A concept of entropy production associated with continuous topolog-ical evolution is deduced (without statistics) from the fact that Cartan-Hilbert 1-form of Action defines a non-equilibrium symplectic system of Pfaff Topological dimension 2n+2. The differential entropy, dS, is composed of the interior product of the non-canonical components of momentum with the components of the differential v...

In the present paper, we introduce a natural extension of AKMtopological entropy for noncompact spaces and prove a variational principle which states that the topological entropy, the supremum of the measure theoretical entropies and the minimum of the metric theoretical entropies always coincide. We apply the variational principle to show that the topological entropy of automorphisms of simply...

In this workwe shall consider a notion of complexity of formal languages L that is inspired by the concept of entropy from dynamical systems. More precisely, we shall define the topological entropy of L to be the exponential growth-rate of the restrictions of the Nerode congruence relation of L to words of length at most n. We shall show that the topological entropy of regular languages is alwa...

We characterise the difficulty of computing topological entropy of one-dimensional subshifts under various dynamical restrictions. Entropy is known to be computable from above for general one-dimensional subshifts, and the aim of the article is to distinguish under which dynamical properties it becomes computable. The considered properties are minimality, unique ergodicity, and various forms of...

We prove the existence of an open and dense subset of maps f 2 Diff 1 ! (S 2) which have positive topological entropy. It follows that these maps have innnitely many hyperbolic periodic points and an exponential growth rate of hyperbolic periodic points. The proof is an application of Pixton's theorem. Topological entropy characterizes the total exponential orbit complexity of a map with a sing...

This paper extends the concept of topological entropy to the case of uncertain dynamical systems. We address problems of observability and optimal control via limited capacity digital communication channels. The main results are given in terms of inequalities between date rate of the communication channel and topological entropy of the open-loop system. Topological entropy is calculated for som...

We consider topological Markov chains (also called Markov shifts) on countable graphs. We show that a transient graph can be extended to a recurrent graph of equal entropy which is either positive recurrent of null recurrent, and we give an example of each type. We extend the notion of local entropy to topological Markov chains and prove that a transitive Markov chain admits a measure of maxima...

The aim of this paper is to introduce a method for computing rigorous lower bounds for topological entropy. The topological entropy of a dynamical system measures the number of trajectories that separate in finite time and quantifies the complexity of the system. Our method relies on extending existing computational Conley index techniques for constructing semi-conjugate symbolic dynamical syst...

We examine the relation between topological entropy, invertability, and prediction in topological dynamics. We show that topological determinism in the sense of Kamińsky Siemaszko and Szymański imposes no restriction on invariant measures except zero entropy. Also, we develop a new method for relating topological determinism and zero entropy, and apply it to obtain a multidimensional analog of ...