Expansiveness, Lyapunov exponents and entropy for set valued maps

Applied General Topology, 2016

We prove that if a uniformly continuous self-map $f$ of a uniform space has topological specification property then the map $f$ has positive uniform entropy, which extends the similar known result for homeomorphisms on compact metric spaces having specification property. An example is also provided to justify that the converse is not true.<br /><br />

Mathematische Nachrichten

In this work, we show that if f is a uniformly continuous map defined over a Polish metric space, then the set of f‐invariant measures with zero metric entropy is a set (in the weak topology). In particular, this set is generic if the set of f‐periodic measures is dense in the set of f‐invariant measures. This settles a conjecture posed by Sigmund (Trans. Amer. Math. Soc. 190 (1974), 285–299), which states that the metric entropy of an invariant measure of a topological dynamical system that satisfies the periodic specification property is typically zero. We also show that if X is compact and if f is an expansive or a Lipschitz map with a dense set of periodic measures, typically the lower correlation entropy for is equal to zero. Moreover, we show that if X is a compact metric space and if f is an expanding map with a dense set of periodic measures, then the set of invariant measures with packing dimension, upper rate of recurrence and upper quantitative waiting time indicator equa...

2012

Co-compact entropy is introduced as an invariant of topological conjugation for perfect mappings defined on any Hausdorff space (compactness and metrizability not necessarily required). This is achieved through the consideration of co-compact covers of the space. The advantages of co-compact entropy include: 1) it does not require the space to be compact, and thus generalizes Adler, Konheim and McAndrew’s topological entropy of continuous mappings on compact dynamical systems, and 2) it is an invariant of topological conjugation, compared to Bowen’s entropy that is metric-dependent. Other properties of co-compact entropy are investigated, e.g., the co-compact entropy of a subsystem does not exceed that of the whole system. For the linear system (R, f) defined by f(x) = 2x, the co-compact entropy is zero, while Bowen’s entropy for this system is at least log 2. More general, it is found that co-compact entropy is a lower bound of Bowen’s entropies, and the proof of this result genera...

2009

Adler, Konheim and McAndrew introduced the concept of topological entropy of a continuous mapping for compact dynamical systems. Bowen generalized the concept to non-compact metric spaces, but Walters indicated that Bowen's entropy is metric-dependent. We propose a new definition of topological entropy for continuous mappings on arbitrary topological spaces (compactness, metrizability, even axioms of separation not necessarily required), investigate fundamental properties of the new entropy, and compare the new entropy with the existing ones. The defined entropy generates that of Adler, Konheim and McAndrew and is metric-independent for metrizable spaces. Yet, it holds various basic properties of Adler, Konheim and McAndrew's entropy, e.g., the entropy of a subsystem is bounded by that of the original system, topologically conjugated systems have a same entropy, the entropy of the induced hyperspace system is larger than or equal to that of the original system, and in particular this new entropy coincides with Adler, Konheim and McAndrew's entropy for compact systems.

Mathematica Slovaca, 1993

We s tudy the continuous functions which m a p a compact real inter­ val back into itself. We investigate the relations between two important concepts of the dynamical systems and real analysis for transitive functions, topological entropy and variation. 0. Introduct ion This paper is concerned with investigation of relations between the topolog­ ical entropy and variation for transitive maps. Topological entropy, denoted ent(-), is a numerical conjugacy invariant of continuous maps. Variation of a function / on the interval 7, denoted Var(/, 7) , is a length of a way of a point f(x) if a point x goes through the interval 7. A continuous map is transitive if some point has a dense orbit. Let 7 = [0,1] be the closed unit interval and C(I, I) be the set of all con­ tinuous functions which map the interval 7 back into itself. MAIN THEOREM. Let (x, y) be a pair of numbers. Then there exists a tran­ sitive function f E (7(7,7) such that (x,y) = (Var(/, 7), ent(/)) if and only if (x,y) e ...

Topology and its Applications, 2012

The generalized entropy (and its properties) of maps f : exp(X) → exp(X), usual functions and multifunctions in a generalized topological space X are studied. Moreover, the connection between generalized entropy and standard entropy for usual functions in a compact metric space is shown.

Earthline Journal of Mathematical Sciences

Topological entropy is used to determine the complexity of a dynamical system. This paper aims to serve as a stepping stone for the study of topological entropy. We review the notions of topological entropy, give an overview on the relation between the notions and fundamental properties of topological entropy. Besides, we cover the topological entropy of the induced hyperspaces and its connection with the original systems. We also provide a summary on the latest research topic related with topological entropy.

arXiv preprint arXiv:1301.1533, 2013

In this work we study the main dynamical properties of the push forward map, a transformation in the space of probabilities P(X) induced by a map T : X → X, X a compact metric space. We also establish a connection between topological entropies of T and of the push forward map.

Entropy, 2013

Co-compact entropy is introduced as an invariant of topological conjugation for perfect mappings defined on any Hausdorff space (compactness and metrizability are not necessarily required). This is achieved through the consideration of co-compact covers of the space. The advantages of co-compact entropy include: (1) it does not require the space to be compact and, thus, generalizes Adler, Konheim and McAndrew's topological entropy of continuous mappings on compact dynamical systems; and (2) it is an invariant of topological conjugation, compared to Bowen's entropy, which is metric-dependent. Other properties of co-compact entropy are investigated, e.g., the co-compact entropy of a subsystem does not exceed that of the whole system. For the linear system, (R, f), defined by f (x) = 2x, the co-compact entropy is zero, while Bowen's entropy for this system is at least log 2. More generally, it is found that co-compact entropy is a lower bound of Bowen's entropies, and the proof of this result also generates the Lebesgue Covering Theorem to co-compact open covers of non-compact metric spaces.

Discrete & Continuous Dynamical Systems - A, 2011

Let f : I = [0, 1] → I be a Borel measurable map and let µ be a probability measure on the Borel subsets of I. We consider three standard ways to cope with the idea of "observable chaos" for f with respect to the measure µ: hµ(f ) > 0 -when µ is invariant-, µ(L + (f )) > 0 -when µ is absolutely continuous with respect to the Lebesgue measure-, and µ(S µ (f )) > 0. Here hµ(f ), L + (f ) and S µ (f ) denote, respectively, the metric entropy of f , the set of points with positive Lyapunov exponent, and the set of sensitive points to initial conditions with respect to µ. It is well known that if hµ(f ) > 0 or µ(L + (f )) > 0, then µ(S µ (f )) > 0, and that (when µ is invariant and absolutely continuous) hµ(f ) > 0 and µ(L + (f )) > 0 are equivalent properties. However, the available proofs in the literature require substantially stronger hypotheses than those strictly necessary. In this paper we revisit these notions and show that the above-mentioned results remain true in, essentially, the most general (reasonable) settings. In particular, we improve some previous results from [2], [6], and .