and bounded operator.  Moreover, for $\mathbf{h}=(h_n)_{n\in \mathbb{Z}}\in X_B$ we have that
\[
\begin{split}
(\mathbf{A}(\mathbf x+\mathbf h)-\mathbf{A}(\mathbf x)-L\mathbf h)_n &=f_{n-1}(x_{n-1}+y_{n-1}+h_{n-1})-f_{n-1}(x_{n-1}+y_{n-1}) \\
&\phantom{=}-d_{x_{n-1}+y_{n-1}}f_{n-1}h_{n-1} \\
&=\int_0^1 d_{x_{n-1}+y_{n-1}+th_{n-1}}f_{n-1}h_{n-1} \, dt \\
&\phantom{=}-\int_0^1 d_{x_{n-1}+y_{n-1}}f_{n-1}h_{n-1}\, dt.
\end{split}
\]
By~\eqref{der},
\begin{align*}
& \lVert (\mathbf{A}(\mathbf x+\mathbf h)-\mathbf{A}(\mathbf x)-L\mathbf h)_n \rVert  \displaybreak[0]\\
&\le \int_0^1 \lVert d_{x_{n-1}+y_{n-1}+th_{n-1}}f_{n-1}h_{n-1}-d_{x_{n-1}+y_{n-1}}f_{n-1}h_{n-1} \rVert \, dt \displaybreak[0]\\
&\le \frac{D}{1+r}\lVert h_{n-1}\rVert^{1+r}.
\end{align*}
Noting that $\lVert h_n \rVert \le \lVert \mathbf h\rVert_B$, we have that
\[
\lVert (\mathbf{A}(\mathbf x+\mathbf h)-\mathbf{A}(\mathbf x)-L\mathbf h)_n \rVert \le \frac{D}{1+r} \lVert \mathbf h\rVert_B^r \cdot \lVert h_{n-1}\rVert,
\]
and consequently
\[
\lVert \mathbf{A}(\mathbf x+\mathbf h)-\mathbf{A}(\mathbf x)-L\mathbf h \rVert_B \le \frac{D}{1+r} \lVert \mathbf h\rVert_B^{r+1}.
\]
Hence,
\[
\lim_{\mathbf h\to 0}\frac{\lVert \mathbf{A}(\mathbf x+\mathbf h)-\mathbf{A}(\mathbf x)-L\mathbf h\rVert_{B}}{\lVert \mathbf h\rVert_B}=0,
\]
which implies the desired conclusion.
\end{proof}
Set $\Gamma:=d_{\mathbf 0}\mathbf{A}$. It follows from Lemma~\ref{L} that
\[
(\Gamma  \boldsymbol{\xi})_n=A_{n-1}\xi_{n-1}+d_{y_{n-1}}f_{n-1}\xi_{n-1},
\]
for $\boldsymbol{\xi}=(\xi_n)_{n\in \mathbb{Z}} \in X_B$ and $n\in \mathbb{Z}$.
\begin{lemma}\label{lem: invertibility}
We have that $\text{\rm Id}-\Gamma$ is an invertible operator on $X_B$. Moreover, there exists a constant $K>0$, independent on the pseudotrajectory $\mathbf y$ such that  \[ \lVert (\text{\rm Id}-\Gamma)^{-1}\rVert \leq K.\]
\end{lemma}

\begin{proof}
The statement follows directly from Theorems~\ref{t1} and~\ref{rob} together with~\eqref{fg} (and the choice of $c$).
\end{proof}

\subsection{Main results} \label{sec: main results}
The following is our  main result.
\begin{theorem}\label{cor: uniqueness}
The system~\eqref{nnd} has an $B$-Lipschitz shadowing property. Furthermore, if $\varepsilon$ is sufficiently small, the sequence $\mathbf x=(x_n)_{n\in \mathbb{Z}}$ satisfying~\eqref{nnd} and~\eqref{wv} is unique.
\end{theorem}

\begin{proof}
We wish to apply Theorem~\ref{FPT}. It follows from~\eqref{der} that
\[
\lVert ((d_{\mathbf z}\mathbf{A}-\Gamma){\boldsymbol \xi})_n\rVert \le D\lVert z_{n-1}\rVert^r \cdot \lVert \xi_{n-1}\rVert \le D \lVert \mathbf z\rVert_B^r \cdot \lVert \xi_{n-1}\rVert ,
\]
and thus
\[
\lVert d_{\mathbf z}\mathbf{A}-\Gamma\rVert \le  D \lVert \mathbf z\rVert_B^r, \quad \text{for $\mathbf z\in X_B$.}
\]
Let $K>0$ be such that $\lVert (\text{\rm Id}-\Gamma)^{-1}\rVert \le K$. As we noted in Lemma \ref{lem: invertibility}, we can choose $K$ independently on $\mathbf y$.

Take now $\varepsilon>0$ such that $DK\varepsilon^r \le 1/2$ and set $L=1/4K$. Thus, for  $\delta=L\varepsilon>0$ we have that $K\delta \le \varepsilon/2$. We conclude that the assumptions of Theorem~\ref{FPT} are satisfied with $\kappa=1/2$ and $\rho=\varepsilon$ and consequently $\mathbf{A}$ has a unique fixed point $\mathbf z\in X_B$ such that $\lVert \mathbf z\rVert_B\le \varepsilon$. By setting $\mathbf x:=\mathbf y+\mathbf z$, we obtain the desired conclusions.
\end{proof}

We have the following simple consequence of Theorem~\ref{cor: uniqueness}.
\begin{corollary}[Expansivity] \label{cor: expansivity}
There exists $\varepsilon>0$ so that if $\mathbf x=(x_n)_{n\in \mathbb{Z}}$ and $\mathbf z=(z_n)_{n\in \mathbb{Z}}$ are sequences satisfying \eqref{nnd} and
\begin{displaymath}
\lVert (x_n-z_n)_{n\in \mathbb{Z}} \rVert_B \leq \varepsilon
\end{displaymath}
then, $\mathbf x=\mathbf z$.
\end{corollary}

\begin{proof}
Choose $\varepsilon>0$  small enough  as in the statement of Theorem~\ref{cor: uniqueness} and corresponding $\delta >0$ as in the definition of the $B$-Lipschitz shadowing property. Obviously,  $\mathbf z$ is a $(\delta,B)$-pseudotrajectory for~\eqref{nnd} which is $(\epsilon, B)$-shadowed by itself and $\mathbf x$. Hence, the uniqueness part in Theorem~\ref{cor: uniqueness} implies that $\mathbf x=\mathbf z$ as claimed.
\end{proof}

Under certain periodicity assumption for system~\eqref{nnd}, we
can  also formulate a version of the Anosov closing lemma in our setting.

\begin{corollary}
Assume that there exists $N\in \mathbb N$ such that $F_{n+N}=F_n$ for each $n\in \mathbb N$. Then, there exists $L>0$ such that for any $\varepsilon >0$ sufficiently small and for every $(L\varepsilon, l^\infty)$-pseudotrajectory $\mathbf y=(y_n)_{n\in \mathbb{Z}}$ for~\eqref{nnd} that satisfies $y_n=y_{n+N}$ for $n\in \mathbb N$, there exists
a solution $\mathbf x=(x_n)_{n\in \mathbb{Z}}$ of~\eqref{nnd} such that $\sup_{n\in \mathbb{Z}} \lVert x_n-y_n\rVert \le \varepsilon$ and $x_n=x_{n+N}$ for $n\in \mathbb N$.

\end{corollary}

\begin{proof}
Applying Theorem~\ref{cor: uniqueness} (for $B=l^\infty$) we obtain the existence of a sequence $\mathbf x=(x_n)_{n\in \mathbb{Z}}$ solving~\eqref{nnd} such that $\sup_{n\in \mathbb{Z}} \lVert x_n-y_n\rVert \le \varepsilon$. Let us define a new sequence $\mathbf x'=(x_n')_{n\in \mathbb{Z}} \subset X$ by
\[
x_n'=x_{n+N} \quad \text{for $n\in \mathbb{Z}$.}
\]
It is easy to verify that $\mathbf x'$ solves~\eqref{nnd} and it obviously satisfies $\sup_{n\in \mathbb{Z}} \lVert x_n'-y_n\rVert \le \varepsilon$. Hence, the uniqueness in Theorem~\ref{cor: uniqueness} implies that $\mathbf x=\mathbf x'$ which immediately yields the desired conclusion.
\end{proof}

\subsection{Shadowing of linear systems}
In this subsection we deal with the system~\eqref{nnd} in the particular case when  $f_n=0$ for each $n\in \mathbb{Z}$, i.e. when $F_n=A_n$ for every $n\in \mathbb{Z}$. Hence, we deal with linear dynamics
\begin{equation}\label{ld}
x_{n+1}=A_n x_n, \quad n\in \mathbb{Z}.
\end{equation}
\begin{corollary}\label{735}
The system~\eqref{ld} has an $B$-Lipschitz shadowing property. Furthermore, for each $\varepsilon >0$, the sequence $\mathbf x=(x_n)_{n\in \mathbb{Z}}$ satisfying~\eqref{wv} and~\eqref{ld} is unique.
\end{corollary}

\begin{proof}
In a view of Theorem~\ref{cor: uniqueness}, it only remains to establish the uniqueness of $\mathbf x$. Assume that $\mathbf x=(x_n)_{n\in \mathbb{Z}}$ and $\mathbf x'=(x_n')_{n\in \mathbb{Z}}$ satisfy~\eqref{ld}, $\lVert \mathbf x-\mathbf y\rVert_B \le \varepsilon$ and $\lVert \mathbf x'-\mathbf y\rVert_B \le \varepsilon$. Hence,
$\lVert \mathbf x-\mathbf x'\rVert \le 2\varepsilon$.

On the other hand, since $\mathbf x$ and $\mathbf x'$ satisfy~\eqref{ld}, we have that \[(\text{\rm Id}-\mathbb A)(\mathbf x-\mathbf x')=0\] and thus Theorem~\ref{t1} implies that $\mathbf x=\mathbf x'$.
\end{proof}

\begin{corollary}\label{755}
For each sequence $\mathbf y=(y_n)_{n\in \mathbb{Z}} \subset X$ such that \[\lim_{\lvert n\rvert \to \infty}\lVert y_{n+1}-A_ny_n\rVert=0,\] there exists a unique sequence $\mathbf x=(x_n)_{n\in \mathbb{Z}}$ that solves~\eqref{ld} and with the property that
\begin{equation}\label{736}
\lim_{\lvert n\rvert\to \infty} \lVert x_n-y_n\rVert=0.
\end{equation}
\end{corollary}

\begin{proof}
Take $B=c_0$ (see Example~\ref{ex2}).
We note that $\mathbf y$ is an $(L\varepsilon, B)$-pseudotrajectory for~\eqref{ld} for some $\varepsilon >0$ (where $L>0$ comes from the definition of $B$-Lipschitz shadowing).  By Corollary~\ref{735}, we can find a sequence $\mathbf x=(x_n)_{n\in \mathbb{Z}}$ that solves~\eqref{ld} and satisfies
$(x_n-y_n)_{n\in \mathbb{Z}} \in B$ (in fact, we also have that $\lVert (x_n-y_n)_{n\in \mathbb{Z}} \rVert_B=\sup_{n\in \mathbb{Z}} \lVert x_n-y_n\rVert \le \varepsilon$), which immediately yields~\eqref{736}.

The uniqueness of $\mathbf x$ can be established by using Theorem~\ref{t1}, as in the proof of Corollary~\ref{735}.
\end{proof}

\begin{corollary}\label{756}
Take $1\le p<\infty$. For each sequence $\mathbf y=(y_n)_{n\in \mathbb{Z}} \subset X$ such that \[\sum_{n\in \mathbb{Z}}\lVert y_{n+1}-A_ny_n\rVert^p <\infty,\] there exists a unique sequence $\mathbf x=(x_n)_{n\in \mathbb{Z}}$ that solves~\eqref{ld} and with the property that
\[
\sum_{n\in \mathbb{Z}}\lVert x_n-y_n\rVert^p <\infty.
\]
\end{corollary}

\begin{proof}
The proof can be obtain by repeating the arguments in the proof of Corollary~\ref{755}  and by using $B=l^p$ (see Example~\ref{ex3}) instead of $B=c_0$.
\end{proof}

\begin{remark}
In a very  particular case when~\eqref{ld} is an autonomous system, i.e. $(A_n)_{n\in \mathbb{Z}}$ is a constant sequence of operators, Corollary~\ref{735} applied to $B=l^\infty$ together with  Corollaries~\ref{755} and~\ref{756} gives us~\cite[Theorem A.]{BCDMP}.
\end{remark}

\section{Applications}\label{A}
In this section we present some applications of our main results.
\subsection{A nonautonomous version of the  Grobman-Hartman theorem}
Let $(A_m)_{m\in \mathbb{Z}}$ be  a sequence of bounded linear operators on $X$ as in Subsection~\ref{sec: setup}.  Let $c>0$ be given by Theorem \ref{rob} and fix $D,r>0$. Associated to these parameters by Theorem \ref{cor: uniqueness} (applied to $B=l^\infty$), consider $\varepsilon>0$ sufficiently small such that $D(3\varepsilon)^r <c/2$ and $\delta=L\varepsilon>0$. Moreover, suppose $\varepsilon$ is so small that $3\varepsilon$ still satisfies Theorem \ref{cor: uniqueness}. Let $(g_n)_{n\in \mathbb{Z}}$ be a sequence of maps $g_n\colon X\to X$ satisfying \eqref{fg}  with $c/2$ instead of $c$ and \eqref{der} and such that
\[\lVert g_n\rVert_{\sup} \le \delta \quad  \text{for each $n\in \mathbb{Z}$.} \] We consider a  difference equation
\begin{equation}\label{j}
y_{n+1}=G_n(y_n) \quad n\in \mathbb{Z},
\end{equation}
where $G_n:=A_n+g_n$. By decreasing $\delta$ (if necessary), we have that $G_n$ is a homeomorphism for each $n\in \mathbb{Z}$ (see~\cite{BV1}). We define
\[
\mathcal{G}(m,n)=
\begin{cases}
G_{m-1}\circ \ldots \circ G_n & \text{if $m>n$,}\\
\text{\rm Id} & \text{if $m=n$,}\\
G_m^{-1}\circ \ldots \circ G_{n-1}^{-1} & \text{if $m<n$.}
\end{cases}
\]
\begin{theorem}\label{NGH}
There exists a unique sequence $h_m \colon X \to X$, $m\in \mathbb{Z}$ of homeomorphisms such that  for each $m\in \mathbb{Z}$,
\begin{equation}\label{644}
h_{m+1}\circ G_m=A_m \circ h_m
\end{equation}
and
\begin{equation}\label{702}
\lVert h_m-\text{\rm Id}\rVert_{\sup}=\sup_{x\in X} \lVert h_m(x)-x\rVert \le \epsilon.
\end{equation}
\end{theorem}

\begin{proof}
Fix $m\in \mathbb{Z}$ and $y\in X$ and define a sequence $\mathbf y=(y_n)_{n\in \mathbb{Z}}$ by $y_n=\mathcal{G}(n, m)y$ for $n\in \mathbb{Z}$. Note that $\mathbf y$ is a solution of~\eqref{j}. Then,
\[
\sup_{n\in \mathbb{Z}}\lVert y_{n+1}-A_n y_n\rVert=\sup_{n\in \mathbb{Z}} \lVert g_n(y_n)\rVert \le \delta.
\]
Hence, it follows from Corollary~\ref{735} (applied to the case when $B=l^\infty$) that there exists a unique sequence $\mathbf x=(x_n)_{n\in \mathbb{Z}}$ such that
$x_{n+1}=A_nx_n$ for $n\in \mathbb{Z}$ and $\sup_{n\in \mathbb{Z}}\lVert x_n-y_n \rVert \le \varepsilon$. Set
\[
 h_m(y)=h_m(y_m):=x_m.
\]
It is easy to verify that~\eqref{644} holds.  Furthermore,
\[
\lVert h_m(y)-y\rVert=\lVert x_m-y_m\rVert \le \epsilon,
\]
which yields~\eqref{702}. We will now prove that each $h_m$ is a homeomorphism.  Let us start with the following simple auxiliary result.
\begin{lemma}\label{lem: unif expansive}
Let $\mathbf x=(x_n)_{n\in \mathbb{Z}}$ and $ \tilde{\mathbf x }=(\tilde{x}_n)_{n\in \mathbb{Z}}$ be two  sequences such that \[x_{n+1}=A_nx_n\quad  \text{and} \quad \tilde{x}_{n+1}=A_n\tilde{x}_n, \]for every $n\in \mathbb{Z}$. Then, for every $\rho >0$ there exists $N\in \mathbb N$ such that if $\lVert x_n -\tilde{x}_n\rVert \le 3\varepsilon$ for every $| n| \leq N$, then we have that $\lVert x_0-\tilde{x}_0\rVert \le \rho$. In addition, analogous property holds for solutions of~\eqref{j}.
\end{lemma}

\begin{proof}[Proof of the lemma]
It follows from~\eqref{P} and~\eqref{ED1} that
\[
\lVert P_0(x_0-\tilde x_0)\rVert =\lVert \EuScript{A}(0, -N)P_{-N}(x_{-N}-\tilde x_{-N})\rVert \le Ce^{-\lambda N}\lVert x_{-N} -\tilde x_{-N}\rVert.
\]
Similarly, it follows from~\eqref{P} and~\eqref{ED2} that
\[
\lVert (\text{\rm Id}-P_0)(x_0-\tilde x_0)\rVert=\lVert \EuScript{A}(0, N)(\text{\rm Id}-P_N)(x_N-\tilde x_N)\rVert \le Ce^{-\lambda N} \lVert x_N-\tilde x_N\rVert.
\]
Hence,
\[
\lVert x_0-\tilde x_0\rVert \le Ce^{-\lambda N} (\lVert x_{-N} -\tilde x_{-N}\rVert+ \lVert x_N-\tilde x_N\rVert),
\]
and therefore choosing $N$ such that $6Ce^{-\lambda N}\varepsilon<\rho$ we obtain desired conclusion.  As for the nonlinear case, let $\mathbf y=(y_n)_{n\in \mathbb{Z}}$ and $\tilde{\mathbf y }=(\tilde{y}_n)_{n\in \mathbb{Z}}$ be sequences satisfying \eqref{j} and suppose $\lVert y_n -\tilde{y}_n\rVert \le 3\varepsilon$ for every $| n| \leq N$. Setting $z_n=y_n -\tilde{y}_n$ we get that
\begin{displaymath}
z_{n+1}=G_n(y_n)-G_n(\tilde{y}_n)=d_{\tilde{y}_n}G_nz_n + G_n(y_n)-G_n(\tilde{y}_n) - d_{\tilde{y}_n}G_nz_n.
\end{displaymath}
Thus,
\begin{displaymath}
z_{n+1}=(L_n+T_n)z_n
\end{displaymath}
where $L_n:=d_{\tilde{y}_n}G_n =A_n+d_{\tilde{y}_n}g_n $ and
\begin{displaymath}
T_n:=\int_0^1 \left(d_{ty_n+(1-t)\tilde{y}_n}G_n -d_{\tilde{y}_n}G_n\right) dt.
\end{displaymath}
Now, using \eqref{der} we get that $\|T_n\|\leq D\|y_n -\tilde{y}_n\|^r\leq D(3\varepsilon)^r<c/2$ for every $| n| \leq N$. Therefore, it follows from Theorem \ref{rob} that the sequence $(B_n)_{n\in \mathbb{Z}}$ given by $B_n=L_n+T_n$ for $| n| \leq N$ and $B_n=L_n$ for $|n|>N$ admits an exponential dichotomy with constants $C$ and $\lambda$ as in Subsection \ref{sec: exp dichot} depending only on $(A_n)_{n\in \mathbb{Z}}$ and $c$ (see Remark~\ref{4:01}).  Thus, choosing $N$ such that $6Ce^{-\lambda N}\varepsilon<\rho$ and proceeding as in the linear case we get the desired result.
\end{proof}

Let us now establish continuity of $h_0$ (the same argument applies for every $h_m$).
For $\rho>0$, take  $N\in \mathbb N$  given by Lemma~\ref{lem: unif expansive}. By continuity of maps $g_n$, there exists $\eta >0$ such that for every $y,z\in X$ which satisfy $\|y-z\|<\eta$, we have that
\begin{displaymath}
\| \mathcal{G}(n,0)y -\mathcal{G}(n,0)z  \| <\varepsilon,
\end{displaymath}
for every  $| n| \leq N$.

Take $y,z\in X$ satisfying $\|y-z\|<\eta$ and consider $y_n=\mathcal{G}(n,0)y$, $z_n=\mathcal{G}(n,0)z$,  $x_n=\mathcal{A}(n,0) h_0(y)$ and $\tilde{x}_n=\mathcal{A}(n,0)h_0(z)$ for  $n\in \mathbb{Z}$. For every $n\in \mathbb{Z}$ satisfying $|n|\leq N$, we have that
\begin{displaymath}
\begin{split}
\| x_n-\tilde{x}_n  \|&=\|  \mathcal{A}(n,0)h_0(y)-\mathcal{A}(n,0)h_0(z) \| \\
&\leq \| h_{n}( \mathcal{G}(n,0)y) - h_{n}( \mathcal{G}(n,0)z)  \| \\
& \leq \| h_{n}( \mathcal{G}(n,0)y) -\mathcal{G}(n,0)y\| + \|\mathcal{G}(n,0)y-\mathcal{G}(n,0)z\|\\
&+\|\mathcal{G}(n,0)z-  h_{n}(\mathcal{G}(n,0)z)  \| \\
& =\|h_{n}(y_{n})-y_{n}\| +\|\mathcal{G}(n,0)y-\mathcal{G}(n,0)z\| \\
&+\| z_{n}-h_{n}(z_n)\| \\
&\le \varepsilon + \varepsilon +\varepsilon =3\varepsilon.
\end{split}
\end{displaymath}
Thus, since $(x_n)_{n\in \mathbb{Z}}$ and $(\tilde{x}_n)_{n\in \mathbb{Z}}$ are solutions of $x_{n+1}=A_nx_n$, $n\in \mathbb{Z}$, it follows from Lemma \ref{lem: unif expansive} that $\|h_0(y)-h_0(z)\|\le \rho$ proving that $h_0$ is continuous.

We now prove that $h_m$ is injective for each $m\in \mathbb{Z}$. Suppose that  there exist $y,z\in X$ such that $h_m(y)=h_m(z)$. We define sequences $(y_n)_{n\in \mathbb{Z}}$, $(z_n)_{n\in \mathbb{Z}}$ and $(x_n)_{n\in \mathbb{Z}}$ by $y_n=\mathcal{G}(n,m)y$, $ z_n= \mathcal{G}(n,m)y$ and $x_n=\mathcal{A}(n,m) h_m(y)=\mathcal{A}(n,m)h_m(z)$, $n\in \mathbb{Z}$. Then, by the definition of $h_m$ we have that
\begin{displaymath}
\sup_{n\in \mathbb{Z}}\lVert x_n-y_n \rVert \le \varepsilon \text{ and } \sup_{n\in \mathbb{Z}}\lVert x_n-z_n \rVert \le \varepsilon.
\end{displaymath}
In particular,
\begin{displaymath}
\sup_{n\in \mathbb{Z}}\lVert y_n-z_n \rVert \le 2\varepsilon.
\end{displaymath}
Then,  Corollary \ref{cor: expansivity} (applied for $B=l^\infty$)  implies that $y_n=z_n$ for every $n\in \mathbb{Z}$. Consequently,  $y=y_m=z_m=z$ and thus $h_m$ is injective.

Let us now establish surjectivity of $h_m$. Take $x\in X$ and consider a sequence $x_n=\mathcal{A}(n,m)x$. Then,
\begin{displaymath}
\sup_{n\in \mathbb{Z}} \| x_{n+1}-G_nx_n\|=\sup_{n\in \mathbb{Z}}\| A_nx_n- G_nx_n \|=\sup_{n\in \mathbb{Z}}\|g_n(x_n)\|\le \delta.
\end{displaymath}
Hence, $(x_n)_{n\in \mathbb{Z}}$ is a $(\delta, l^\infty)$-pseudotrajectory for \eqref{j}. In particular, by Theorem \ref{cor: uniqueness} applied to $(G_n)_{n\in \mathbb{Z}}$ there exists a unique sequence $(y_n)_{n\in \mathbb{Z}}$ satisfying \eqref{j} and such that $\sup_{n\in \mathbb{Z}}\lVert x_n-y_n \rVert \le \varepsilon$. Therefore, $h_m(y_m)=x_m=x$ proving that $h_m$ is surjective.

The proof of continuity of $h_m^{-1}$ is completely analogous to the proof of continuity of $h_m$ and therefore we omit it.

Finally, it remains to establish uniqueness of $h_m$. Indeed,  suppose that $(\tilde{h}_m)_{m\in \mathbb{Z}}$ is  a sequence satisfying~\eqref{644} and~\eqref{702}. Take $y\in X$, $m\in \mathbb{Z}$  and consider  a sequence $y_n=\mathcal{G}(n,m)y$, $n\in \mathbb{Z}$. Then,
\begin{displaymath}
h_{n+1}(y_{n+1})=A_n h_n(y_n)  \text{ and } \tilde{h}_{n+1}(y_{n+1})=A_n \tilde{h}_n (y_n).
\end{displaymath}
Moreover, $\|h_n(y_n)-y_n\|\le \varepsilon$ and $\|\tilde{h}_n(y_n)-y_n\|\le \varepsilon$ for every $n\in \mathbb{Z}$. Thus, by the uniqueness in Theorem \ref{cor: uniqueness} it follows that $h_n(y_n)=\tilde{h}_n(y_n)$ for every $n\in \mathbb{Z}$. Consequently, $h_m(y)=\tilde h_m(y)$.  Since $y$ was arbitrary we conclude that $h_m=\tilde h_m$ and the proof of the theorem is completed.
\end{proof}

\begin{remark}\label{10:53}
Theorem~\ref{NGH} can be described  as a nonautonomous version of the classical Grobman-Hartman theorem~\cite{Hart60}. The first result of this type has been established by Palmer~\cite{Palmer}  for finite-dimensional dynamics with continuous time. Subsequent generalizations for infinite-dimensional  dynamics  that admits a nonuniform exponential dichotomy are due to Barreira
and Valls~\cite{BV1, BarVal-DCDS07} (see also~\cite{BDV1} for simple proof).

When compared with main results in~\cite{BV1,  BarVal-DCDS07, BDV1}, our Theorem~\ref{NGH} works under stronger assumptions that maps $g_n$ are differentiable and that~\eqref{der} holds.

However, our goal was not to refine results from those papers (which in fact seem to be rather optimal)  but rather to offer a new approach to the problem of linearization of a nonautonomous dynamics based on the shadowing theory we developed in previous section.
\end{remark}

\subsection{Preservation of positive Lyapunov exponents}
Using our main results, we can also formulate conditions under which positive Lyapunov exponents associated with a linear dynamics remain unchanged under small nonlinear perturbations.  For more general results related to the preservation of Lyapunov exponents under perturbations we refer to~\cite[Section 7.]{BDV2} and references therein.
We continue to use the same notation as in the previous subsection.
\begin{corollary}
Assume that $(y_n)_{n\in \mathbb{Z}}$ is a solution of~\eqref{j} such that
\[
\lambda:=\limsup_{n\to \infty} \frac 1 n \log \lVert y_n\rVert>0.
\]
Then, there exists a solution $(x_n)_{n\in \mathbb{Z}}$ of~\eqref{ld} such that
\begin{equation}\label{800}
\lambda=\limsup_{n\to \infty} \frac 1 n \log \lVert x_n\rVert.
\end{equation}
\end{corollary}

\begin{proof}
Observe that
\[
\sup_{n\in \mathbb{Z}}\lVert y_{n+1}-A_n y_n\rVert=\sup_{n\in \mathbb{Z}} \lVert g_n(y_n)\rVert \le \delta.
\]
Hence, it follows from Corollary~\ref{735} (applied to the case when $B=l^\infty$) that there exists a unique sequence $\mathbf x=(x_n)_{n\in \mathbb{Z}}$ such that
$x_{n+1}=A_nx_n$ for $n\in \mathbb{Z}$ and $\sup_{n\in \mathbb{Z}}\lVert x_n-y_n \rVert \le \varepsilon$. It now follows readily that~\eqref{800} holds.
\end{proof}

\medskip{\bf Acknowledgements.} L. B. was partially supported by a CAPES-Brazil postdoctoral fellowship under Grant No. 88881.120218/2016-01 at the University of Chicago. D. D. was supported by the Croatian Science Foundation under the project IP-2014-09-2285.

redpajama_set_name: RedPajamaArXiv


Welcome to the luxurious lifestyle of Cape Harbour. This beautiful yachting community is full of life and fun under the sun! Endless options with: tennis & basketball courts, restaurants, shopping, farmers market (in season), and swimming. All within walking distance from your front door. Bring your furry friend along as the community is extremely pet friendly. When it's time to relax you come home to this amazing two story, four bedroom, three bathroom, plus den home that welcomes you with water views from nearly every room. Luxurious diagonal tile installed throughout the first level with an underlayment proving to be a flawless installation. Bedrooms and upstairs area have carpet. Lower your boat into the water from your private dock with boat lift, one lock leads you to open water. No bridges! Have more than one boat? Options to store them at the Marina as well. Watch the dolphins play in the water from your private tiki hut, witness the majestic manatees swim by while you paddle board. The time is now for you to live in paradise! A neighboring vacant lot sold for $650k- the value is absolutely MASSIVE in this property!!!
Listing courtesy of RE/MAX Radiance. Selling Office: RE/MAX Radiance.


Abstract: Itamar Radai's micro history of the experiences of the Palestinians in Jerusalem and Jaffa in the closing months of the British Palestine Mandate is a scholarly and informed comparative study. The focus is narrow but the book's significance is broad, and it augments our understanding of how and why the al-Nakhba, 'the catastrophe,' unfolded for the Palestinians. The book is rich on detail on the Palestinian community, based on excellent use of Arabic sources. Radai avoids taking sides in the polarized literature on 1948, but his conclusions reflect poorly on the Palestinian middle class, who fled the shock of war. The book's human and engaged text vividly brings to life the Palestinians' lives and their struggle to remain in their homes as Palestine went to war from late 1947.


| General Discussion (Anxiety Free Zone) | TV / Movies | Topic: Actor Bill Paxton has died... 0 Members and 1 Guest are viewing this topic.
...after complications of heart surgery. How tragic.
Another one that has left too soon, may he rest in peace.


text:
using std::string;
using std::map;

class BrainfuckCompileAndGo : public BrainfuckRunner {
 public:
  BrainfuckCompileAndGo();
  virtual bool init(string::const_iterator start,
                    string::const_iterator end);
  virtual void* run(BrainfuckReader reader,
                    void* reader_arg,
                    BrainfuckWriter writer,
                    void* writer_arg,
                    void* memory);

  virtual ~BrainfuckCompileAndGo();

 private:
  int executable_size_;
  void* executable_;
  int exit_offset_;

  void add_jne_to_exit(string* code);
  void add_jl_to_exit(string* code);
  void add_jmp_to_offset(int offset, string* code);
  void add_jmp_to_exit(string* code);
  void emit_offset_table(map<int8_t, uint8_t>* offset_to_change,
                         int8_t* offset,
                         string *code);
  bool generate_sequence_code(string::const_iterator start,
                              string::const_iterator end,
                              string* code);
  bool generate_loop_code(string::const_iterator start,
                          string::const_iterator end,
                          string* code);
  void generate_read_code(string* code);
  void generate_write_code(string* code);
};

#endif  // BF_COMPILE_AND_GO_H_

redpajama_set_name: RedPajamaGithub


text:
Motorola Rumored To Be Working On A 5.9-inch Nexus Smartphone
Updated on Jul 30, 2014 by Guest Authors
Motorola's decline over the years has been well documented. The once giant is now forced to live in the shadow of the likes of Samsung, Apple and HTC. Although their adaption to Android was quick, it wasn't quite impactful as its rivals.
Motorola released a winner in the form of its latest flagship, the Moto X, but it didn't quite pick up the traction and sales figure Motorola hoped it would. Moto G was another great release, but it wasn't a game changer, which Motorola direly needs.
It seems Big G has finally heard the cry, as the rumor mill has brought a new bit claiming a new Motorola made Nexus phone is on the way. To make it spicier, the release date is touted for November.
The rumored Nexus device is codenamed 'Shamu', which fits Google's sea creature theme it uses to assign codenames to Nexus devices. The active development has been more or less confirmed by the filing of a bug which references in Google's issue tracker.
The Shamu will purportedly boast a gigantic 5.9-inch touchscreen, making it the biggest Nexus phone ever, and also Google & Motorola's first venture in the already crowded phablet market. With Apple also rumored to launch a 5.5-inch iPhone 6 this fall, it makes sense for Google to take the big step also. The Shamu is also touted to boast a fingerprint sensor, although its placement is a mystery. Even still, it makes for an interesting smartphone.
Apart from its screen size and fingerprint sensor, nothing is known about the Shamu yet. But if it indeed turns out to be the next Nexus smartphone, the rumor mill will get bonkers pretty soon, and we'll start hearing about it. Still, you might want to take this bit with a pinch of salt, though, even if Motorola is making a handset codenamed 'Shamu' it might be a different phone entirely, or may not launch at all. Regardless, rumors do have a funny way of becoming true.
If it does pan out, we'll be seeing the Shamu launched alongside the rumored HTC Nexus tablet, and when Android L is finally released.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: How do I recreate /var/log/audit with the right permissions? Since my disk space was running out and I saw it was using up several gigabytes, I decided to delete everything in /var/log/*, thinking they're just log files and it should be OK to delete.
I manually ran a sudo mkdir /var/log/audit. However, when running an ausearch command, I'm getting the following output.
Error opening /var/log/audit/audit.log (No such file or directory)

How do I recreate this folder and file with the right permissions, attributes, etc?
I'm using Fedora 34.

A: ausearch searches through the audit log at /var/log/audit/audit.log. It not being present causes the error.
Probably, the sudo mkdir /var/log/audit didn't set the correct permissions. It should be
drwx------.   root root system_u:object_r:auditd_log_t:s0      /var/log/audit

A sudo chown root:root /var/log/audit && chmod 0700 /var/log/audit should take care of that.
You can force the correct SELinux permissions with restorecon -Rv /var/log/audit/.
Restart the audit daemon afterwards with systemctl restart auditd.service.

A: Can't comment so will note that Edward's info above is SLIGHTLY incorrect:  auditd cannot be restarted with systemctl.
Restart Auditd with "service auditd restart" (or start, stop, etc.)

redpajama_set_name: RedPajamaStackExchange


Ayurvedic Herbs and Home Remedies for Children By Vishnu Dass, NTS, so I will just go over some home remedies, Ayurvedic herbs, and herbal compounds that can be safely used in most cases of common childhood as well as lowering a fever by promoting sweating.
Harmony Remedies: An Overview of Adaptogens ©David Winston, AHG, 2004 and spermatorrhea) and to reduce excessive sweating. Dang Shen root (Codonopsis pilosula) – Sweet, warm, moist herbal traditions.
Prosthetist's Column: Don't Sweat It! by Tony van der Waarde, CP(c) Another possibility is that excessive sweating inside the socket is causing skin irritation. decided to check out what was available in the way of natural remedies for skin problems.
INVESTIGATION DATA Infant's Information: Last: Excessive sweating i) Stool changes c) Lethargy or sleeping more than usual j) Herbal remedies f) Other Currently, does any caregiver use any of the following?


Christi and I were given tickets to the IMAX Theater in Balboa Park. We went on Sunday afternoon with the intent of watching whatever was showing. She wanted to see National Parks and I wanted to see Jerusalem but we ended up watching Journey to Space. The best part for me was images from the Hubble telescope. To see the mysteries of deep space on such a large screen was powerful. It makes me appreciate the size and scope of God and God's creation.
I've been introduced to a new word. It's actually an ancient Hebrew word, "Tzintzum" (zimzum) which means to contract/constrict/condensate. These English definitions incorporate physics and chemistry as their basis and are used in childbirth, breathing and turning invisible vapor into visible water. Zimzum is a powerful concept. When used in talking about God, zimzum becomes cosmic.
The theological notion of this word is that God contracted/constricted in order to create the physical world. Think of it like good parenting. Prior to children, "you-yourself" filled your entire life. In order to allow a child into your life, you had to contract and make room. I am amazed at families with multiple children because not only do the parents have to zimzum, but so does each child in order to make room for another sibling. Zimzum does not come natural to humans. We tend to do the opposite – we expand and take over everything. Like water in the bathtub, we spread out, covering everything. Imagine the couple that moves from a studio apartment to a three-bedroom house. It doesn't take long to fill the house. Yet, zimzum is like the grandparents that occupy a 10,000 square foot home but choose to constrict into an 8 by 10 bedroom so that many others can have a roof over their heads. Theologically, this is what God did to make creation; except that the house is the size of the Boeing Everett Factory and the living space is the size of a mini-fridge.
Imagine the zimzum it takes for God to be born as a human. It's a little like Robin Williams' cartoon portrayal of Aladdin describing what it's like to live as a genie in a lamp. He said, "PHENOMENAL COSMIC POWERS!!! … Itty-bitty living space." That's zimzum for Jesus and explains why the tomb couldn't stay occupied for too long. I wonder if Jesus upon ascending into heaven felt as if he could finally breathe again.
Zimzum shows me the depth, length and breadth of God's love. It is strange to imagine someone with phenomenal cosmic powers willingly constricting for a creation that will eventually turn against its creator. It takes a strangle type of love that is willing to take a huge volume of vapor and condense into a single drop of baptismal water. Or that the King of the Universe is willing to be born in a barn to humble parents and to eventually condense onto a cross. That is some kind of love.


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