On the zero-temperature limit of one-dimensional Gibbs measures
This paper has been withdrawn by the authors due to an error in the main theorem.
math.DS↗
arXiv subjects
J. -R. Chazottes
Publications and source records attributed to J. -R. Chazottes.
Epsilon-Distortion Complexity for Cantor Sets
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 complexity of various central Cantor sets on the line generated by iterated function systems (IFS's). 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.
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Concentration inequalities for random fields via coupling
We present a new and simple approach to concentration inequalities for functions around their expectation with respect to non-product measures, i.e., for dependent random variables. Our method is based on coupling ideas and does not use information inequalities. When one has a uniform control on the coupling, this leads to exponential concentration inequalities. When such a uniform control is no more possible, this leads to polynomial or stretched-exponential concentration inequalities. Our abstract results apply to Gibbs random fields, in particular to the low-temperature Ising model which is a concrete example of non-uniformity of the coupling.
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Testing the irreversibility of a Gibbsian process via hitting and return times
We introduce estimators for the entropy production of a Gibbsian process based on the observation of a single or two typical trajectories. These estimators are built with adequate hitting and return times. We then study their convergence and fluctuation properties. This provides statisticals test for the irreversibility of Gibbsian processes.
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Occurrence, repetition and matching of patterns in the low-temperature Ising model
We continue our study of exponential law for occurrences and returns of patterns in the context of Gibbsian random fields. For the low temperature plus phase of the Ising model, we prove exponential laws with error bounds for occurrence, return, waiting and matching times. Moreover we obtain a Poisson law for the number of occurrences of large cylindrical events and a Gumbel law for the maximal overlap between two independent copies. As a by-product, we derive precise fluctuation results for the logarithm of waiting and return times. The main technical tool we use, in order to control mixing, is disagreement percolation.
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Devroye Inequality for a Class of Non-Uniformly Hyperbolic Dynamical Systems
In this paper, we prove an inequality, which we call "Devroye inequality", for a large class of non-uniformly hyperbolic dynamical systems (M,f). This class, introduced by L.-S. Young, includes families of piece-wise hyperbolic maps (Lozi-like maps), scattering billiards (e.g., planar Lorentz gas), unimodal and H{é}non-like maps. Devroye inequality provides an upper bound for the variance of observables of the form K(x,f(x),...,f^{n-1}(x)), where K is any separately Holder continuous function of n variables. In particular, we can deal with observables which are not Birkhoff averages. We will show in \cite{CCS} some applications of Devroye inequality to statistical properties of this class of dynamical systems.
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Statistical Consequences of Devroye Inequality for Processes. Applications to a Class of Non-Uniformly Hyperbolic Dynamical Systems
In this paper, we apply Devroye inequality to study various statistical estimators and fluctuations of observables for processes. Most of these observables are suggested by dynamical systems. These applications concern the co-variance function, the integrated periodogram, the correlation dimension, the kernel density estimator, the speed of convergence of empirical measure, the shadowing property and the almost-sure central limit theorem. We proved in \cite{CCS} that Devroye inequality holds for a class of non-uniformly hyperbolic dynamical systems introduced in \cite{young}. In the second appendix we prove that, if the decay of correlations holds with a common rate for all pairs of functions, then it holds uniformly in the function spaces. In the last appendix we prove that for the subclass of one-dimensional systems studied in \cite{young} the density of the absolutely continuous invariant measure belongs to a Besov space.
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Large deviations for empirical entropies of Gibbsian sources
The entropy of an ergodic finite-alphabet process can be computed from a single typical sample path x_1^n using the entropy of the k-block empirical probability and letting k grow with $n$ roughly like log n. We further assume that the distribution of the process is a g-measure; g-measures form a large class of Gibbs measures. We prove large deviation principles for conditional, non-conditional and relative k(n)-block empirical entropies.
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Entropy estimation and fluctuations of Hitting and Recurrence Times for Gibbsian sources
Motivated by entropy estimation from chaotic time series, we provide a comprehensive analysis of hitting times of cylinder sets in the setting of Gibbsian sources. We prove two strong approximation results from which we easily deduce pointwise convergence to entropy, lognormal fluctuations, precise large deviation estimates and an explicit formula for the hitting-time multifractal spectrum. It follows from our analysis that the hitting time of a $n$-cylinder fluctuates in the same way as the inverse measure of this $n$-cylinder at "small scales", but in a different way at "large scales". In particular, the Renyi entropy differs from the hitting-time spectrum, contradicting a naive ansatz. This phenomenon was recently numerically observed for return times that are more difficult to handle theoretically. The results we obtain for return times, though less complete, improve the available ones.
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Multifractals via recurrence times ?
This letter is a comment on an article by T.C. Halsey and M.H. Jensen in Nature about using recurrence times as a reliable tool to estimate multifractal dimensions of strange attractors. Our aim is to emphasize that in the recent mathematical literature (not cited by these authors), there are positive as well as negative results about the use of such techniques. Thus one may be careful in using this tool in practical situations (experimental data).
nlin.CD↗
Finite type approximations of Gibbs measures on sofic subshifts
Consider a Hölder continuous potential $ϕ$ defined on the full shift $A^\nn$, where $A$ is a finite alphabet. Let $X\subset A^\nn$ be a specified sofic subshift. It is well-known that there is a unique Gibbs measure $μ_ϕ$ on $X$ associated to $ϕ$. Besides, there is a natural nested sequence of subshifts of finite type $(X_m)$ converging to the sofic subshift $X$. To this sequence we can associate a sequence of Gibbs measures $(μ_ϕ^m)$. In this paper, we prove that these measures weakly converge at exponential speed to $μ_ϕ$ (in the classical distance metrizing weak topology). We also establish a strong mixing property (ensuring weak Bernoullicity) of $μ_ϕ$. Finally, we prove that the measure-theoretic entropy of $μ_ϕ^m$ converges to the one of $μ_ϕ$ exponentially fast. We indicate how to extend our results to more general subshifts and potentials. We stress that we use basic algebraic tools (contractive properties of iterated matrices) and symbolic dynamics.
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Birkhoff averages of Poincare cycles for Axiom-A diffeomorphisms
We study the time of $n$th return of orbits to some given (union of) rectangle(s) of a Markov partition of an Axiom A diffeomorphism. Namely, we prove the existence of a scaled generating function for these returns with respect to any Gibbs measure (associated to a Holderian potential). As a by-product, we derive precise large deviation estimates and a central limit theorem for Birkhoff averages of Poincare cycles. We emphasize that we look at the limiting behavior in term of number of visits (the size of the visited set is kept fixed). Our approach relies on the spectral properties of a one-parameter family of induced transfer operators on unstable leaves crossing the visited set.
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Exponential distribution for the occurrence of rare patterns in Gibbsian random fields
We study the distribution of the occurrence of rare patterns in sufficiently mixing Gibbs random fields on the lattice $\mathbb{Z}^d$, $d\geq 2$. A typical example is the high temperature Ising model. This distribution is shown to converge to an exponential law as the size of the pattern diverges. Our analysis not only provides this convergence but also establishes a precise estimate of the distance between the exponential law and the distribution of the occurrence of finite patterns. A similar result holds for the repetition of a rare pattern. We apply these results to the fluctuation properties of occurrence and repetition of patterns: We prove a central limit theorem and a large deviation principle.
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Projection of Markov measures may be Gibbsian
We study the induced measure obtained from a 1-step Markov measure, supported by a topological Markov chain, after the mapping of the original alphabet onto another one. We give sufficient conditions for the induced measure to be a Gibbs measure (in the sense of Bowen) when the factor system is again a topological Markov chain. This amounts to constructing, when it does exist, the induced potential and proving its Holder continuity. This is achieved through a matrix method. We provide examples and counterexamples to illustrate our results.
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Multi-valued mappings in generalized chaos synchronization
The onset of generalized synchronization of chaos in directionally-coupled systems corresponds to the formation of a continuous mapping which enables one to persistently define the state of the response system from the trajectory of the drive system. The recently developed theory of generalized synchronization of chaos deals only with the case where this synchronization mapping is a single-valued function. In this paper, we explore generalized synchronization in a regime where the synchronization mapping can become a multi-valued function. Specifically, we study the properties of the multi-valued mapping which occurs between the drive and response systems when the systems are synchronized with a frequency ratio other than one-to-one, and address the issues of the existence and continuity of such mappings. The basic theoretical framework underlying the considered synchronization regimes is then developed.
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