Related papers: Konzepte der abstrakten Ergodentheorie. Zweiter Te…

Our conception of a generalized ergodic theory shall exceed the generality of general topology: In this first part of the generalized ergodic theory we investigate the logical constitution of the conception of attractors. We prepare a…

In the framework of statistical mechanics the properties of macroscopic systems are deduced starting from the laws of their microscopic dynamics. One of the key assumptions in this procedure is the ergodic property, namely the equivalence…

A discrete quantum process is represented by a sequence of quantum operations, which are completely positive maps that are not necessarily trace preserving. We consider quantum processes that are obtained by repeated iterations of a quantum…

We describe the approximation of a continuous dynamical system on a p. l. manifold or Cantor set by a tractable system. A system is tractable when it has a finite number of chain components and, with respect to a given full background…

The stability against perturbations of a dynamical system conserving a generalized phase-space volume is studied by exploiting the similarity between statistical physics formalism and that of ergodic theory. A general continuity theorem is…

The generic behavior of quantum systems has long been of theoretical and practical interest. Any quantum process is represented by a sequence of quantum channels. We consider general ergodic sequences of stochastic channels with arbitrary…

We start by reviewing recent probabilistic results on ergodic sums in a large class of (non-uniformly) hyperbolic dynamical systems. Namely, we describe the central limit theorem, the almost-sure convergence to the gaussian and other stable…

For every positive integer $n\geq 2$, we introduce the concept of measure-theoretic $n$-sensitivity for measure-theoretic dynamical systems via finite measurable partitions, and show that an ergodic system is measure-theoretically…

We establish connections between several properties of topological dynamical systems, such as: - every point is generic for an ergodic measure, - the map sending points to the measures they generate is continuous, - the system splits into…

We characterize the points that satisfy Birkhoff's ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space…

The ergodic decomposition theorem is a cornerstone result of dynamical systems and ergodic theory. It states that every invariant measure on a dynamical system is a mixture of ergodic ones. Here we formulate and prove the theorem in terms…

The purpose of this note is twofold. In the first part we observe that two finitely generated non-amenable groups are quasi-isometric if and only if they admit topologically orbit equivalent Cantor minimal actions. In particular, free…

We construct classes of two-dimensional aperiodic Lorentz systems that have infinite horizon and are 'chaotic', in the sense that they are (Poincar\'e) recurrent, uniformly hyperbolic, ergodic, and the first-return map to any scatterer is…

The main result of this paper is that two large collections of ergodic measure preserving systems, the Odometer Based and the Circular Systems have the same global structure with respect to joinings. The classes are canonically isomorphic…

Downarowicz and Maass (2008) proposed topological ranks for all homeomorphic Cantor minimal dynamical systems using properly ordered Bratteli diagrams. In this study, we adopt this definition to the case of all essentially minimal…

The present note reviews recent results on resonances for one-dimensional quantum ergodic systems constrained to a large box. We restrict ourselves to one dimensional models in the discrete case. We consider two type of ergodic potentials…

This study describes such a situation that a Cantor set emerges as a result of the exploration of sufficient conditions for the property which is generalized from fundamental chaotic maps, and the Cantor set even guarantees infinitely many…

This survey is focused on the results related to topologies on the groups of transformations in ergodic theory, Borel, and Cantor dynamics. Various topological properties (density, connectedness, genericity) of these groups and their…

For a dynamical system satisfying the approximate product property and asymptotically entropy expansiveness, we characterize a delicate structrue of the space of invariant measures: The ergodic measures of intermediate entropies and…

We study the time evolution operator in a family of local quantum circuits with random fields in a fixed direction. We argue that the presence of quantum chaos implies that at large times the time evolution operator becomes effectively a…