Topological entropy and variation for transitive maps

Studia Mathematica, 2002

We answer affirmatively Coven's question [PC]: Suppose f : I → I is a continuous function of the interval such that every point has at least two preimages. Is it true that the topological entropy of f is greater than or equal to log 2? 2000 Mathematics Subject Classification: 37E05, 37B40.

arXiv (Cornell University), 2024

We generalize the definition of topological entropy given by Adler, Konheim, and McAndrew (AKM) for piecewise continuous self-maps defined on a compact interval (pc-maps). For this notion of entropy, we prove that the properties of the AKM-entropy in the compact-continuous setting get naturally extended, including that it can be estimated using Bowen's formula disregarding the metric used to compute it. Additionally, for piecewise strictly monotonic pc-maps, we prove that the Misiurewicz-Szlenk formula based on the asymptotic behavior of the number of monotony pieces for iterations of the map coincides with our notion of topological entropy. In this way, we unify the notions of entropy commonly used for piecewise continuous interval maps, proving that they all coincide under compactness assumptions.

Topology and its Applications, 1998

Let f : II -II be a continuous function where II is the unit interval. Let (JI, f) be the inverse limit space obtained from the inverse sequence all of whose maps are f and all of whose spaces are 1. This paper addresses the question of when (II, f) has the property that every homeomorphism of (1, S) has zero topological entropy. An obvious necessary condition for this is that f itself has zero topological entropy. In this paper it is proved that if f is piecewise monotone and has only finitely many periods. then every homeomorphism of (II, f) has zero entropy. 0 1998 Elsevier Science B.V.

Physics Letters A, 1982

A method for computing the growth number and the topological entropy in maps of the interval, given the kneading sequence, is presented. This technique is applied to the study of the parameter dependence and universal propties of the topological entropy in the approach of aperiodic regimes. In particular we show that period doubling does not change entropy.

Nonlinearity, 2002

For piecewise monotone interval maps we show that Kolmogorov-Sinai entropy can be obtained from order statistics of the values in a generic orbit. A similar statement holds for topological entropy.

Journal of Applied Analysis, 2000

We investigate the topological entropy of a green interval map. Defining the complexity we estimate from above the topological entropy of a green interval map with a given complexity. The main result of the paper-stated in Theorem 0.2-should be regarded as a completion of results of [4].

Ergodic Theory and Dynamical Systems, 2012

We study dynamics of continuous maps on compact metrizable spaces containing a free interval (i.e. an open subset homeomorphic to an open interval). Special attention is paid to relationships between topological transitivity, weak and strong topological mixing, dense periodicity and topological entropy as well as to the topological structure of minimal sets. In particular, a trichotomy for minimal sets and a dichotomy for transitive maps are proved.

Topology and its Applications, 2011

We prove some results concerning the entropy of continuous and almost continuous functions. We first introduce the notions of bundle entropy and (strong) entropy points and then we study properties of these notions in connection with the theory of multifunctions. Based on these facts we give theorems about approximation of functions defined and assuming their values on compact manifold by functions having strong entropy points.

International Journal of Bifurcation and Chaos, 1999

We introduce the notion of the center of a point for discrete dynamical systems and we study its properties for continuous interval maps. It is known that the Birkhoff center of any such map has depth at most 2. Contrary to this, we show that if a map has positive topological entropy then, for any countable ordinal α, there is a point x α ∈ I such that its center has depth at least α. This improves a result by . *

2003

Let M be the set of all pairs (T;g ) such that T R is compact, g : T ! T is continuous, g is minimal on T and has a piecewise monotone extension to conv T.T wo pairs (T;g ); (S; f )f romM are equivalent { (T;g ) (S; f ){i f the map h :o rb(minT;g ) ! orb(min S; f) dened for each m 2 N0 by h(gm(min T )) = = fm(min S) is increasing on orb(min T;g ). An equivalence class of this relation is called a minimal (oriented) pattern. Such a pattern A 2M is strictly ergodic if for some (T;g ) 2 A there is exactly one g-invariant normalized Borel measure satisfying supp = T. A pattern A is exhibited by a continuous interval map f : I ! I if there is a set T I such that (T;f jT )=( T;g ) 2 A. Using the fact that for two equivalent pairs (T;g ); (S; f) 2 A their topological entropies ent(g; T )a nd ent(f; S) equal we can dene the lower topological entropy entL(A) of a minimal pattern A as that common value. We show that the topological entropy ent(f; I )o f a continuous interval map f : I ! I is ...