Related papers: On some dynamical aspects of NIP theories
A probability measure is a characteristic measure of a topological dynamical system if it is invariant to the automorphism group of the system. We show that zero entropy shifts always admit characteristic measures. We use similar techniques…
We study stable like behaviour in first order theories without the independence property. We introduce generically stable measures, give characterizatiions, and show their ubiquity. We also introduce generic compact domination. We also…
This is an expository article on properties of actions on Lie groups by subgroups of their automorphism groups. After recalling various results on the structure of the automorphism groups, we discuss actions with dense orbits, invariant and…
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…
In this work we are going to study the dynamics of the linear automorphisms of a measure convolution algebra over a finite group, $T(\mu)=\nu * \mu$. In order to understand an classify the asymptotic behavior of this dynamical system we…
We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$…
We study invariant types in NIP theories. Amongst other things: we prove a definable version of the (p,q)-theorem in theories of small or medium directionality; we construct a canonical retraction from the space of M-invariant types to that…
We develop a fine-scale local analysis of measure entropy and measure sequence entropy based on combinatorial independence. The concepts of measure IE-tuples and measure IN-tuples are introduced and studied in analogy with their…
We discuss some of the issues that arise in attempts to classify automorphisms of compact abelian groups from a dynamical point of view. In the particular case of automorphisms of one-dimensional solenoids, a complete description is given…
We study the dynamics of compositions of a sequence of holomorphic mappings in projective space. We define ergodicity and mixing for non-autonomous dynamical systems, and we construct totally invariant measures for which our sequence…
We study the dynamics of the metrics generated by measure preserving transformations. We consider a sequence of average metrics and define the corresponding sequence of $\epsilon$-entropies ({\it scaling sequence}) of the measure with…
It is shown how to model any automorphism of a totally disconnected, locally compact group by a symbolic dynamical system. The model is an inverse limit of a product of a full-shift, on a finite number of symbols, with one of two types of…
The article presents a new perspective on the isomorphism problem for non-ergodic measure-preserving dynamical systems with discrete spectrum which is based on the connection between ergodic theory and topological dynamics constituted by…
This paper concerns the dynamics of polynomial automorphisms of ${\bf C}^2$. One can associate to such an automorphism two currents $\mu^\pm$ and the equilibrium measure $\mu=\mu^+\wedge\mu^-$. In this paper we study some geometric and…
This essay advocates the view that any problem that has a meaningful empirical content, can be formulated in constructive, more definitely, finite terms. We consider combinatorial models of dynamical systems and approaches to statistical…
In dynamical systems composed of interacting parts, conditional exponents, conditional exponent entropies and cylindrical entropies are shown to be well defined ergodic invariants which characterize the dynamical selforganization and…
We here describe the possibility of a synthetic description of the onset of Chaos in many degrees of freedom dynamical systems within the framework of the geometric description of dynamics. We show how this approach to instability helps to…
Dynamical systems generated by $d\ge2$ commuting homeomorphisms (topological $\mathbb{Z}^d$-actions) contain within them structures on many scales, and in particular contain many actions of $\mathbb{Z}^k$ for $1\le k\le d$. Familiar…
We give algebraic characterizations of the properties of autonomy and of controllability of behaviours of spatially invariant dynamical systems, consisting of distributional solutions, that are periodic in the spatial variables, to a system…
The axiomatic theory of ordinary differential equations, owing to its simplicity, can provide a useful framework to describe various generalizations of dynamical systems. In this study, we consider how dynamical properties can be…