A bstract In classical chaotic systems the entropy, averaged over initial phase space distributions, follows a universal behavior. While approaching thermal equilibrium it passes through stage where grows linearly, while growth rate, Kolmogorov-Sinai entropy (rate), is given by sum all positive Lyapunov exponents. natural question whether similar relation valid for quantum systems. We argue tha...

We propose a simple complexity indicator of classical Liouvillian dynamics, namely the separability entropy, which determines the logarithm of an effective number of terms in a Schmidt decomposition of phase space density with respect to an arbitrary fixed product basis. We show that linear growth of separability entropy provides a stricter criterion of complexity than Kolmogorov-Sinai entropy,...

This report compares a variety of computable information measures for finite strings. These include Shannon’s n-block entropy, the three best known versions of the Lempel-Ziv production complexity (LZ-76, LZ-77, and LZ-78), and the lesser known T-entropy. We apply these measures to strings of known entropy, each derived from the logistic map. Pesin’s identity allows us to deduce corresponding S...

Our paper analyzes here some new lines to introduce the evolving concept of an important Uncertainty Measure, the so-called Entropy. We need to obtain these new ways to model adequate conditions, departing from vague pieces of information. For this, it will be very necessary to analyze certain type of such measures, with very interesting applications, as Graph Entropy, Metric Entropy and Topolo...

We revisit the notion of Kolmogorov-Sinai entropy for classical dynamical systems in terms of an algebraic formalism. This is the starting point for deening the entropy for general non-commutative systems. Hereby typical quantum tools are introduced in the statistical description of classical dynamical systems. We illustrate the power of these techniques by providing a simple, self-contained pr...

The entropy of a finite probability space X measures the observable cardinality of large independent products X of the probability space. If two probability spaces X and Y have the same entropy, there is an almost measure-preserving bijection between large parts of X and Y . In this way, X and Y are asymptotically equivalent. It turns out to be challenging to generalize this notion of asymptoti...

Kolmogorov (1958) recognised that by appropriately encoding phase-space trajectories of dynamical systems, one could in principle compute the corresponding Shannon entropies (in physics, referred to as the Kolmogorov-Sinai entropy) for each. Pesin [1] (1977) proved that for certain classes of dissipative non-linear systems, including one dimensional maps, the KS-entropy is precisely the sum of ...

We continue the study of Rokhlin entropy, an isomorphism invariant for p.m.p. actions of countable groups introduced in Part I. In this paper we prove a non-ergodic finite generator theorem and use it to establish subadditivity and semi-continuity properties of Rokhlin entropy. We also obtain formulas for Rokhlin entropy in terms of ergodic decompositions and inverse limits. Finally, we clarify...

In this work we discuss the structure of ordinal pattern distributions obtained from orbits of dynamical systems. In particular, we consider the extreme cases of systems with a singular pattern distribution and of realizing each ordinal pattern of any order, respectively. Finally, we review results relating the Kolmogorov-Sinai entropy and the topological entropy of one-dimensional dynamical sy...