Related papers: Effective symbolic dynamics, random points, statis…
Measuring the average information that is necessary to describe the behaviour of a dynamical system leads to a generalization of the Kolmogorov-Sinai entropy. This is particularly interesting when the system has null entropy and the…
The Thermodynamic Formalism provides a rigorous mathematical framework to study quantitative and qualitative aspects of dynamical systems. At its core there is a variational principle corresponding, in its simplest form, to the Maximum…
Different notions of entropy play a fundamental role in the classical theory of dynamical systems. Unlike many other concepts used to analyze autonomous dynamics, both measure-theoretic and topological entropy can be extended quite…
A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new…
The existence of the {\em typical set} is key for data compression strategies and for the emergence of robust statistical observables in macroscopic physical systems. Standard approaches derive its existence from a restricted set of…
In this paper we prove estimates on the behaviour of the Kolmogorov-Sinai entropy relative to a partition for randomly perturbed dynamical systems. Our estimates use the entropy for the unperturbed system and are obtained using the notion…
Dynamical system properties give rise to effects in Statistical Mechanics. Topological index changes can be the basis for phase transitions. The Euler characteristic is a versatile topological invariant that can be evaluated for model…
There are two main approaches to non-equlibrium statistical mechanics: one using stochastic processes and the other using dynamical systems. To model the dynamics during inflation one usually adopts a stochastic description, which is known…
This article reviews a generous sampling of both classical and more recent results on the interplay between measurable and topological dynamics. In the first part we have surveyed the strong analogies between ergodic theory and topological…
We consider the collection of uniformly discrete point sets in Euclidean space equipped with the vague topology. For a point set in this collection, we characterise minimality of an associated dynamical system by almost repetitivity of the…
We describe classes of ergodic dynamical systems for which some statistical properties are known exactly. These systems have integer dimension, are not globally dissipative, and are defined by a probability density and a two-form. This…
Using tools from computable analysis we develop a notion of effectiveness for general dynamical systems as those group actions on arbitrary spaces that contain a computable representative in their topological conjugacy class. Most natural…
Finding the correct encoding for a generic dynamical system's trajectory is a complicated task: the symbolic sequence needs to preserve the invariant properties from the system's trajectory. In theory, the solution to this problem is found…
In this paper, we study the complicated dynamics of infinite dimensional random dynamical systems which include deterministic dynamical systems as their special cases in a Polish space. Without assuming any hyperbolicity, we proved if a…
Random substitutions are a natural generalisation of their classical `deterministic' counterpart, whereby at every step of iterating the substitution, instead of replacing a letter with a predetermined word, every letter is independently…
In a recent paper, K.Keller has given a characterization of the Kolmogorov-Sinai entropy of a discrete-time measure-preserving dynamical system on the base of an increasing sequence of special partitions. These partitions are constructed…
We numerically test the method of non-sequential recursive pair substitutions to estimate the entropy of an ergodic source. We compare its performance with other classical methods to estimate the entropy (empirical frequencies, return…
We introduce amorphic complexity as a new topological invariant that measures the complexity of dynamical systems in the regime of zero entropy. Its main purpose is to detect the very onset of disorder in the asymptotic behaviour. For…
Ergodic optimization is the study of problems relating to maximizing orbits, maximizing invariant measures and maximum ergodic averages. An orbit of a dynamical system is called f-maximizing if the time average of the real-valued function f…
Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. For a wide class of intrinsically ergodic subshifts over a finite alphabet, we show that the space of…