text:
\section{Introduction}
A basic question in the theory of discrete dynamical systems, and in statistical mechanics, is whether a chaotic iterated function $M:X\rightarrow X$, that maps a space $X\subseteq\mathbb{R}^d$ back onto $X$, has an equilibrium distribution, with probability density function (PDF) $\rho(x)$~\footnote{The density function is with respect to some underlying measure, typically Lebesgue. Throughout this paper we will use the same symbol for the distribution and associated PDF, with meaning taken from context.}. A necessary condition is that the distribution $\rho$ is \emph{invariant} under $M$, i.e. if $x\sim \rho$ ($x$ is distributed as $\rho$) then so is $M(x)$, and further that the orbit of almost all points $x\in X$ defined as $\mathcal{O}^+(x)=\left\{x,M(x),M^2(x),M^3(x),\ldots\right\}$ tends in distribution to $\rho$.
Then, under mild conditions, the map is ergodic for $\rho$, that is, expectations with respect to $\rho$ may be replaced by averages over the orbit~\cite{dorfman1999introduction,LasotaMackey}.

For example, it is well known~\cite{UlamvonNeumann1947,may1976simple,dorfman1999introduction,LasotaMackey}  that the logistic map $M_\text{log}(x)=4x(1-x)$, for $x\in [0,1]$, is chaotic with the equilibrium distribution having PDF
\[ \rho_\text{log}(x) = \frac{1}{\pi \sqrt{x(1-x)}}, \]
implying that $M_\text{log}$ is ergodic for $\rho_\text{log}$.

Our motivating interest is in using this ergodic property to implement sample-based inference for Bayesian analysis or machine learning. In those settings, the target distribution $\rho$ is defined by the application. Generating a sequence $\left\{x_0,x_1,x_2,x_3,\ldots\right\}$ that is ergodic for $\rho$ is useful because then  expectations of any quantity of interest can be computed as averages over the sequence, i.e.,
\begin{equation}
 \lim_{N\to\infty}\frac{1}{N}\sum_{i=0}^{N-1} g\left(x_i\right) =\int_X g(x)\rho(x)\,\mathrm{d} x .
 \label{eq:expect}
\end{equation}
Commonly in statistics, such ergodic sequences are generated using a \emph{stochastic} iteration that simulates a Markov chain targeting $\rho$~\cite{mcmc_liu}; here we explore \emph{deterministic} iterations that generate an orbit that is ergodic for $\rho$.

The equilibrium distribution for a given iterated function, if it exists, can be approximated by computing the orbit of the map for some starting point then performing kernel density estimation, or theoretically by seeking the stationary distribution of the  Frobenius--Perron (FP) operator that is the transfer~\cite{KlusKoltaiSchutte2016}, aka push-forward, operator induced by a deterministic map~\cite{dorfman1999introduction,LasotaMackey}; we present the FP  equation in Section~\ref{sec:LEFP}.

The inverse problem that we consider here, of determining a map that gives a prescribed equilibrium distribution, is the \emph{inverse Frobenius--Perron problem} (IFPP) and has been studied extensively~\cite{grossmann1977invariant,ershov1988solution,diakonos1996construction,diakonos1999stochastic,pingel1999theory,bollt2000controlling,nie2013new,nie2018matrix}. Summaries of previous approaches to the IFPP are presented in~\cite{bollt2000controlling,rogers2008synthesis} that characterize approaches as based on conjugate functions (see \cite{grossmann1977invariant} for details) or the matrix method (see~\cite{rogers2008synthesis} for details), and~\cite{wei2015solutions} that also lists the differential equation approach. Existing work almost solely considers the IFPP in one-variable, $d=1$, the exception being the development of a two-dimensional solution in~\cite{bollt2000controlling} that is also presented in~\cite{rogers2008synthesis}.

The matrix method, first suggested by Ulam~\cite{Ulam1960}, solves the IFPP for a piecewise-constant approximation to the target density, using a transition matrix representation of the approximated FP operator. Convergence of the discrete approximation is related to Ulam's conjecture, and has been proved for the multidimensional  problem; see~\cite{wei2015solutions} and references therein. While the matrix method allows construction of solutions, at least in the limit, existing methods only offer a limited class of very non-smooth solutions, so are not clearly useful for characterizing all solutions, as we do here. We do not further consider the matrix methods.

The development in this paper starts with the differential equation approach in which the IFPP for restricted forms of distributions and maps is written as a differential equation that may be solved. We re-derive some existing solutions to the IFPP this way in Sec.~\ref{sec:IFPP}. The main contribution of this paper is to show that the form of these solutions may be generalized to give the general solution of the IFPP for any probability distribution in any dimension $d$, as presented in the factorization theorem of Sec.~\ref{sec:fact}.  This novel factorization represents solutions of the IFPP in terms of the Rosenblatt transformation~\cite{rosenblatt-1952} and a \emph{uniform map}, that is a map on $[0,1]^d$ that leaves the uniform distribution invariant. In particular, we show that the conjugating functions in~\cite{grossmann1977invariant} are exactly the inverse Rosenblatt transformations. For a given Rosenblatt transformation, there is a one-to-one correspondence between solution of the IFPP and choice of a uniform map.

This reformulation of the IFPP in terms of two well-studied constructs leads to practical analytic and numerical solutions by exploiting existing, well-developed methods for Rosenblatt transformations, and for deterministic iterations that target the uniform distribution. The factorization also allows us to establish equivalence of solutions of the IFPP and other methods that employ a deterministic map within the generation of ergodic sequences.  This standardizes and simplifies existing solution methods by showing that they are special cases of constructing the Rosenblatt transformation (or its inverse), and selection of a uniform map.

This paper starts with definitions of the IFPP and the Lyuapunov exponent in Sec.~\ref{sec:LEFP}. Solutions of the IFPP in one-dimension, $d=1$, are developed in Sec.~\ref{sec:IFPP}. These solutions for $d=1$ motivate the factorization theorem in Sec.~\ref{sec:fact} that presents a general solution to the IFPP for probability distributions with domains in $\mathbb{R}^d$ for any $d$. Sec.~\ref{sec:onedim}  presents further examples of univariate, $d=1$, solutions to the IFPP based on the factorization theorem  in Sec.~\ref{sec:fact}.
Two 2-dimensional numerical examples are presented in Section~\ref{sec:num} to demonstrate that the theoretical constructs may be implemented in practice. A summary and discussion of results is presented in Section~\ref{sec:sad}, including a discussion of some existing computational methods that can be viewed as implicitly implementing the factorization solution of the IFPP presented here.

\section{Inverse Frobenius--Perron Problem and Lyapunov Exponent}
\label{sec:LEFP}

In this section we define the forward and inverse Frobenius--Perron problems, and also the Lyuapunov exponent that measures chaotic behaviour.

\subsection{Frobenius--Perron Operator}
\label{sec:FPop}
A deterministic map $x_{n+1} = M(x_n)$ defines a map on probability distributions over state, called the transfer operator~\cite{KlusKoltaiSchutte2016}. Consider the case where the initial state $x_0\sim\rho_0(\cdot)$ ($x_0$ is distributed as $\rho_0$) for some distribution $\rho_0$ and let $\rho_n$ denote the $n$-step distribution, i.e., the distribution over $x_n=M^n(x_0)$ at iteration $n$. The transfer operator that maps $\rho_{n}\mapsto\rho_{n+1}$ induced by $M$  is given by the Frobenius--Perron operator associated with $M: x\mapsto y$~\cite{dorfman1999introduction,LasotaMackey,Gaspard1998}
\begin{equation}
  \rho_{n+1}(y) = \sum_{x\in M^{-1}(y)} \frac{\rho_{n}\left(x \right)}{|J(x)|}
  \label{eq:FP}
\end{equation}
where $|J(x)|$ denotes the Jacobian determinant~\footnote{We have used the language of differential maps, as all the maps that we display in this paper are differentiable almost everywhere~\cite{varberg1971change}. More generally, $|J(x)|^{-1}$ denotes the density of $\rho_{n} M^{-1}$ with respect to $\rho_{n+1}$, see, e.g., \cite[Remark 3.2.4.]{LasotaMackey}.} of $M$ at $x$, and the sum is over inverse images of $y$.

The equilibrium distribution $\rho$ of $M$ satisfies
\begin{equation} \label{eq:FPinvar}
  \rho(y) = \sum_{x\in M^{-1}(y)} \frac{\rho(x)}{|J(x)|}
\end{equation}
and we say that $\rho$ is invariant under $M$.

\subsection{Inverse Frobenius--Perron Problem}

The inverse problem that we address is finding an iterative map $M$ that has a given distribution $\rho$ as its equilibrium distribution. We do this by performing the inverse Frobenius--Perron problem (IFPP) of finding $M$ that satisfies~\eqref{eq:FPinvar} to ensure that $\rho$ is an invariant distribution of $M$. Establishing chaotic hence ergodic behaviour is a separate calculation.

We will assume throughout that $\rho$ is absolutely continuous with respect to the underlying measure so that a probability density function $\rho(x)$ exists, and, furthermore, that $\rho(x)>0$, $\forall x\in X$.

\subsection{Lyapunov Exponent}

The Lyapunov exponent $h$ of an iterative map gives the average exponential rate of divergence of trajectories.
We define the (maximal) Lyapunov exponent $h$ as~\cite{dorfman1999introduction,LasotaMackey}
\begin{equation}
   h = \lim_{N\to\infty} \frac{1}{N} \log \left|\frac{\mathrm{d} x_N}{\mathrm{d} x_0}\right|
   = \lim_{N\to\infty}\frac{1}{N} \sum^{N-1}_{n=0} \log|J(x_n)|
   \label{eq:Lyap}
\end{equation}
that features the starting value $x_0$. For ergodic maps the dependency on $x_0$ is lost as $N\to\infty$  and the Lyapunov exponent may be written
\begin{equation}
  h = \int_X \log|J(x)| \rho(x)\, \mathrm{d} x \label{eq:LyapExpect}
\end{equation}
where $\rho(x)$ is the invariant density. A positive Lyapunov  exponent $h$ indicates that the map is chaotic.

The theoretical value for the Lyapunov exponent may be obtained using~\eqref{eq:LyapExpect}, while~\eqref{eq:Lyap} provides an empirical value obtained by iterating the map $M$. For example, the Lyapunov exponent of the logistic map evaluated by~\eqref{eq:LyapExpect} is $h_\text{log} = \log 2\approx 0.693147$ while~\eqref{eq:Lyap} evaluated over an orbit with $10000$ iterations gave $h_\text{log}\approx 0.693140$.

For chaotic maps, any uncertainty in initial value  means that an orbit cannot be precisely predicted, since initial states with any separation become arbitrarily far apart, within $X$, as iterations increase. Hence it is useful to characterize the orbit statistically, in terms of the equilibrium distribution over states in the orbit.

It is interesting to note that theoretical chaotic and ergodic behaviour does not necessarily occur when iterations of a map are implemented on a finite-precision computer. For example, when the logistic map, above, on $[0,1]$ is iterated on a binary computer the multiplication by $4$ corresponds to a shift left by two bits and all subsequent operations maintain lowest order bits that are zero. Repeated iterations eventually produce the number zero, no matter the starting value. While it is simple to correct this non-ideal behaviour, as was done to give the numerical Lyapunov exponent, above, it is important to note that computer implementation can have very different dynamics to the mathematical model.

\section{Solution of the IFPP in $1$-dimension}
\label{sec:IFPP}

In this Section we develop solutions of the IFPP in one-dimension, $d=1$. Without loss of generality we consider distributions on the unit interval $X=[0,1]$, as the domain of any univariate distribution may be transformed by a change of variable to $X=[0,1]$, including when the domain is the whole real line $(-\infty,\infty)$.

For distributions over a scalar random variable, the FP equation for the invariant density~\eqref{eq:FPinvar} simplifies to
\begin{equation}
  \rho(y) = \sum_{x\in M^{-1}(y)} \frac{\rho\left(x \right)}{|M'(x)|}.
  \label{eq:FP1dim}
\end{equation}

\subsection{The Simplest Solution}
\label{sec:simplest}
We first note, almost trivially, that the identity map $M=I$, where $I(x)=x$, has $\rho$ as an invariant distribution, so solves the IFPP for any $\rho$. Somewhat less trivial is to derive this simplest solution by assuming that $M$ is monotonic increasing and $M(0)=0$, so that there is just one inverse image in~\eqref{eq:FP1dim}. Writing $|M'(x)|=\mathrm{d} M/\mathrm{d} x$ gives the differential equation with separated variables
\begin{equation}
  \rho(M) \mathrm{d} M = \rho(x) \mathrm{d} x
  \label{eq:sepde}
\end{equation}
that has solution
\[ F(M) = F(x) \]
where $F(x)=\int_0^x\rho(x')\,\mathrm{d} x'$ is the cumulative distribution function (CDF) for $\rho$.
If $F$ is invertible denote the inverse by $F^{-1}$, called the the inverse distribution function (IDF), otherwise let $F^{-1}$ denote the generalized inverse distribution function,
$ \displaystyle F^{-1}(p)=\inf\{x\in X :F(x)\geq p\}$. Then, $M=F^{-1}(F(x)) = x$, or $M(x)=x$, almost everywhere. Hence the identity map is the \emph{unique} monotonic increasing map that has $\rho$ as its invariant distribution. Clearly, the identity map is not ergodic for $\rho$.

We may generalize this solution by setting $M(0)= k$, for some $k\in [0,1)$, and also only requiring $M$ to be piecewise continuous. Allowing one discontinuity in $M$ we write the integral of the separated differential equation~\eqref{eq:sepde} as
\[ F(M) = F(x) +k  \mod 1 \]
giving the solution to the IFPP
\begin{equation}
  M(x) = \left(F^{-1} \circ T^c \circ F\right)(x)
  \label{eq:shift}
\end{equation}
where $c=F^{-1}(k)$. Here $T^c$ denotes the operator that translates by $c$ with wrap-around on $[0,1)$~\footnote{Hence $T^c$ is the translation operator on the unit circle $S^1$}
\begin{equation}
 T^c(y) = y + c - \lfloor y + c \rfloor ,
 \label{eq:Tc}
\end{equation}
where $\displaystyle \lfloor \,\ \rfloor $ denotes the floor function; see Figure~\ref{fig:Tct1} (left). The identity map is recovered when $k=c=0$.

\begin{figure} \centering
     \includegraphics[width=0.49\linewidth]{Tc_0_7.png}
     \includegraphics[width=0.49\linewidth]{t1.png}
     \caption{Translation operator $T^c$ for $c=0.3$ (left) and triangle map $t_1$ (right).}
     \label{fig:Tct1}
\end{figure}

It is easily seen that the Lyapunov exponent of the map in~\eqref{eq:shift} is $h=0$, so the map is not chaotic. However, interestingly, this map can generate a sequence of states that produce a numerical integration rule with respect to $\rho$, since appropriate choices of $c$ and number of iterations $N$ can produce a rectangle rule quadrature or a quasi Monte Carlo lattice rule; see Sec.~\ref{sec:prop}.

\subsection{Exploiting Symmetry in $\rho(x)$}
When the PDF $\rho$ has reflexive symmetry about $1/2$, i.e. $\rho(x)=\rho(1-x)$, we can simplify the FP equation~\eqref{eq:FP1dim} by assuming that the map $M$ has the same symmetry. Specifically, we write the triangle map (see Figure~\ref{fig:Tct1} (right))
\begin{equation}
 t_1(x) = 1-2\left|x-1/2\right|
 \label{eq:t1}
\end{equation}
that has reflexive symmetry about $1/2$, and write
\begin{equation}
 M(x) = m(t_1(x))
 \label{eq:doubsym}
\end{equation}
where $m(x):[0,1]\to [0,1]$ is a monotonic increasing map with $m(0)=0$ (and, it will turn out, $m(1)=1$).
Hence the FP equation simplifies to
\begin{equation}
  \rho(y) = 2\frac{\rho\left(x \right)}{|M'(x)|}, \quad  x \in M^{-1}(y),
  \label{eq:FPsym}
\end{equation}
that we can write as the separated equations
\begin{eqnarray*}
  \rho(M)\, \mathrm{d} M = 2 \rho(x) \mathrm{d} x, \quad & x<1/2, \\
  \rho(M)\, \mathrm{d} M = -2 \rho(x) \mathrm{d} x, \quad & x>1/2,
\end{eqnarray*}
that has the continuous solution
\[ F(M) = t_1(F(x)) \]
giving the continuous solution to the IFPP
\begin{equation}
  M(x) = \left(F^{-1} \circ t_1 \circ F\right)(x).
  \label{eq:sym}
\end{equation}
One can solve for $m(x) = F^{-1}(2F(x/2))$, though we do not further consider the function $m$.

The approach we have used here simplifies the approach in~\cite{diakonos1996construction}, while  `doubly symmetric' maps of the form~\eqref{eq:doubsym} were considered in~\cite{gyorgyi1984fully}, and again in~\cite{pingel1999theory,diakonos1999stochastic}.

\subsection{Symmetric Triangular Distribution}
\label{sec:tent}
To give a concrete example of the solution in~\eqref{eq:sym}, we consider the symmetric triangular distribution on $[0,1]$ with PDF
\begin{equation}
 \rho_\text{tri}(x) = 2-4\left| x-\frac{1}{2} \right|
\end{equation}
that has reflexive symmetry about $x=\frac{1}{2}$. The CDF is
\[ F_\text{tri}(x) = \begin{cases}
                        2x^2                    & 0\le x \le \frac{1}{2} \\
                        \frac{1}{2} + 4x - 2x^2 & \frac{1}{2} \le x \le 1
                      \end{cases}
\]
giving the unimodal map, after substituting into~\eqref{eq:sym},
\begin{equation}
  \label{eq:Mtent}
  M_\text{tri}(x) = \left\{ \begin{array}{rl}
  \sqrt{2}x & 0\le x \le \frac{1}{\sqrt{8}} \\
  1-\sqrt{\frac{1}{2}-2x^2} & \frac{1}{\sqrt{8}} \le x \le \frac{1}{2} \\
  1-\sqrt{\frac{1}{2}-2(1-x)^2} & \frac{1}{2} \le x \le 1 - \frac{1}{\sqrt{8}} \\
  \sqrt{2}(1-x) & 1 - \frac{1}{\sqrt{8}} \le x \le 1
  \end{array} \right.
 \end{equation}
shown in Figure~\ref{fig:Mtent} (left). The same map was derived in~\cite{grossmann1977invariant}. Figure~\ref{fig:Mtent} (right) shows a normalized histogram of $10^6$ iterates of $M_\text{tri}$ starting at $x=0.3$, confirming that the orbit of $M_\text{tri}$ converges to the desired triangular distribution. The numerical implementation avoids finite-precision effects, as discussed later.
\begin{figure} \centering
     \includegraphics[width=0.49\linewidth]{Mtri.png}
     \includegraphics[width=0.49\linewidth]{MtentOrbit.png}
     \caption{Iterative map $M_\text{tri}$ in~\eqref{eq:Mtent} (left) and a histogram of $1\times 10^6$ iterates of the map $M_\text{tri}$ (right).}
     \label{fig:Mtent}
\end{figure}

The theoretical Lyapunov exponent for $M_\text{tri}$ is $h_\text{tri}=\log 2\approx 0.693147$ while~\eqref{eq:Lyap} evaluated over an orbit with $10^6$ iterations gave $h_\text{tri}\approx 0.693148$.

\section{Solutions of the IFPP for General Multi-Variate Target Distributions}
\label{sec:fact}
The solutions to the $1$-dimensional IFPP with special structure in Eqs~\eqref{eq:shift} and \eqref{eq:sym} are actually examples of a general solution to the IFPP for multi-variate probability distributions with no special structure. We state that connection via a theorem that establishes a factorization of all possible solutions to the IFPP, and that also provides a practical means of solving the IFPP.

We first introduce the forward and inverse Rosenblatt transformations, that is the multi-variate generalization the CDF and IDF for univariate distributions.

\subsection{Forward and Inverse Rosenblatt Transformations}
\label{sec:rosen}
A simple transformation of an absolutely continuous $d$-variate distribution into the uniform distribution on the $d$-dimensional hypercube was introduced by Rosenblatt~\cite{rosenblatt-1952}, as follows.
The joint PDF can be written as a product of conditional densities,
\[
 \rho(x_1,\ldots,x_d) = \rho_1(x_1) \rho_2(x_2|x_1) \cdots \rho_d(x_d | x_1\ldots,x_{d-1}),
\]
where $\rho_k(x_k| x_1\ldots,x_{k-1})$ is a conditional density given by
\begin{equation}
\rho_k(x_k| x_1\ldots,x_{k-1}) = \frac{p_k(x_1,\ldots,x_k)}{p_{k-1}(x_1\ldots,x_{k-1})},
\label{eq:rho_k}
\end{equation}
in terms of the marginal densities,
\begin{equation}
 p_k = \int \rho(x_1, \ldots,x_d)\, \mathrm{d} x_{k+1} \cdots \mathrm{d} x_d,
 \label{eq:mdf}
\end{equation}
where $k=1,\ldots,d$.

Let $z=(z_1,\ldots,z_d)=R(x_1,\ldots,x_d)$ where $R$ is the Rosenblatt transformation~\cite{rosenblatt-1952} from the state-space $X \subseteq\mathbb{R}^d$ of $\rho$ to the $d$-dimensional unit cube $[0,1]^d$, defined in terms of the (cumulative) distribution function $F$ by
\begin{align*}
 z_1 &=  F_1(x_1) = \int_{-\infty}^{x_1}\rho_1(x_1')\, \mathrm{d} x_1' ,\\
 z_2 &=  F_2(x_2|x_1) = \int_{-\infty}^{x_2}\rho_2(x_2'|x_1)\, \mathrm{d} x_2' ,\\
     &\vdots \\
 z_d &=  F_d(x_2|x_1,\ldots,x_{d-1}) = \int_{-\infty}^{x_d}\rho_d(x_d'|x_1,\ldots,x_{d-1})\, \mathrm{d} x_d' .
\end{align*}

As noted in~\cite{rosenblatt-1952}, there are $d!$ transformations of this type, corresponding to the $d!$ ways of ordering the coordinates. Further multiplicity is introduced by considering coordinate transformations, such as rotations.

Notice that in $1$-dimension, the Rosenblatt transformation $R(x)$ is simply the CDF $F(x)$.

It follows that if $x\sim \rho$ then $z=R(x)\sim \operatorname{Unif}([0,1]^d)$, i.e., $z$ is uniformly distributed on the $d$-dimensional unit cube~\cite{rosenblatt-1952}. When $\rho(x)>0$, $\forall x\in X$ the distribution functions are strictly monotonic increasing and the inverse of the Rosenblatt transformation $R^{-1}$ is well defined, otherwise let $R^{-1}$ denote the generalized inverse as in Sec.~\ref{sec:simplest}. Then, if $z\sim \operatorname{Unif}([0,1]^d)$ it follows that $x= R^{-1}(z)\sim \rho$, i.e., $x$ is distributed as the desired target distribution $\rho$~\cite{johnson1987multivariate}; this is the basis of the \emph{conditional distribution method} for generating multi-variate random variables, that generalizes the inverse cumulative transformation method for univariate distributions~\cite{devroye-rvgen-1986,johnson1987multivariate,hormann-rvgen-2004,dolgov2020approximation}. These results may also be established by substituting $R$ or $R^{-1}$ into the the FP equation~\eqref{eq:FP}, noting that there is a single inverse image and that the Jacobian determinant of $R$ equals the target PDF $\rho(x)$.

In the remainder of this paper, we refer to \emph{any} map $R$ satisfying $x\sim \rho \Rightarrow R(x)\sim \operatorname{Unif}([0,1]^d)$ as a Rosenblatt transformation, with (generalized) inverse as defined above.

\subsection{Factorization Theorem}
The following theorem characterizes solutions to the IFPP.
\begin{theorem}
\label{theo:fact}
 Given a probability distribution $\rho$ in $d$-dimensions, a map  $M(x)$ is a solution of the IFPP, that is, $M(x)$ satisfies the FP equation~\eqref{eq:FPinvar}, if and only if
 \begin{equation}
   M(x) = (R^{-1} \circ U \circ R)(x),
   \label{eq:fact}
 \end{equation}
 where $R$ is a Rosenblatt transformation and $U$ is a `uniform map' on the unit $d$-dimensional hypercube, i.e. a map that has $\operatorname{Unif}([0,1]^d)$ as invariant distribution~\footnote{When $U$ satisfies the stronger condition that $\operatorname{Unif}([0,1]^d)$ is the equilibrium distribution, $U$ is called an exact map~\cite{LasotaMackey}.}.
\end{theorem}
\noindent \textbf{Proof:}
We will show that $\rho$ is an invariant distribution of $M$ iff $M$ has the form~\eqref{eq:fact}. ($\Rightarrow$) Assume $M$ has the form~\eqref{eq:fact}. If $x\sim\rho$ then $R(x)\sim \operatorname{Unif}([0,1]^d)$ hence $U(R(x))\sim \operatorname{Unif}([0,1]^d)$, as $\operatorname{Unif}([0,1]^d)$ in invariant under $U$, hence  $M(x)=R^{-1}(U(R(x)))\sim\rho$, as desired. ($\Leftarrow$) If $\rho$ is invariant under $M$ then
\( U = R \circ M \circ R^{-1} \)
is a uniform map, since if $z\sim \operatorname{Unif}([0,1]^d)$ then $R^{-1}(z)\sim\rho$, $M(R^{-1}(z))\sim\rho$, and $R(M(R^{-1}(z)))\sim\operatorname{Unif}([0,1]^d)$. Inserting this $U$ into~\eqref{eq:fact} gives the desired factorization.
\quad\qed

The first part of the proof shows that any uniform map $U$ induces a solution to the IFPP, though the particular solution depends on the particular Rosenblatt transformation. The second part of the proof shows that different solutions to the IFPP effectively differ only by the choice of the uniform map $U$, once the Rosenblatt transformation is determined, that is, a coordinate system is chosen with an ordering of those coordinates.

Grossmann and Thomae~\cite{grossmann1977invariant} called dynamical systems $M$ and $U$ related by a formula of the form~\eqref{eq:fact} as `related by conjugation', or just `conjugate', and the map $R^{-1}$ in~\eqref{eq:fact} is a `conjugating function'. Thus, in the language of~\cite{grossmann1977invariant}, Theorem~\ref{theo:fact} shows that the IFPP for any distribution $\rho$ has a solution (actually, it shows that there are infinitely many solutions), every solution map is conjugate to a uniform map, and the conjugating function is precisely the inverse Rosenblatt transformation.

Notice that both the translation operator $T^c$ in~\eqref{eq:Tc} and the triangle map in~\eqref{eq:t1} are uniform maps on the unit interval $[0,1]$. Thus, the solutions to the IFPP given in Eqs~\eqref{eq:shift} and \eqref{eq:sym} are examples of the general solution form in~\eqref{eq:fact}. In particular, while the solution to the IFPP in~\eqref{eq:sym} was derived assuming symmetry of the target density $\rho(\cdot)$, \eqref{eq:sym} actually gives a solution of the IFPP for \emph{any} density $\rho(\cdot)$. Unimodal maps of this form were derived in~\cite{ershov1988solution}.

Computed examples of solutions to the IFPP given by the factorization~\eqref{eq:fact} are presented in Sec.~\ref{sec:onedim}, in one dimension, and in Sec~\ref{sec:twodim} in two dimensions. High dimensional calculations are discussed in Sec.~\ref{sec:sad}.

\subsection{Properties of $M$ from $U$}
\label{sec:prop}
Many properties of the map $M$ are inherited from the uniform map $U$.

When $R$ and $R^{-1}$ are continuous, $M$ is continuous iff $U$ is continuous. In one dimension, monotonicity of the CDF and IDF imply that the number of modes of $U$ equals the number of modes of $M$; in particular, $M$ is unimodal iff $U$ is unimodal.

Constructing iterative maps with specific periodicity of the orbit is possible through the use of translation operators $T^c$ as uniform maps, defined in Eq.~\eqref{eq:Tc}. Consider, first, maps in one dimension on $[0,1]$.
If the shift $c\neq0$ and $c\notin\mathbb{Q}$, the map is aperiodic. However, in the case that $c\neq0$ and $c\in\mathbb{Q}$ such that
 \begin{equation} c = \frac{N}{D} \end{equation}
with $N,D\in\mathbb{N}$ and $\gcd(N,D)=1$, then the map is periodic with periodicity $D$, and iterative maps constructed with $U=T^c$ exhibit the same periodicity. These properties may be extended to multi-dimensional settings when the translation constant $c$ is a vector of shifts in each coordinate direction, as used in rank-one lattice rules for quasi-Monte Carlo integration~\cite{dick2013high}.

The factorization in Theorem~\ref{theo:fact} also shows that performing an iteration $x_{n+1}=M(x_n)$ with an iterative map $M$ on the space $X$  is equivalent to applying the corresponding uniform map $z_{n+1}=U(z_n)$ on the space $[0,1]^d$ through the transformations $R$ and $R^{-1}$, as indicated in the following (commuting) diagram.
\[
 \begin{tikzcd}
   x_n \arrow[r, "\displaystyle R" ']  \arrow[d, left, "\displaystyle M" ']  & z_n \arrow[d, "\displaystyle U" ] \arrow[l, shift right=1.5ex, "\displaystyle R^{-1}" '] \\
   x_{n+1} \arrow[r,  shift right=1.5ex, "\displaystyle R" ']  & z_{n+1} \arrow[l, "\displaystyle R^{-1}" ']
 \end{tikzcd}
\]
Using this commuting property, it is straightforward to prove the following lemma.
\begin{lemma} \label{lem:commute}
 For given distribution $\rho$, let $R$ be a Rosenblatt transformation for $\rho$.
 Let $M=R^{-1}\circ U\circ R$ be a solution of the IFFP, as guaranteed by Theorem~\ref{theo:fact}, where $U$ is a uniform map.
 Then
 \begin{equation}
  \mathcal{O}^+_M(x_0)= R^{-1}\left(\mathcal{O}^+_U(R(x_0)) \right).
  \label{eq:commute}
 \end{equation}
\end{lemma}
Hence, instead of iterating $M$ on the space $X$ to produce the sequence $\{x_1,x_2,x_3,\ldots\}$, Eq.~\eqref{eq:commute} shows that one can iterate the map $U$ on the space  $[0,1]^d$ to produce the sequence $\{z_1,z_2,z_3,\ldots\}$ and then evaluate $x_n=R^{-1}z_n$, $n=1,2,\ldots$, to produce exactly the same sequence on $X$. Since the map $M$ is mixing or ergodic iff the uniform map $U$ is mixing or ergodic, respectively, in this sense mixing and ergodicity of $M$ is inherited from $U$.

Using the expansion in Eq.~\eqref{eq:commute}, we see that $M$ is deterministic or stochastic iff $U$ is deterministic or stochastic, respectively. Even though we are not considering stochastic maps here, we note that, for stochastic maps, iterations of $M$ are correlated or independent iff iterations of $U$ are correlated or independent, respectively.

Some other properties that are, and are not, preserved by the transformation from $U$ to $M$ are discussed in~\cite{grossmann1977invariant}.

\section{Examples in One Dimension}
\label{sec:onedim}

\subsection{Uniform Maps on $[0,1]$}
We have already encountered three uniform maps on the interval $[0,1]$, namely the
identity map $I(x)=x$ (Figure~\ref{fig:uniform} (top, left)) and the translation operator~\eqref{eq:Tc} (Figure~\ref{fig:Tct1} (left)), that have Lyapunov exponent $h=0$, and the triangle map (Figure~\ref{fig:Tct1} (right)) with Lyapunov exponent $h=\log 2$.

Some further elementary uniform maps on $[0,1]$, and associated Lyapunov exponents, are:
\begin{itemize}
 \item $\ell$ periods of a sawtooth function  on $[0,1]$ (Figure~\ref{fig:uniform}, top-right, for $l=3$ periods)~\footnote{The two period sawtooth map $s_2$ is also called the Bernoulli map, and its orbit $\mathcal{O}^+(x)$ is the dyadic transformation.}
       \begin{equation} s_\ell(x) = \ell*x-\lfloor \ell*x\rfloor ,\label{eq:saw}\end{equation}
       with Lyapunov exponent $h=\log\ell$,
 \item $\ell$ periods of a triangle function on $[0,1]$ (Figure~\ref{fig:uniform} (bottom, left) for $l=3$ periods)~\footnote{This is the `broken linear transformation' in~\cite{grossmann1977invariant} of order $p=2\ell$.}
 \begin{equation} t_\ell(x) = 1-2\left|s_\ell(x)-1/2\right| .\label{eq:tri}\end{equation}
 with Lyapunov exponent $h=\log 2\ell$,
 \item and the asymmetric triangle, for $c\in (0,1)$ (Figure~\ref{fig:uniform} (bottom, right) for $c=0.3$)
 \begin{equation} t^c(x) = \begin{cases}
                    \displaystyle    \frac{x}{c}         & 0\le x \le c \\[1em]
                    \displaystyle    \frac{1-x}{1-c}     & c \le x \le 1
              \end{cases}
 \label{eq:asym}\end{equation}
 with Lyapunov exponent $0\leq h= -c\log c - (1-c)\log(1-c) \leq \log 2$.
\end{itemize}
\begin{figure} \centering
     \includegraphics[width=0.49\linewidth]{I.png}
     \includegraphics[width=0.49\linewidth]{sl.png}
     \includegraphics[width=0.49\linewidth]{tl.png}
     \includegraphics[width=0.49\linewidth]{tc.png}
     \caption{Four examples of uniform maps on $[0,1]$. Top-left: identity map. Top-right: $\ell$ periods of a sawtooth wave for $\ell=3$. Bottom-left: $\ell$ periods of a triangle wave for $\ell=3$. Bottom-right: asymmetric triangle for $c=0.3$.}
     \label{fig:uniform}
\end{figure}

Obviously, many more uniform maps are possible. Further examples can be formed by partitioning the domain and range of any uniform map, and then permuting the subintervals. Many existing `matrix-based' methods for constructing solutions to the IFPP, can be viewed as examples of such a partition-and-permute of an elementary uniform map~\cite{rogers2004synthesizing}. Uniform maps of other forms are developed in~\cite{huang2005characterizing} from two-segmental Lebesgue processes, producing uniform maps that are curiously non-linear.

\begin{lemma}
 \label{lem:comp}
 The composition of uniform maps is also a uniform map, i.e., if $U_1$ and $U_2$ are uniform maps then so is $U = U_1 \circ U_2$.
\end{lemma}

An example is $t_\ell$, that can be constructed as the composition $t_\ell = s_\ell \circ t_1$.

We mentioned the numerical artifacts that can occur with finite-precision arithmetic, particularly when implementing maps on a binary computer and when the endpoints of the interval $X$  and constants in the maps have exact binary representations. We avoided these artifacts by composing the stated uniform map with the translation $T^c$ for $c=1/3\times10^{-9}$ that does not have finite binary representation~\footnote{Computation was performed in MatLab that implements IEEE Standard 754 for double-precision binary floating-point format.}. This small shift is indiscernible in the graphs of the maps.

\subsection{Ramp Distribution}
\label{sec:ramp}
To give a concrete example of the solution in~\eqref{eq:sym} for a distribution without reflexive symmetry, we consider the ramp distribution with PDF
\begin{equation}
 \rho_\text{ramp}(x) = 2x
\end{equation}
that has CDF
\[ F_\text{ramp}(x) = x^2.
\]
We produce a unimodal, continuous map by choosing the uniform map $t_1$, as in~\eqref{eq:sym}, to give
\begin{equation}
  \label{eq:Mramp}
  M_\text{ramp} =
  \begin{cases}
    \sqrt{2}x                  & 0\le x \le \frac{1}{\sqrt{2}} \\
    \sqrt{2}\sqrt{1-x^2}       & \frac{1}{\sqrt{2}} \le x \le 1
  \end{cases}
\end{equation}
shown in Figure~\ref{fig:Mramp} (left). Figure~\ref{fig:Mtent} (right) shows a normalized histogram of $10^6$ iterates of $M_\text{ramp}$ starting at $x=0.3$, confirming that the orbit of $M_\text{ramp}$ converges to the desired ramp distribution, as guaranteed by Theorem~\ref{theo:fact}. The numerical implementation avoids finite-precision effects, as discussed earlier.
\begin{figure} \centering
     \includegraphics[width=0.49\linewidth]{Mramp.png}
     \includegraphics[width=0.49\linewidth]{MrampOrbit.png}
     \caption{Iterative map $M_\text{ramp}$ in~\eqref{eq:Mramp} (left) and a normalized histogram of $1\times 10^6$ iterates  that approximates the equilibrium PDF (right).}
     \label{fig:Mramp}
\end{figure}
The estimated Lyapunov exponent for this map is $h\approx 1.040035$, which is greater than the Lyapunov exponent for the inducing triangular map $t_1$ which is $\log 2\approx  0.693147$.

Using a different uniform map gives a different solution to the IFPP. For example, choosing $s_3$ gives the map $M=F^{-1}\circ s_3\circ F$ shown in Figure~\ref{fig:MrampS} (left). A normalized histogram over an orbit of $10^6$ iterations is shown in Figure~\ref{fig:MrampS} (right), confirming that this map is also ergodic for $\rho_\text{ramp}$.
\begin{figure} \centering
     \includegraphics[width=0.49\linewidth]{MrampS.png}
     \includegraphics[width=0.49\linewidth]{MrampSOrbit.png}
     \caption{Iterative map $M_\text{ramp}$ in~\eqref{eq:Mramp} (left) and a normalized histogram of $1\times 10^6$ iterates  that approximates the equilibrium PDF (right).}
     \label{fig:MrampS}
\end{figure}
The estimated Lyapunov exponent for this map is $h\approx 1.098612$, which is the same numerical value as the Lyapunov exponent for the sawtooth map $s_3$  which is $\log 3\approx  1.098612$. Comparing with the result for the map induced by $t_1$, this shows that the Lyapunov exponent of a map is not generally equal to the Lyapunov exponent of the inducing uniform map.

\subsection{The Logistic Map and Alternatives}
\label{sec:altlog}
The logistic map, mentioned in the introduction, is
\begin{equation}
  M_\text{log}(x) = 4x(1-x).
  \label{eq:logisticmap}
\end{equation}
The equilibrium distribution of this map
\begin{equation} \rho_\text{log}(x)=\frac{1}{\pi\sqrt{x(1-x)}}, \label{eq:logisticdist} \end{equation}
can be easily verified by substituting into the FP equation~\eqref{eq:FP1dim}. The CDF of $\rho_\text{log}(x)$ is
\begin{equation}
 F_\text{log}(x) = \int_0^x \rho_\text{log}(x') dx' = \frac{2}{\pi}\arcsin(\sqrt{x}),
 \label{eq:logCDF}
\end{equation}
and the IDF is
\begin{equation}
  F_\text{log}^{-1}(x) = \sin^2(\frac{\pi x}{2}) = \frac{1}{2}(1-\cos(\pi x)).
  \label{eq:logIDF}
\end{equation}

The logistic map~\eqref{eq:logisticmap} is induced by the factorization~\eqref{eq:fact} by choosing the triangle map $t_1$ as uniform map, i.e., substituting the CDF~\eqref{eq:logCDF} and IDF~\eqref{eq:logIDF} into~\eqref{eq:sym}; see Figure~\ref{fig:logmaps} (left). Equivalently, one may note that the logistic map~\eqref{eq:logisticmap} is transformed into the triangle map $t_1$ by the change of variables $z=F^{-1}(x)$; in the language of~\cite{grossmann1977invariant}, $M_\text{log}$ and $t_1$ are \emph{conjugate} dynamical laws.

Other iterative maps which preserves the same equilibrium distribution~\eqref{eq:logisticdist} can be constructed by choosing another uniform map, such as $\ell$ periods of a triangle function~\eqref{eq:tri}. This gives the iterative maps
\begin{equation}
 M_\ell=F_\text{log}^{-1}\circ t_\ell \circ F_\text{log} = \sin^2(2\ell \arcsin(\sqrt{x})), \ell\geq 1,
 \label{eq:logalt}
\end{equation}
that coincide with the $n^\text{th}$ power of the logistic map~\eqref{eq:logisticmap} for $\ell=2^{n-1}$.
Figure~\ref{fig:logmaps} (right) shows the map which preserves the same equilibrium distribution as the logistic map but induced by the uniform map $t_3$. Since $3$ is not of the form $2^{n-1}$, this map is not simply a power of the logistic map.
\begin{figure} \centering
     \includegraphics[width=0.49\linewidth]{Mlog_t_1.png}
     \includegraphics[width=0.49\linewidth]{Mlog_t_3.png}
     \caption{Logistic map, that equals $M_\text{log}=F_\text{log}^{-1}\circ t_1 \circ F_\text{log}$ (left), and the map $M=F_\text{log}^{-1}\circ t_3 \circ F_\text{log}$ that has the same equilibrium distribution (right).}
     \label{fig:logmaps}
\end{figure}

The theoretical value of the Lyapunov exponent of the map in~\eqref{eq:logalt} is $\log 2\ell$, using~\eqref{eq:LyapExpect}.
Table~\ref{tab:1} gives the theoretical values of the Lyapunov exponent and experimentally calculated values using $10000$ iterations, as in~\eqref{eq:Lyap}, for some values of $\ell$.
\begin{table}[h]
\begin{tabular}{|l|l|l|}
\hline
 $\ell$   & $h_\text{e}$ & $h_\text{t}$ \\ \hline
 $1$      & $0.692819$   & $0.693147$   \\ \hline
 $2$      & $1.386284$   & $1.386294$   \\ \hline
 $4$      & $2.079430$   & $2.079442$   \\ \hline
 $2^{24}$ & $17.328594$  & $17.328680$  \\ \hline
 $2^{39}$ & $27.725713$  & $27.725887$  \\ \hline
\end{tabular}
\caption{Experimental $h_\text{e}$ and theoretical $h_\text{t}$ Lyapunov exponents for the maps in~\eqref{fig:logmaps} for a range of $\ell$, given to $6$ decimal places. \label{tab:1}}
\end{table}

\section{Two Examples in Two Dimensions}
\label{sec:twodim}
\subsection{Uniform Maps on $[0,1]^2$}
Two well-known examples of uniform maps in the two-dimensional unit square, $X=[0,1]^2$, are the baker's map
\begin{equation}
  U_\text{b}(x_1,x_2) = \left( 2x_1\mod 1, \frac{x_2+u\left(x_1-\frac{1}{2}\right)}{2} \right) ,
  \label{eq:bakers}
\end{equation}
where $u$ is the unit step function, and the Arnold cat map
\begin{equation}
  U_\text{A}(x_1,x_2) = \left( (2x_1+x_2)\mod 1, (x_1+x_2)\mod 1 \right).
  \label{eq:arnold}
\end{equation}
Other uniform maps in $d>1$ dimensions may be formed by $1$-dimensional uniform maps acting on each coordinate, giving the coordinate-wise uniform map
\begin{equation}
  U(x) = \left(U_1(x_1), U_2(x_2),\ldots, U_n(x_d)\right)
  \label{eq:compunif}
\end{equation}
where $U_i(x)$, $i=1,2,\ldots,d$, are uniform maps in one dimension. We will use the baker's map~\eqref{eq:bakers} and a coordinate-wise uniform map in the $2$-dimensional examples that follow.

\subsection{Checker-Board Distribution}

This example demonstrates construction of a map in two-dimensions that targets the checker board distribution, shown in Figure~\ref{fig:checker_baker} (bottom-left) and Figure~\ref{fig:checker_twintent} (bottom-left), using the factorization~\eqref{eq:fact}.

The first step in constructing a solution to the IFPP for this distribution is to construct the forward and inverse Rosenblatt transformations, which requires the marginal density functions~\eqref{eq:mdf} that may be evaluated analytically in this case. A plot of the two components of the functions $R$ and $R^{-1}$ is shown in Figure~\ref{fig:checker_FI}.
\begin{figure} \centering
     \includegraphics[width=1\linewidth]{E_87_FI.pdf}
     \caption{Plots of the Rosenblatt transformation $R$ and its inverse $R^{-1}$ for the checker-board distribution. Top row shows the components of $R$: first component (left) and second component (right). Bottom row shows the components of $R^{-1}$: first component (left) and second component (right).}
     \label{fig:checker_FI}
\end{figure}

We construct two solutions to the IFPP, each induced by choosing a particular uniform map: the first is the baker's map~\eqref{eq:bakers}; the second is a component-wise uniform map~\eqref{eq:compunif} with an asymmetric triangle map~\eqref{eq:asym} acting on each component,
 \begin{equation} U(x_1,x_2) = \left( U_1(x_1), U_2(x_2) \right) \end{equation}
where $U_1 =t^c$ for $c=0.3$ and $U_2=t^c$ with $c=0.9$.

Figure~\ref{fig:checker_baker} (top row) shows the two components of the map induced by the baker's map, the checker-board distribution (bottom-left), and a histogram of $10^6$ iterates (bottom-right) showing that the map does indeed converge in distribution to the desired distribution.
\begin{figure} \centering
     \includegraphics[width=1\linewidth]{E_85_Checker.pdf}
     \caption{Iterated function constructed with $U$ being the baker's transformation, a histogram of iterates, and the checker board distribution. Shown is: the $x_1$ part of the constructed map (top-left), the $x_2$ part of the constructed map (top-right), the checker board distribution (bottom-left), and a histogram of iterates of the constructed map (bottom-right).}
     \label{fig:checker_baker}
\end{figure}

Figure~\ref{fig:checker_twintent} (top row) shows the two components of the map constructed using the two component-wise asymmetric triangular maps, the checker-board distribution (bottom-left), and a histogram of $10^6$ iterates (bottom-right) showing that the map also converges in distribution to the desired distribution, and hence is also a solution to the IFPP.
\begin{figure} \centering
     \includegraphics[width=1\linewidth]{E_84_Checker.pdf}
     \caption{Same as Figure~\ref{fig:checker_baker} except $M$ is constructed with $U$ being component-wise asymmetric triangular maps.}
     \label{fig:checker_twintent}
\end{figure}

\subsection{A Numerical Construction}
\label{sec:num}
Numerical implementation of the factorized solution~\eqref{eq:fact} is not difficult in few dimensions. In this section we present an example of numerical implementation using a normalized greyscale image of a coin~\footnote{A pre-2006 New Zealand $50$ cent coin.}, piecewise constant over pixels, as the target distribution; see Figure~\ref{fig:coins} (left). The marginal distributions~\eqref{eq:mdf} are evaluated as linear interpolation of cumulative sums over pixel values, and hence the CDF and then forward and inverse Rosenblatt transformations follow as in section~\ref{sec:rosen}. The uniform map was produced as component-wise univariate translation maps, specifically
\[ U(x_1,x_2) = \left( U_1(x_1), U_2(x_2) \right) \]
where $U_1 =T^c$ for $c=0.6$ and $U_2=T^c$ with $c=0.2$.
The resulting map is given by Eq.~\eqref{eq:fact}.
\begin{figure} \centering
   \includegraphics[width=0.4\linewidth]{coinpict.png}
   \hspace*{0.15\linewidth}
   \includegraphics[width=0.4\linewidth]{coinhist.png}
     \caption{A normalized greyscale image of a coin used as the target distribution, and a normalized histogram of iterations of the map targeting this distribution. Original image (left), and normalized histogram of $10^6$ iterations (right).}
     \label{fig:coins}
\end{figure}

Figure~\ref{fig:coins} (right) shows a normalized histogram, binned to pixels, of $10^6$ iterations of this map. As can be seen, the estimated PDF from the orbit of this map does reproduce the image of the coin. However, there are also obvious artifacts near the edge of the image showing that mixing could be better. We conjecture that a chaotic uniform map would produce better mixing and fewer numerical artifacts.

\section{Summary and Discussion}
\label{sec:sad}
We have shown how the solution of the IFPP, of finding an iterative map with a given invariant distribution, can be constructed from uniform maps through the factorization established in Theorem~\ref{theo:fact},
 \[ M = R^{-1} \circ U \circ R \]
where $R$ denotes the Rosenblatt transformation that has Jacobian determinant equal to density function of the invariant distribution. In one-dimension, $R$ is exactly the CDF of the given distribution, so the factorization generalizes existing one-dimensional solutions to the setting of arbitrary multi-variate distributions. The factorization also shows the relationship between arbitrary iterative maps and uniform maps, i.e., given a Rosenblatt transformation the solution of the IFPP is equivalent to the choice of a uniform map that has $\operatorname{Unif}([0,1]^d)$ as invariant distribution.

We find the factorization~\eqref{eq:fact} appealing as it shows that solution of the IFPP for arbitrary distributions, and in multi dimensions, is reduced to two standard and well-studied problems, i.e., constructing the Rosenblatt transformation (or CDF in one dimension) and designing a uniform map. It is therefore surprising, to us, that the factorization~\eqref{eq:fact}, and more generally the Rosenblatt transformation, appears to not be widely used in the study of chaotic iterated functions and the IFPP.  Grossmann and Thomae~\cite{grossmann1977invariant}, in one of the earliest studies of the IFPP, essentially derived the factorization~\eqref{eq:fact} by introducing conjugate maps and establishing the relation (in their notation)  that $\rho^*(x)=\mathrm{d} h^{-1}(x)/\mathrm{d} x$ where $\rho^*$ is the invariant distribution and $h$ is the conjugating function; see~\cite[Figure 3]{grossmann1977invariant}. It is a small step to identify that $h$ is the IDF, generalized in multi-dimensions by the inverse Rosenblatt transformation. However, the connection was not made in~\cite{grossmann1977invariant}, despite the Rosenblatt transformation having been already known, in statistics, for some decades~\cite{rosenblatt-1952}.

We constructed solutions to the IFPP for distributions with special reflexive symmetry structure, and then with no special structure, by constructing the Rosenblatt transformation, and its inverse, for some examples in one and two dimensions. For simple distributions with analytic form the Rosenblatt transformation may be constructed analytically, while numerically-defined distributions required calculation of the marginal distributions~\eqref{eq:mdf} using numerical techniques.

Although this factorization and construction is applicable to high-dimensional problems, the main difficulty is obtaining all necessary marginal densities, which requires the high-dimensional integral over $x_{k+1}\ldots x_d$ in \eqref{eq:mdf}. In general, this calculation can be extremely costly. Even a simple discretization of the PDF $\rho$, or of the argument of the marginal densities~\eqref{eq:mdf}, leads to exponential cost with dimension.

To overcome this cost, Dolgov \emph{et al.}~\cite{dolgov2020approximation} precomputed an approximation of $\rho(x_1,\ldots,x_d)$ in a compressed tensor train representation that allows fast computation of integrals in~\eqref{eq:mdf}, and subsequent simulation of the inverse Rosenblatt transformation $R^{-1}$ from the conditionals in \eqref{eq:mdf}, and showed that computational cost scales \emph{linearly} with dimension $d$. Practical examples presented in~\cite{dolgov2020approximation}, in dimension $d\leq 32$, demonstrate that operation by the forward and inverse Rosenblatt transformations is computationally feasible for multivariate problems with no special structure.

Finding a solution of the IFPP with desired properties is reduced to a standard problem of designing a uniform map on $[0,1]^d$, for which there are many existing efficient options. For example, standard computational uniform random number generators, that produce pseudo-random sequences of numbers, are one such existing uniform map, as are the quasi-Monte Carlo rules that we mentioned earlier~\cite{dick2013high}. These induce pseudo-random and quasi-Monte Carlo sequences, respectively, on the space $X$ via the inverse Rosenblatt transformation $R^{-1}$~\cite{gentle2003random}.  Both these schemes were demonstrated in practical high-dimensional settings in~\cite{dolgov2020approximation}.

The RHS of Eq.~\eqref{eq:commute} in $d=1$ dimension is exactly the standard computational route for implementing inverse cumulative transformation sampling from $\rho$, since computational uniform pseudo-random number generators perform a deterministic iteration on $[0,1]$ to implement a uniform map~\cite{gentle2003random}. For $d>1$, the RHS of Eq.~\eqref{eq:commute} is the conditional distribution method that  generalizes the inverse cumulative transformation method, as mentioned above~\cite{devroye-rvgen-1986,johnson1987multivariate,hormann-rvgen-2004,dolgov2020approximation}. Hence Lemma~\ref{lem:commute} shows that standard computational implementation of both the inverse cumulative transformation method in $d=1$ and the conditional distribution method in $d>1$ is equivalent to implementing a solution to the IFFP. In this sense, computational inverse cumulative transformation sampling from $\rho$ can be viewed as the prototype for all iterative maps that target the distribution $\rho$, with each ergodic sequence corresponding to a particular choice of uniform map.

We mentioned that the Rosenblatt transformation associated with a given distribution $\rho$ is not unique. Actually, any two Rosenblatt transformations for $\rho$ are related by a uniform map, as shown in the following Lemma.
\begin{lemma}
 If $R_1$ is a Rosenblatt transformation for $\rho$ then $R_2$ is a Rosenblatt transformation for $\rho$ if and only if
 \[ R_2 = U\circ R_1\]
 for some uniform map $U$.
\end{lemma}
\noindent \textbf{Proof:}
($\Rightarrow$) Since $R_1$ and $R_2$ are Rosenblatt transformations for $\rho$, then $U=R_2\circ R_1^{-1}$ is a uniform map and $R_2 = U\circ R_1$. ($\Leftarrow$) If $R_2 = U\circ R_1$ then if $x\sim\rho$, $R_1(x)\sim \operatorname{Unif}([0,1]^d)$ and $U\circ R_1(x)\sim \operatorname{Unif}([0,1]^d)$ so $R_2$ is a Rosenblatt transformation.
\quad\qed

Hence, any Rosenblatt transformation $R$ may be written as $R = U\circ R_0$ for some uniform map $U$ and a fixed Rosenblatt transformation $R_0$.

The Rosenblatt transformations that map any distribution to the uniform distribution on the hypercube may also be used to understand mappings between spaces that are designed to transform one distribution to another, such as the `transport maps' developed in~\cite{parno2018transport}. Consider distributions $\rho_A$ and $\rho_B$, with Rosenblatt transformations $R_A$ and $R_B$, respectively, that may be related as in the following diagram.
\[
 \begin{tikzcd}
   \rho_A \arrow[r, shift left=1.0ex, "\displaystyle R_A"]
   & \operatorname{Unif}([0,1]^d) \arrow[l, shift left=1.0ex, "\displaystyle R_A^{-1}"]
   \arrow[r,  shift left=1.0ex, "\displaystyle R_B^{-1}" ]
   \arrow[loop above]{}{\displaystyle U}
   & \rho_B  \arrow[l, shift left=1.0ex, "\displaystyle  R_B" ]
 \end{tikzcd}
\]
The diagram suggests a proof of the following lemma, that generalizes the factorization Theorem~\ref{theo:fact}.
\begin{lemma}
 A map $M$  satisfies $x\sim \rho_A \Rightarrow M(x)\sim\rho_B$ iff it can be written as $M= R_B^{-1}\circ U \circ R_A$, where $R_A$ and $R_B$ are Rosenblatt transformations for $\rho_A$ and $\rho_B$, respectively, and $U$ is a uniform map.
\end{lemma}

Hence, for given Rosenblatt transformations, the choice of a map, that maps samples from $\rho_A$ to samples from $\rho_B$, is equivalent to the choice of a uniform map. Alternatively, if a fixed uniform map is selected, such as the identity map, the choice of map $M$ is completely equivalent to the choice of Rosenblatt transformations. This factorization also shows that the equivalence class of conjugate maps, noted in~\cite{grossmann1977invariant}, for each dimension $d$, is generated by the uniform maps, and each member of the equivalence class contains maps that target each distribution, when the associated Rosenblatt transformation satisfies the mild conditions to be a conjugating function as defined in~\cite{grossmann1977invariant}.

\bibliographystyle{unsrt}

redpajama_set_name: RedPajamaArXiv


text:
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Aug 16, 2022 Argentina soccer player, Best Lionel Messi Quotes, FC Barcelona, Lionel Messi Quotes
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He began his career at Newell's Old Boys in 1995. Later, he joined the Barcelona Under 14s. At each of his advancing ages, he made a significant change. He set a calendar-year goal-scoring record and was named Europe's Ballon d'Or winner for the sixth time in 2019.
We are huge fans of his, and if you see the following quotes by Lionel Messi.
"There's nothing more satisfying than seeing a happy and smiling child. I always help in any way I can, even if it's just by signing an autograph. A child's smile is worth more than all the money in the world."― Lionel Messi
"YOU HAVE TO FIGHT TO REACH YOUR DREAM. YOU HAVE TO SACRIFICE AND WORK HARD FOR IT."― LIONEL MESSI
"I prefer to win titles with the team ahead of individual awards or scoring more goals than anyone else. I'm more worried about being a good person than being the best football player in the world. When all this is over, what are you left with? When I retire, I hope I am remembered for being a decent guy."― Lionel Messi
"IN FOOTBALL AS IN WATCHMAKING, TALENT AND ELEGANCE MEAN NOTHING WITHOUT RIGOUR AND PRECISION."― LIONEL MESSI
"I always thought I wanted to play professionally, and I always knew that to do that I'd have to make a lot of sacrifices. I made sacrifices by leaving Argentina, leaving my family to start a new life. I changed my friends, my people. Everything. But everything I did, I did for football, to achieve my dream."― Lionel Messi
Go Through- Leah Ashe Bio, Age, Net worth, Parents, Height, Ethnicity, Relationship
(Lionel Messi psg) Picture from: Pinterest
"The day you think there are no improvements to be made is a sad one for any player."― Lionel Messi
Top 40 Inspiring Lionel Messi Quotes
"MONEY IS NOT A MOTIVATING FACTOR. MONEY DOESN'T THRILL ME OR MAKE ME PLAY BETTER BECAUSE THERE ARE BENEFITS TO BEING WEALTHY. I'M JUST HAPPY WITH A BALL AT MY FEET. MY MOTIVATION COMES FROM PLAYING THE GAME I LOVE. IF I WASN'T PAID TO BE A PROFESSIONAL FOOTBALLER I WOULD WILLINGLY PLAY FOR NOTHING."― LIONEL MESSI
"It's not so easy for us when we play teams who have a different mindset, like Chelsea or Inter Milan, because they have the intention of trying to stop us rather than playing a game that is more attractive for the spectators to enjoy."― Lionel Messi
"The truth is that I don't have a favorite goal. I remember important goals more than I do favorite goals, like goals in the Champions League where I had the opportunity to have scored in both finals I have played in. Finals in the World Cup or Copa del Rey are the ones that have stayed with me for longer or that I remember more."― Lionel Messi
"EVERY YEAR I TRY TO GROW AS A PLAYER AND NOT GET STUCK IN A RUT. I TRY TO IMPROVE MY GAME IN EVERY WAY POSSIBLE. BUT THAT TRAIT IS NOT SOMETHING I'VE WORKED ON, IT'S PART OF ME."― LIONEL MESSI
"There are more important things in life than winning or losing a game."― Lionel Messi
Go Through- Aaron Doh Bio, Age, Net worth, Parents, Height, Ethnicity, Relationship
"Nah, I've always had a great relationship with my two brothers, I have always had their support in my football and in everything. They've been very close to me and we have a great relationship."― Lionel Messi
"IT'S SCORING GOALS THAT'S GREAT, WHETHER AGAINST BRAZIL OR ANYBODY ELSE."― LIONEL MESSI
"For my part, I try to do my bit to make people's lives more bearable, in particular children across the globe who are having problems."― Lionel Messi
"On a personal note, I think it won't be until after I've retired that I'm fully aware of what I've done or what I've gone on to achieve in my career."― Lionel Messi
(Lionel Messi PSG transfer) Picture from: Pinterest
"I've never stopped being Argentine, and I've never wanted to. I feel very proud of being Argentine, even though I left there. I've been clear about this since I was very young, and I never wanted to change. Barcelona is my home because both the club and the people here have given me everything, but I won't stop being Argentine."― Lionel Messi
"I REPEAT WHAT I ALWAYS SAY: I WANT THE BEST FOR ARGENTINA IN EVERY WAY. I NEVER TRY TO MAKE TROUBLE FOR ANYBODY."― LIONEL MESSI
"I have many years to get better and better, and that has to be my ambition. The day you think there are no improvements to be made is a sad one for any player."― Lionel Messi
"I NEVER THINK ABOUT THE PLAY OR VISUALIZE ANYTHING. I DO WHAT COMES TO ME AT THAT MOMENT. INSTINCT. IT HAS ALWAYS BEEN THAT WAY."― LIONEL MESSI
"The truth is my idea has been to always stay at Barcelona and see out the rest of my career here. Like I always say, one doesn't know what can happen in the future, but if it were up to me to decide, I would stay at Barcelona for the rest of my career."― Lionel Messi
"BEING NAMED AMONG THE BEST AT SOMETHING IS SPECIAL AND BEAUTIFUL. BUT IF THERE ARE NO TITLES, NOTHING IS WON."― LIONEL MESSI
"I am competitive and I feel bad when we lose. You can see it in me when we've lost. I'm in a bad way. I don't like to talk to anyone."― Lionel Messi
"I'm lucky to be part of a team who help to make me look good, and they deserve as much of the credit for my success as I do for the hard work we have all put in on the training ground."― Lionel Messi
"TODAY I AM THE HAPPIEST MAN IN THE WORLD, MY SON WAS BORN AND THANKS TO GOD FOR THIS GIFT."― LIONEL MESSI
"Sometimes you have to accept you can't win all the time."― Lionel Messi
"What I do is play soccer, which is what I like."― Lionel Messi
"YOU CANNOT ALLOW YOUR DESIRE TO BE A WINNER TO BE DIMINISHED BY ACHIEVING SUCCESS BEFORE AND I BELIEVE THERE IS ROOM FOR IMPROVEMENT IN EVERY SPORTSMAN."― LIONEL MESSI
"I have fun like a child in the street. When the day comes when I'm not enjoying it, I will leave football."― Lionel Messi
20 Jesse Owens Quotes that proves his greatness
"You have to show up in the World Cup, and in the World Cup anything can happen."― Lionel Messi
"WE WILL NOT TAKE ARSENAL LIGHTLY FOR ONE MINUTE, AS THEY SHOWED IN THE TWO MATCHES AGAINST US LAST SEASON THAT THEY CAN BE DANGEROUS OPPONENTS IF WE LOSE OUR CONCENTRATION FOR A MOMENT."― LIONEL MESSI
"It seems like people want to blame me for everything. Whenever any issue arises, I'm said to have been involved even if I've had nothing to do with it. That's why I always focus on what I know, which is playing football, and try to be very careful with what I say because people always try and twist things."― Lionel Messi
"When the year starts the objective is to win it all with the team, personal records are secondary."― Lionel Messi
"I WANT TO CONCENTRATE ON WINNING THINGS WITH BARCELONA AND ARGENTINA. THEN IF PEOPLE WANT TO SAY NICE THINGS ABOUT ME WHEN I HAVE RETIRED, GREAT. RIGHT NOW, I NEED TO CONCENTRATE ON BEING PART OF A TEAM – NOT JUST ON ME."― LIONEL MESSI
"I've never stopped being Argentine, and I've never wanted to. I feel very proud of being Argentine, even though I left there. I've been clear about this since I was very young, and I never wanted to change."― Lionel Messi
"Whether it's a friendly match, or for points, or a final, or any game – I play the same. I'm always trying to be my best, first for my team, for myself, for the fans, and to try and win."― Lionel Messi
"Like I've said many times before, I'm always more likely to remember goals for their importance rather than if they're beautiful or not. Goals scored in finals, for example."― Lionel Messi
"I've always really just liked football, and I've always devoted a lot of time to it. When I was a kid, my friends would call me to go out with them, but I would stay home because I had practiced the next day."― Lionel Messi
"I've always really just liked football, and I've always devoted a lot of time to it. When I was a kid, my friends would call me to go out with them, but I would stay home because I had practiced the next day. I like going out, but you have to know when you can and when you can't."― Lionel Messi
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redpajama_set_name: RedPajamaCommonCrawl


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This is the summary of the whole process in Facebook Marketing, according to my mentor, Jomar Hilario. This model is what you call the Infinite Wealth Diamond. The process is interconnected to each other and by using this will maximize the potential of marketing in Facebook.
The whole process is getting prospects. Prospects are people who will buy the products or services in the future. This is sometimes ignored by business people because they think prospects do not have the possibility of being a buyer in the future. Just imagine yourself that you see long line of people waiting to buy the best donuts in town. You happen to see it and you wonder why long line of people waiting to have that donuts. Maybe their donuts taste good so you plan to buy it sometime. It has something to do with marketing strategy on how to capture the prospects because they have already the potential to buy.
To show the process, the first step in Getting Prospects is showing and giving the Promise of the product or services or business.This is an indirect way of saying to prospects to buy the product or service. Did you know that most of the decisions of the buyers are based on emotions? The best way to market is to show the Promise of the product or service. This is to be further discussed later.
The Posts are also called Facebook Ads. Here Promise is shown with image and text supporting the promise, call to action and heading showing again the Promise. Whatever promise shown in Posts must also be the same with the promise in Pages.
Pages are from the website. These consist of Email Catcher page and the Thank You page. The main purpose of these pages is to get email addresses of prospects. Our main goal is to have email addresses list.
Pixels are from Facebook. The main purpose of this is to track what Posts get the most email addresses. The Pixels are pasted to pages of email catcher and thank you page. This is the best way to measure success.
The process of changing posts from time to time is to personalize.


VOD services are provided by CDN edge servers that deliver static pre-encoded video files to client video players. RTMPE and MMS. but workarounds have been introduced. client and network status are utilized in negotiating video traffic between the client and server and also take into account the overall server load and performance. BASIC CDN VIDEO FUNCTIONS CDN video services are divided into two basic types: ―Video On Demand‖ (VOD) and ―live‖. and network components that acts as one large distributed computer. STATEFUL CONNECTIONS Stateful connections are physical socket connections that exist between an encoder and a CDN or a player and a CDN. among others. CDNs Demystified – A Streaming Technology Primer 3 . The network protocols used for these types of connections are usually proprietary and specific to the media server and player used. Read on to gain a working knowledge of how a CDN operates. These can be difficult to emulate in stateless connections. Server. most CDNs provide both analytic and provisioning control panels for video suppliers to configure and monitor their usage. WHAT IS A CDN? A CDN is a system of software. hardware. these processes are transparent to the encoder source and client player. Inc. such as ChinaCache. they are being replaced by stateless based protocols primarily based on HTTP. These protocols utilize sophisticated client and server message stacks to optimize performance and user experience. server co-location and cloud processing. CDNs frequently use specialized edge servers running media server software. They remain connected throughout the duration of an encoder or player viewing session. This brief focuses predominantly on CDN services related to ingest and delivery of digital video content. file storage. become familiar with terminology common to CDNs. to handle these types of connections. or deployed worldwide. and streaming delivery architectures. quicker startup. such as Akamai. In addition to these services. PROTOCOLS AND CONNECTIONS All interactions between video encoders and CDNs or video players and CDNs are defined by network protocols. Examples of these include RTMP. Some of the key benefits of stateful connections include: the ability to track unique user sessions in viewing video content.OVERVIEW The definition of a Content Delivery Network (CDN) is evolving as these providers expand the range of services they deliver and video suppliers they serve. Live services are provided by CDN edge servers that deliver live content—dynamically generated video files or network data streams— that originates from video encoders and is transported to client video players through the backbone network of the CDN. lower latency. Services performed by a CDN may include digital media delivery. In most cases. Network connection protocols fall into two categories: stateful and stateless connections. such as those provided by Elemental. This brief describes the fundamental capabilities and practices of a CDN as they relate to video processing functions. Copyright © 2012 Elemental Technologies. such as Adobe's Media Server or Wowza Media Server. network transit. such as Apple HLS or Microsoft Smooth Streaming. and learn about the interactions that take place between the components that make up a CDN video delivery system. such as HTTP or RTMP. CDNs can be regional. Although these protocols have been used for over a decade. and better error control/handling. website acceleration.
It is the most inefficient method because it does not support multi-bit rate delivery. or a type of manifest/file segment paradigm. but is being replaced by newer methods of delivery described below. which makes it difficult to map specific file requests to individual user sessions for reporting or analytics. The most common implementation of HTTP range request video delivery has been the initial implementation of HTML5 players with WebM content. a browser will download the entire video file regardless of what the user is viewing.   Copyright © 2012 Elemental Technologies. Others include Flash player. many CDNs are optimized for small object delivery of content and now are able to move video around in small segments instead of large files.STATELESS CONNECTIONS Stateless connections use the HTTP protocol to deliver video content in one of three modes: progressive download. This decreases their network load and storage requirements while increasing efficiency and utilizing many of the cache management features in their existing HTTP delivery infrastructure. a video buffer exists in the range of 3-5 seconds. Microsoft Smooth Streaming (MSS). or slightly enhanced HTTP platforms. but differs in that the player makes multiple GET requests for sections of the video via the HTTP 1. Additionally. The player does not require the entire file to be downloaded before displaying video. logic. it is simultaneously requesting the next range of data from within the video file. and Apple HTTP Live Streaming (HLS). Flowplayer. Progressive download is the lowest common denominator in video delivery. The three most common implementations are Adobe HTTP Dynamic Streaming (HDS).0 edge server to service the request. HTTP range request delivery is similar to progressive download. devices and formats are supported and it only requires the CDN to provide the most basic HTTP 1. and in many cases. Inc. Typically. As the player is displaying video from one request. and Microsoft Smooth Streaming. to deliver all of their content instead of relying on a fleet of media servers. there have been issues requiring smart players or smart HTTP servers to handle segment requests that fail to result in leading reference/Iframes. such as Apple Streaming. this type of delivery is very attractive to CDNs since it allows them to use existing generic servers.1 support on it edge servers. such as Netflix. tablets and web platforms. In most cases. This type of delivery is characterized by repetitive HTTP requests or pushes of video data chunks in the form of individual file segments. Adobe HDS. HTTP range request.  Progressive download delivery is characterized by a video player requesting a file and playing it as it downloads. If a player requests a range request of data. That said. Progressive download was the first method used to present video on the internet. Almost all players. CDNs Demystified – A Streaming Technology Primer 4 . primarily due to the explosion of video via mobile devices. Although stateless connections may reuse a connection for each request. Most player scenarios require a manifest file that is used by the player as a map of the video segment files. the encoder simply pushes 2 – 10 second sequential file segments to an HTTP origin server and periodically updates the manifest file. such as IIS. such as Apache. which may include third party licensing fees and usually require specialized resources to configure and maintain. This still works well for short video clips. the files are written to disk and then proceed to be pushed to the CDN for distribution and delivery.1 range request/partial content mechanism. no special functionality is required of the CDN other than basic HTTP 1. These stateless connections have a very simple HTTP protocol in which all request. Most implementations attempt to reuse connections through keep-alive mechanisms to promote for speed and efficiency. If the player can handle these cases. and optimization is performed by the player and there is no evaluation or awareness of server state. it is possible the request will result in a video segment that does not have a leading reference/I-frame. when a request for media file types is made. Some CDNs have enhanced HTTP servers that. this is not guaranteed. In all three of these scenarios. will guarantee that the first frame returned in the video segment begins with a reference/I-frame. In practice. JW Player and several other popular players. usually in response to a seek function. Each connection may or may not use the same physical network socket. Manifest/file segment delivery has become the fastest growing method of delivery.
Pull strategies are characterized by the CDN pulling a file from a HTTP or FTP server based on a request the CDN received for content. The video supplier is then required to explicitly introduce each piece of content onto the CDN before it can be viewed or downloaded.  PHYSICAL TOPOLOGY OF A CDN A CDN consists of two tangible pieces: Points of Presence (PoP) and the Backbone Network (BBN). Most traditional CDNs were built for hardwired delivery of content via ISPs. the CDN will retrieve (pull) the content from the video supplier's origin. In many cases. Copyright © 2012 Elemental Technologies. CDN PoPs are co-located in MSO/ISP back office installations. managed and monitored network resources. Most of the services that video suppliers use are hosted on servers that reside within PoPs. In a pull scenario. the CDN simultaneously proxies the request back to the requesting client and propagates the content into the CDN's content management system. such as Akamai. a specific predefined origin must be set up in the video supplier's CDN profile before a request is made. CDNs operate under the strategy of owning PoPs in areas of dense population and close geographical proximity to the large number of viewers that are accessing content from video suppliers. Large CDNs.000 CDN places in a physical location. Not all PoPs contain all types of servers or serve 100 PoPs worldwide. the same purposes. Recently. Typically. the proximity of these PoPs to the viewers making content request. When the CDN receives a request for content that it does not have in its inventory. while in other cases. they may be shared peering arrangements. whereby they are not dedicated to CDN traffic but are also not used as part of the public internet. have hundreds of PoPs worldwide. the BBN's connection of the PoPs and the efficiency. as opposed to relying on the public internet. The reason CDNs use a dedicated backbone is to provide transfers between PoPs using high speed. This mechanism may be as simple as an FTP client pushing to a CDN FTP server. CDNs have re-architected for delivery via mobile wireless network providers. The BBN connects the PoPs together. CDNs Demystified – A Streaming Technology Primer 5 . A CDN's performance is dependent on a number of factors: the number of PoPs they own. In some cases. content from the video supplier is physically moved and prepositioned onto the CDN. A non-CDN company may own dedicated links between cities that are not fully subscribed. but in many cases the push mechanism is located on a CDN customer control panel. reliability and performance of the CDN software that is used to deliver and manage content. The file and data transfer that occurs between PoPs usually transits over the BBN. A Backbone Network (BBN) is the system that CDNs use to connect PoPs. dedicated. The server types and roles are described below in the ―Elements of a CDN‖ servers in more than section. and may peer or lease access to these pipes. Frequently. The main value that CDNs provide is placing content in close proximity to the viewer requesting the content and reliable high-speed transfer of data between the PoPs.  Push strategies are characterized by a computer outside the CDN pushing content to the CDN. Inc. as long as an origin has been defined. however. Points of Presence (PoP) are a collection of servers that a Akamai has 84. these are dedicated private data pipes between PoPs.CDN CONTENT ONRAMP STRATEGIES There are two main strategies that are used to populate a CDN with customer content: push and pull. In this scenario.
though a physical server may serve multiple purposes.  Copyright © 2012 Elemental Technologies. it is stateful in that it uses one long running post request. In the case of live content.  Origin (Ingest) servers are the entry point to place content onto the CDN. or. FMS or Wowza. Inc. Edge Servers deliver content to the viewer's client players. or possibly within their entire network. authentication credentials and encryption information is usually provided to the customer by the CDN. The purpose of these servers is to provide a fanned-out distribution of content from a single point or points within a PoP/CDN to all of the edge servers within that PoP.  Distribution Servers tend to be deployed within each geographical Point of Presence (PoP). and not necessarily an entire file. As previously stated. these are typically FTP sites. Below are some of the common roles and their functional descriptions. In addition to providing live and VOD content. there are usually publishing points defined in the media server. For VOD or HLS content. CDNs Demystified – A Streaming Technology Primer 6 . an HTTP web server. For stateful protocols such as RTMP or RTSP. logging. There are usually multiple distribution servers to provide failover. such as IIS. This caching is frequently chunks or segments of individual files. but in general the last/least requested cache object is the first to be removed.ELEMENTS OF A CDN Several elements comprise a CDN and they are effectively described by the roles they play in delivering video content. a CDN will have tens or hundreds of edge servers in a PoP. In the case of Microsoft Smooth Streaming. Typically. Some CDNs use multicast network delivery as a distribution system within a PoP. The publishing point. they offer client authentication. not all PoPs will play all these roles. access control. these can either be media servers. and many other functions related to the client's interaction with the CDN. This information may also be obtained through the CDN's configuration control panel. HTTP servers or HTTP upload functions from a UI control panel. The cache is tunable. The actual content that is served from an edge server may come from a variety of sources: o Software Cache – Recently requested content is usually cached within the edge delivery software that handles incoming requests. although it uses the stateless HTTP protocol. in the case of HTTP Live Streaming (HLS). Live encoders have to correctly define these entry points before they will be able publish live content to a CDN.
or in remote network storage. These requests should instead be funneled into a single request and proxied out through the edge servers. Origin pull servers are used by CDNs to retrieve content from a video supplier's origin server in the case of a pull request. CDNs Demystified – A Streaming Technology Primer 7 . SAN Network Storage – Many CDNs have a SAN within a PoP that an edge server may use to request content. This type of storage may have an NFS or some other mount point. This is especially true of dedicated HLS edge servers. Network DVR Archive Servers are used to record live content and create an archive VOD representation for later viewing.o Local File System – Many CDNs will store recently requested files on the local disk file system. a CDN will remove content from an edge server if it is inactive. content will be removed after a defined period of time or inactivity. This shields the video supplier's origin server from being overwhelmed by requests. This is where a video supplier will place content and it usually requires authenticated access. if a viewer joins a live event after the broadcast has started. These origins typically have to be defined in the video supplier's CDN control panel so the CDN will know where to find particular content and how to authenticate it. o o  File servers are the near-line storage that CDNs use to hold content. o Video supplier origin servers are usually web or FTP servers outside of the CDN that reside on the video supplier's network and are controlled by the video supplier. Similar to caching. Not all CDNs use local file disk storage. the last/least requested file is usually the first to be removed. CDNs use these as mezzanine storage devices for content supplied by customers. If a request is made for content that has been aged out. A key point here is that a flood of requests for an object not already within the CDN does not—or should not—generate a flood of requests to the video supplier's origin server. Based on the CDN and individual customer policy rules. but use their edge servers in cache/proxy-only roles. they may use HTTP. In many cases. Origin Servers are typically of two types in any CDN implementation: customer origin servers and origin pull servers. NNTP or some other protocol for file request. it will make a request to a CDN origin server that will retrieve the file from the video supplier. In many cases. In many cases. The edge server will simultaneously retrieve the file from the CDN origin server and proxy it back to the requesting client. the CDN software will retrieve it from the file server. on disk. In this case. the origin server will also place the file into the CDN's content management and inventory system. These servers are not usually accessible to customers of the CDN. they are able to rewind to the beginning or any other part of the live event. as long as it has already been delivered by the CDN. and place it back at the edge. A request for content a CDN does not have in storage will result in an origin pull server making a request to a customer origin server to retrieve this content and propagate it throughout the network.   o Copyright © 2012 Elemental Technologies. Inc. Origin Pull – If an edge server receives a request that it does not have in cache. Some systems provide DVR capabilities that allow viewing of the recorded segment before the live event is actually completed. This is usually an HTTP or FTP request.
Because of this.cdn. URL request resolvers accept a request.cdn. or many other rules to lead to a specific edge server. quality of service. CDNs provide critical infrastructure and play a crucial role in the global distribution of video content from encoding sources such as those supplied by Elemental. inventory availability.) These requests may be redirected based on geography. video supplier origins. Copyright © 2012 Elemental Technologies. The request resolver services the request for content as fast as possible by directing the request to an edge server that has the content and is physically closest to the client that is requesting the content. and can go a long way to impact a video supplier's purchasing decision. log reporting definitions.ny1. Configuration Control Panel In the early days of the CDN industry. In an expanding video universe with an ever widening number of playback devices. As CDN software and customer-facing services evolved. cost of service. CDNs Demystified – A Streaming Technology Primer 8 . video suppliers increasingly rely on CDN technologies and/or services to effectively deliver video content to end users. all configuration and management was accomplished by back office service request systems similar to a phone company's service provisioning system. such as videosupplier. CDNs provided control panels for customer access and account management. configurable guidelines for bandwidth and usage. content cache control. and many other attributes that are used to define a CDN's account and content profile. content authentication. (Typically. The control panel is the visible user interface for the CDN. These are usually bandwidth and storage statistics that directly relate to billing.edge033/path/content. Inc. this tends to be the ―secret sauce‖ that differentiates CDNs and defines their performance. Video suppliers did not have the ability to directly manage their accounts and all changes were implemented through this service provisioning method.   CONCLUSION With a broad range of capabilities and service offerings. URL Request Resolvers are the front door of the CDN for all requests. Example account settings include live event definitions. which may result in a request to videosupplier. While the concept is relatively straightforward. network routing. CDN load.com/path/content. Analytics/Billing Control Panel Most CDNs today provide video suppliers with raw logging data and analytics to access and review usage metrics. there are many patents and lawsuits concerning request resolution.


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Kelly Wiley, I-TEAM reporter
Erik Avanier, Reporter
Tags: News, Jacksonville
Woman, boyfriend subject of police search found safe in Volusia County
Jacksonville police had suggested woman was being held against her will
JACKSONVILLE, Fla. – Two days after police asked for help Wednesday night locating a Tennessee woman who they said might be in danger, the Jacksonville Sheriff's Office announced she and the man she was with were found.
Police had said the woman might have been held against her will by her boyfriend after a reported carjacking. They released photos of the two along with a car with Tennessee license plates in which they might be traveling.
Based on initial information, the two might have been staying in the Jacksonville Beach area. A viewer sent News4Jax cellphone video recorded Wednesday in Edgewood that appears to show the vehicle that police were looking for, but that was not confirmed by investigators. The viewer said it appears two people were fighting inside.
According to the police report, the woman's husband got a text message from her phone saying the boyfriend, "has carjacked me with me in the car and is threatening to kill me. We're in Jacksonville and will be heading to Interlachen, from what he says."
"It is a red flag because it prompts them to look into it and see if this is, in fact, a legitimate abduction," News4Jax crime and safety expert Ken Jefferson said while the couple was missing.
On Thursday, police said the couple was seen in Volusia County.
On Friday afternoon, JSO announced they were located in Volusia. There's no word on if there was a carjacking or any other criminal activity involved, but police said the investigation was ongoing.
"Thank you to everyone for helping us get the word out and for the abundance of tips received," the JSO tweeted.
Copyright 2019 by WJXT News4Jax - All rights reserved.
Kelly Wiley
Kelly Wiley, an award-winning investigative reporter, joined the News4Jax I-Team in June 2019.
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Erik Avanier
Award-winning broadcast and multimedia journalist with 20 years experience.


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Q: Earth destroyed, survivor, aliens, reincarnation, classical music I'm looking for an old story, maybe from Analog or Omni (print versions) 25-30 years ago.
The story is about a guy who's stranded orbiting (?) an alien planet after humanity has been wiped out. The aliens are sympathetic to his plight and take him in.
Toward the end of the story, you find out there's a "cult" springing up among the alien youth that human souls from the destroyed Earth are transmigrating to the alien world and reincarnating there. The man also listens to classical music a lot - nothing with human voices though.

A: "Listening to Brahms" by Suzy McKee Charnas, first published in the September 1986 Omni (scanned copy here), reprinted under these covers.
About a guy who's stranded orbiting (?) an alien planet after humanity has been wiped out.
The narrator (and a couple of fellow astronauts) was in cryosleep orbiting Earth when the alien Kondrai came:

I sit looking through the viewplate at Earth, such as it is. I know what the lizards say is true, but I don't think I really believe it. I think mostly that I'm dead or having a terrible dream.

The aliens take them back to their home planet:

Entry 3: Captain Midnight says now that we're all up he would be honored beyond expression if we would consent to come back to Kondra with him and his crew in their ship. Kondra is their name for their world. Chu says she's worked out where and what it is in our terms, and she keeps trying to show me on the star charts. I don't care. I came up here to do studies on cryogenic nutrition in space, not to look at star charts.

Toward the end of the story, you find out there's a "cult" springing up among the alien youth that human souls from the destroyed Earth are transmigrating to the alien world and reincarnating there.

Entry 20: They are saying that the reason there's so much crime and violence on Kondra isn't because of the population explosion at all. Some snake who calls himself Swami Nanda has worked out how the demographic growth is only a sign of the underlying situation.
According to him Kondra made an "astral agreement" to take in not only us living human survivors but the souls of all the dead of Earth. Earth souls on the astral plane seeing that there were soon going to be no more human bodies on Earth to get born into, sent out a call for new bodies and a new world to inhabit. The Kondran souls on the astral plane, having pretty much finished their work on the material world of Kondra, agreed to let human souls take over the physical plant here, as it were. Now the younger generation is all Earth souls reborn as Kondrai on this planet, and they're recreating conditions familiar to them from Earth.

The man also listens to classical music a lot -- nothing with human voices though.

Entry 5: Sibelius, Symphony No. 2 in D, Op. 43; Tchaikovsky, Variations on a Rococo Theme, Op. 33; Rachmaninoff, Symphonic Dances, Op. 45; Mozart, Clarinet Quintet in A Major, K581; Sibelius, Symphony No. 2 in D, Op. 43; Sibelius, Symphony No. 2 in D, Op. 43

redpajama_set_name: RedPajamaStackExchange


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Home » Data Breach, Equifax, Federal Trade Commission, privacy
Equifax to pay $650 million for the 2017 data breach
Aditi Agrawal
By Aditi Agrawal (	@aditi_muses aditi@medianama.com ) July 23, 2019
Equifax, the credit consumer reporting agency, will have to pay at least $650 million to resolve most claims stemming from the 2017 data breach that exposed the personal data of more than 147 million consumers, the New York Times reported. A federal judge in Atlanta approved the settlement on July 22. This is the largest data breach settlement thus far, surpassing the $148 million penalty Uber agreed to last year, BBC reported.
The settlement is the result of investigations by two US federal agencies, Consumer Financial Protection Bureau and Federal Trade Commission, and 48 state attorney generals. Massachusetts and Indiana sued the company separately; their suits haven't been resolved yet.
The company had set aside $690 million last quarter to cover the potential costs. The current settlement figure of $650 million is roughly what the company earns in sales in a quarter. In 2018, the company earned $300 million on sales of $3.4 biliion.
Who is compensated?
Victims who lost money
People who suffered through the hassles of phone bank trees
Credit card customer service lines can bill Equifax $25 for their time
However, it is not clear who will be approved for compensation because their identities were stolen.
Terms of the settlement
According to settlement documents,
Almost half the amount, $300 million, will go to American consumers harmed by the breach; Equifax will add up to $125 million if this fund is depleted
Consumers who can show they suffered direct costs following the breach, will be eligible for restitution, capped at $20,000/person
$175 million in fines to end investigations 48 states, District of Columbia, and Puerto Rico, and $100 million to the CFPB, Reuters reported
Equifax will provide up to 10 years of free credit monitoring services to victims of the breach in the US; it is paying Experian, a competitor, to provide the service for first four years (settlement assumes that only 7 million people will sign up). Every million consumers who opt in will cost Equifax more than $16 million. If all 147 million victims of the breach opt in, it would cost Equifax more than $2 billion
Review of its security policies by a government-appointed third party
Equifax's board must annually certify that it complied with the settlement terms, or be fined
Information for consumers will be posted at equifaxbreachsettlement.com, set up by the group that will handle claims.
What was the Equifax breach?
Equifax is one of the three largest credit bureaus in the US. Experian and TransUnion are the two others. In July 2017, Equifax discovered that attackers had been stealing information from company servers for 76 days. The company disclosed the breach after more than a month of discovery.
The FTC said that Equifax had been warned in March 2017 that one of its databases suffered from a critical vulnerability, BBC reported. The company failed to get the vulnerability patched and as a result, multiple hackers were able to exploit the flaw and steal consumers' personal details.
The stock price of the company tumbled after the breach, but it has since recovered most losses.
The hackers reportedly copied:
At least 147 million names and dates of birth
About 145.5 million Social Security Numbers
209,000 payment card numbers and expiration dates
It has been difficult to quantify the harm caused by the breach to consumers because cybersecurity experts have not seen the stolen data surface in online marketplaces where such stolen information is trafficked.
The UK's Information Commissioner's Office had already fined Equifax £500,000 for failing to protect the personal information of up to 15 million British citizens during the same attack.
Aditi Agrawal	Senior Research Associate
TikTok has removed over 60 lakh videos between July 2018 to April 2019
India had 989.5M active mobile connections, Vodafone Idea and Airtel lost 12.82M active connections in May 2019: TRAI
'Bloatware on Android devices can leave users' data vulnerable': 50+ privacy orgs write to Google
MEITY to brief Joint Committee on Personal Data Protection Bill on January 14
NCRB extends deadline to submit bids for Automated Facial Recognition System, again
Facial recognition based surveillance systems to be installed at 983 railway stations across India
Xiaomi says that 'very few Indians' might have been affected by flaw that allowed users to see strangers' images
More than 2,600 passengers have signed up for getting their faces scanned at Delhi airport
California Consumer Privacy Act has come into effect, starting Jan 1, 2020
Brazil slaps $1.6 million penalty on Facebook in Cambridge Analytica scandal
Key aspects of the Personal Data Protection Bill, 2019