Asymptotic behaviour of the entropy of interval maps (original) (raw)

Asymptotic Behavior of the Entropy of Interval Maps

2015

Abstract. We obtain upper estimates on the entropy of interval maps of given modality and Sharkovskii type. Following our results we formulate a conjecture on asymptotic behavior of the entropy of interval maps. 1.

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.

Entropy of interval maps via permutations

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.

On entropy of patterns given by interval maps

Fundamenta Mathematicae, 1999

Defining the complexity of a green pattern exhibited by an interval map, we give the best bounds of the topological entropy of a pattern with a given complexity. Moreover, we show that the topological entropy attains its strict minimum on the set of patterns with fixed eccentricity m/n at a unimodal X-minimal case. Using a different method, the last result was independently proved in [11].

On the Topological Entropy of Green Interval Maps

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].

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.

Entropy of twist interval maps

Israel Journal of Mathematics, 1997

We investigate the recently introduced notion of rotation numbers for periodic orbits of interval maps. We identify twist orbits, that is those orbits that are the simplest ones with given rotation number. We estimate from below the topological entropy of a map having an orbit with given rotation number. Our estimates are sharp: there are unimodal maps where the equality holds. We also discuss what happens for maps with larger modality. In the Appendix we present a new approach to the problem of monotonicity of entropy in one-parameter families of unimodal maps.

Monotonicity of entropy and positively oriented transversality for families of interval maps

2016

In this paper we will develop a very general approach which shows that critical relations of holomorphic maps on the complex plane unfold transversally in a positively oriented way. We will mainly illustrate this approach to obtain transversality for a wide class of one-parameter families of interval maps, for example maps with flat critical points, piecewise linear maps, maps with discontinuities but also for families of maps with complex analytic extensions such as certain polynomial-like maps.

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 ...