{natural two-sided measures with the initial condition $\bvec p_0$} on the symbolic and phase space dynamics, respectively.
\end{definition}
The natural two-sided Markov measure has already been used implicitly in the literature on so-called \emph{multibaker maps} \cite{Gaspard2005,Vollmer_etal1997,Vollmer_etal1998,Breymann_etal1998,Tel+Vollmer2000,Vollmer2002,Colangeli_etal2011}.
In Chapter~\ref{chap:information-st} we discuss the role of the two-sided natural measure for \emph{network multibaker maps}, which are the most general variant of multibaker maps.
Additionally, we discuss the (stochastic) thermodynamic properties of and the notion of entropy for such maps in a general information-theoretic framework.

\section{Discussion}
\label{sec:mms-discussion}
After the technical considerations above, we put our findings in the context of previous mathematical and physical results.
Firstly, we revisit the concept of natural measures $\pms_\Phi$, which we used to define the natural transition matrix.
In particular, we discuss their relation to the natural stationary Markov measure $\fmms\tind{\infty}(\ptmat,\bvec p\tind{\infty})$.

After that, we relate our abstract results to the \emph{operational} framework outlined at the beginning of Section~\ref{sec:symbolic-stochastic-dynamics}.
There, we see how our abstract mathematical results obtain natural physical interpretations.


\subsection{Connection to ergodic theory}
\label{sec:discussion-math}
An ergodic natural measure with full support on the whole phase space is the attractor of measures that are absolutely continuous with respect to the Lebesgue measure.
Then, any probability measure with a density will converge to the natural measure.
Further, the natural transition matrix is well-defined, \textit{cf.\,} Def.~\ref{def:natural-tmat}.
Note that generically the natural measure is \emph{not} absolutely continuous.

For the remainder of this subsection, we stick with the above assumptions.
Then, the Markov chain defined by the natural transition matrix is both ergodic and aperiodic with unique stationary distribution $\bvec p\tind{\infty}$.
Hence, any extensions of a one-sided natural Markov measure eventually converges to the unique stationary Markov measure $\fmms\tind{\infty}(\tmat):=\fmms\tind{\infty}(\tmat,\bvec p\tind{\infty})$.

Let us now assume that the partition $\parti$, which we use to define the natural transition matrix, generates the Borel $\sigma$-algebra.
Further, let $\mms$ be an extension of a (non-stationary) natural one-sided Markov measure $\mms(\ptmat,\bvec p)$.
In that case, $\mms^\pspace:= \mms\circ\mtraj$ is a Borel measure on phase space.
Two questions naturally arise:
\begin{itemize}
  \item What is the relation of the natural measure $\pms_\Phi$ and
    \begin{align*}
      \mms^\pspace\tind{\infty} := \lim_{t\to\infty} (\Phi^t)_*\mms^\pspace,
    \end{align*}
    given the latter exists?
  \item Under which conditions is a non-stationary measure $\mms^\pspace$ absolutely continuous with respect to the Lebesgue measure?
\end{itemize}

Let us first assume a positive answer to the second question.
In that case, $\mms^\pspace$ necessarily converges to the natural measure under push-forward by $\Phi$ and we have $\mms^\pspace\tind{\infty}=\pms_\Phi$.
Moreover, $\mms$ converges to the unique natural stationary measure $\fmms\tind{\infty}(\ptmat)$ defined on the whole shift, \textit{i.e.\,} for $\timdom =\integers$.
Because $\mtraj \circ \Phi = \shiftmap \circ \mtraj$, we find that the natural stationary measure on phase space actually \emph{agrees} with the natural measure, \textit{i.e.\,}
\begin{align}
  \fmms^\pspace\tind{\infty}(\ptmat) := \fmms\tind{\infty}(\ptmat)\circ\mtraj =  \mms\tind{\infty}^\pspace = \pms_\Phi. 
  \label{eq:natural-measure-agrees}
\end{align}

However, this fact crucially depends on the fact that $\mms^\pspace$ was absolutely continuous in the first place.
Remember that we demanded that $\mms^\pspace$ is a pull-back to phase space of an (arbitrary) extension of a natural one-sided measure $\fmms(\ptmat)$.
The latter is defined on the symbolic $\sigma$-algebra $\salg^\naturals$ for the forward sequences.
This $\sigma$-algebra uniquely defines the restricted $\sigma$-algebra $\palg^\naturals = \mtraj^{-1}(\salg^\naturals)$.

The most refined sets in $\palg^{\naturals}$ are the pre-images of semi-infinite sequences $\traj\omega \in \alphabet^\naturals$ under $\mtraj$.
That is, they are sets containing points with the same \emph{symbolic future}.
Such sets lie on the same (local) stable manifold.
Specifying a measure on $\palg^\naturals$ can thus be understood at defining a \emph{marginalized} measure, where the local stable manifold is integrated out.
What remains is the topological structure that governs the \emph{future} of points, which are also known as the (local) unstable manifolds.
One also says that a measure on $\palg^\naturals$ defines a \emph{conditional measure on the unstable manifolds} \cite{Young2002}.

The stable and unstable manifolds always intersect transversally.
In a sense, they define a curvilinear transversal coordinate system \cite{Adler1998}.
Extending a one-sided Markov measure to the bi-infinite sequences thus means specifying a structure along the stable directions.
Or to put it differently:
Specifying the \emph{past} of a system defines a density along the stable directions, specifying the \emph{future} does the same along the unstable directions.

Let us now choose the measure $\mms$ as an extension of $\fmms(\ptmat)$ such that along the stable direction we have a density with respect to the (restricted) Borel measure.
Then, the question whether $\mms$ converges to the natural measure can be reformulated:
Is the measure $\fmms^\pspace(\ptmat)$ (restricted to $\palg^\naturals$) absolutely continuous with respect to $\lms$ (restricted to $\palg^\naturals$)?
Or equivalently:
Does the measure $\mms^\pspace$, constructed as an extension of a \emph{natural} one-sided Markov measure, have absolutely continuous conditional measures on the unstable manifolds?

A positive answer to that question for \emph{invariant} measures defines (a generalization) of a so-called SRB measure \cite{Young2002}.
The latter were discovered by Sinai, Ruelle and Bowen in 1970s in the context of Anosov systems and Axiom-A flows \cite{Sinai1972,Bowen+Ruelle1975}.
The natural measure has been proposed to be the proper generalization of the SRB measure in a general context \cite{Young2002,Blank+Bunimovich2003}.

These facts suggest that it is useful to think of natural measures as stationary Markov measures defined for a generating Markov partition.
In fact, the SRB measure can be defined using generating Markov partitions for hyperbolic systems~\cite{Sinai1972,Gallavotti+Cohen1995}.
Hence, it seems natural to ask whether under the present assumptions, the invariant natural measure on phase space $\mms\tind{\infty}$ \emph{agrees} with the natural measure $\pms_\Phi$.

What we know is that both measures (by definition) agree on sets of the form $\pcyl\tind{t}[(\omega,\omega')] \equiv \pcyl\tind{t}[\omega] \cap \pcyl\tind{t+1}[(\omega')]$.
However, as the family of sets
\begin{align*}
  \set{\pcyl\tind{t}[(\omega,\omega')] \,\middle\vert\,t \in \integers,\, \omega,\omega'\in\alphabet}
\end{align*}
is not closed under intersections, this does not imply that the measures are the same, \textit{cf.\,} Lemma~\ref{theo:pi-system}.

Further, it is possible to construct ergodic invariant measures with full support on $\pspace$ which are not (pull-backs) of Markov measures.
The easiest example is provided by considering the pull-back of a stationary $m$-step Markov measure \cite{Bowen+Chazottes1975}.
The latter is a Markov measure constructed for an alphabet $\al