A triangular map of type with positive topological entropy on a minimal set (original) (raw)

Entropy and periodic points for transitive maps

1999

Abstract: The aim of this paper is to investigate the connection between transitivity, density of the set of periodic points and topological entropy for low dimensional continuous maps. The paper deals with this problem in the case of the $ n $-star and the circle among the one-dimensional spaces and in some higher dimensional spaces. Particular attention is paid to triangular maps and to extensions of transitive maps to higher dimensions without increasing topological entropy.

A CHARACTERIZATION OF THE SET Ω (f)\ ω (f) FOR CONTINUOUS MAPS OF THE INTERVAL WITH ZERO TOPOLOGICAL ENTROPY

International Journal of Bifurcation and Chaos, 1995

We give a characterization of the set of nonwandering points of a continuous map f of the interval with zero topological entropy, attracted to a single (infinite) minimal set Q. We show that such a map f can have a unique infinite minimal set Q and an infinite set B ⊂ Ω(f ) \ ω(f ) (of nonwandering points that are not ω-limit points) attracted to Q and such that B has infinite intersections with infinitely many disjoint orbits of f .

The topological entropy versus level sets for interval maps (part II)

Studia Mathematica, 2005

Let f : [a, b] → [a, b] be a continuous function of the compact real interval such that (i) card f −1 (y) ≥ 2 for every y ∈ [a, b]; (ii) for some m ∈ {∞, 2, 3,. . .} there is a countable set L ⊂ [a, b] such that card f −1 (y) ≥ m for every y ∈ [a, b] \ L. We show that the topological entropy of f is greater than or equal to log m. This generalizes our previous result for m = 2. 0. Introduction. The aim of this paper is to demonstrate a relationship of two characteristics of an interval map: its topological entropy and cardinalities of level sets. Our main result states that for an interval map-as opposed to circle maps or some maps on higher dimensional manifolds [Ma]-the cardinalities of level sets strongly determine the value of entropy. Elaborating our approach from [B1] we focus on the case of m-preimages for fixed m ∈ N \ {1} or m equal to infinity. In particular, Theorem 4.3 shows that if we forbid an exceptional case of one-point level sets, the dependence between entropy and the cardinalities of level sets is rather regular. Based on that and several known (always) non-transitive counterexamples we conjecture that this should be the case for a wider variety of transitive maps on compact topological manifolds. Let [a, b] be a compact real interval. We denote by C([a, b]) the set of all continuous functions which map [a, b] into itself. Any element of C([a, b]) is called an interval map. For m ∈ {∞, 2, 3, 4,. . .} let L(m, [a, b]) be the subset of C([a, b]) maps satisfying (1 m) ∀y ∈ [a, b]: card f −1 (y) ≥ m. From [B1] we know that the topological entropy of any f ∈ L(2, [a, b]) is greater than or equal to log 2. In this paper we extend that result as follows.

On homeomorphisms of (, f) having topological entropy zero

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.

Strictly Ergodic Patterns and Entropy for Interval Maps

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 ...

A Note on Topological Entropy of Continuous Self-Maps

Journal of Mathematics and System Science, 2015

Topological entropy can be an indicator of complicated behavior in dynamical systems. It is first introduce by Adler, Konheim and McAndrew by using open covers in 1965. After that it is still an active research by many researchers to produce more properties and applications up to nowadays. The purpose of this paper is to review and explain most important concepts and results of topological entropies of continuous self-maps for dynamical systems on compact and non-compact topological and metric spaces. We give proofs for some of its elementary properties of the topological entropy. Slight modification on Adler's topological entropy is also presented.

The topological entropy versus level sets for interval 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.

Triangular maps with all periods and no infinite ω-limit set containing periodic points

Topology and its Applications, 2005

Our main result is an example of a triangular map of the unite square, F (x, y) = (f (x), g x (y)), possessing periodic orbits of all periods and such that no infinite ω-limit set of F contains a periodic point. We also show that there is a triangular map F of type 2 ∞ monotone on the fibres such that any recurrent point of F is uniformly recurrent and F restricted to the set of its recurrent points is chaotic in the sense of Li and Yorke.