Our goal is to present the basic results on one-dimensional Gibbs and equilibrium states viewed as special invariant measures on symbolic dynamical systems, and then to describe without technicalities a sample of results they allowed to obtain for ce
rtain differentiable dynamical systems. We hope that this contribution will illustrate the symbiotic relationship between ergodic theory and statistical mechanics, and also information theory.
We obtain moment and Gaussian bounds for general Lipschitz functions evaluated along the sample path of a Markov chain. We treat Markov chains on general (possibly unbounded) state spaces via a coupling method. If the first moment of the coupling tim
e exists, then we obtain a variance inequality. If a moment of order 1+epsilon of the coupling time exists, then depending on the behavior of the stationary distribution, we obtain higher moment bounds. This immediately implies polynomial concentration inequalities. In the case that a moment of order 1+epsilon is finite uniformly in the starting point of the coupling, we obtain a Gaussian bound. We illustrate the general results with house of cards processes, in which both uniform and non-uniform behavior of moments of the coupling time can occur.
This paper has been withdrawn. It will be split into two separate papers. New results will be added in both papers.
Let $DeltasubsetneqV$ be a proper subset of the vertices $V$ of the defining graph of an irreducible and aperiodic shift of finite type $(Sigma_{A}^{+},S)$. Let $Sigma_{Delta}$ be the subshift of allowable paths in the graph of $Sigma_{A}^{+}$ which
only passes through the vertices of $Delta$. For a random point $x$ chosen with respect to an equilibrium state $mu$ of a Holder potential $phi$ on $Sigma_{A}^{+}$, let $tau_{n}$ be the point process defined as the sum of Dirac point masses at the times $k>0$, suitably rescaled, for which the first $n$-symbols of $S^k x$ belong to $Delta$. We prove that this point process converges in law to a marked Poisson point process of constant parameter measure. The scale is related to the pressure of the restriction of $phi$ to $Sigma_{Delta}$ and the parameters of the limit law are explicitly computed.
Let $DeltasubsetneqV$ be a proper subset of the vertices $V$ of the defining graph of an aperiodic shift of finite type $(Sigma_{A}^{+},S)$. Let $Delta_{n}$ be the union of cylinders in $Sigma_{A}^{+}$ corresponding to the points $x$ for which the fi
rst $n$-symbols of $x$ belong to $Delta$ and let $mu$ be an equilibrium state of a Holder potential $phi$ on $Sigma_{A}^{+}$. We know that $mu(Delta_{n})$ converges to zero as $n$ diverges. We study the asymptotic behaviour of $mu(Delta_{n})$ and compare it with the pressure of the restriction of $phi$ to $Sigma_{Delta}$. The present paper extends some results in cite{CCC} to the case when $Sigma_{Delta}$ is irreducible and periodic. We show an explicit example where the asymptotic behaviour differs from the aperiodic case.
For a map of the unit interval with an indifferent fixed point, we prove an upper bound for the variance of all observables of $n$ variables $K:[0,1]^ntoR$ which are componentwise Lipschitz. The proof is based on coupling and decay of correlation pro
perties of the map. We then give various applications of this inequality to the almost-sure central limit theorem, the kernel density estimation, the empirical measure and the periodogram.
This paper has been withdrawn by the authors due to an error in the main theorem.
We define the epsilon-distortion complexity of a set as the shortest program, running on a universal Turing machine, which produces this set at the precision epsilon in the sense of Hausdorff distance. Then, we estimate the epsilon-distortion complex
ity of various central Cantor sets on the line generated by iterated function systems (IFSs). In particular, the epsilon-distortion complexity of a C^k Cantor set depends, in general, on k and on its box counting dimension, contrarily to Cantor sets generated by polynomial IFS or random affine Cantor sets.
