The minimum tree for a given zero-entropy period

1991

Let $X$ be a compact tree, $f$ be a continuous map from $X$ to itself, $End(X)$ be the number of endpoints and $Edg(X)$ be the number of edges of $X$. We show that if $n>1$ has no prime divisors less than $End(X)+1$ and $f$ has a cycle of period $n$, then $f$ has cycles of all periods greater than $2End(X)(n-1)$

Proceedings of the American Mathematical Society

This paper deals with the relationship between the periodic orbits of continuous maps on graphs and the topological entropy of the map. We show that the topological entropy of a graph map can be approximated by the entropy of its periodic orbits.

Transactions of the American Mathematical Society, 2016

Consider, for any n ∈ N, the set Posn of all n-periodic tree patterns with positive topological entropy and the set Irrn Posn of all n-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families Posn and Irrn. Let λn be the unique real root of the polynomial x n − 2x − 1 in (1, +∞). We explicitly construct an irreducible n-periodic tree pattern Qn whose entropy is log(λn). For n = m k , where m is a prime, we prove that this entropy is minimum in the set Posn. Since the pattern Qn is irreducible, Qn also minimizes the entropy in the family Irrn.

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.

International Journal of Bifurcation and Chaos, 2003

We study the set of periods of tree maps f : T → T which are monotone between any two consecutive points of a fixed periodic orbit P. This set is characterized in terms of some integers which depend only on the combinatorics of f|P and the topological structure of T. In particular, a typep ≥ 1 of P is defined as a generalization of the notion introduced by Baldwin in his characterization of the set of periods of star maps. It follows that there exists a divisor k of the period of P such that if the set of periods of f is not finite then it contains either all the multiples of kp or an initial segment of the kp≥ Baldwin's ordering, except for a finite set which is explicitly bounded. Conversely, examples are given where f has precisely these sets of periods.

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].

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.

Proyecciones (Antofagasta), 2018

The Lorenz Attractor has been a source for many mathematical studies. Most of them deal with lower dimensional representations of its first return map. An one dimensional scenario can be modeled by the standard two parameter family of contracting Lorenz maps. The dynamics, in this case, can be modeled by a subshift in the Lexicographical model. These subshifts are the maximal invariant set for the shift map in some interval. For some of them the extremes, of the interval, are a minimal periodic sequence and a maximal periodic sequence which is an iteration of the lower extreme (by the shift map). For some of these subshifts the topological entropy is zero. In this case the dynamics (of the respective Lorenz map) is simple. Associated to any of these subshifts (let call it Λ) we consider an extension (let call it Γ) of it that contains Λ which also can be constructed by using an interval whose extremes can be defined by the extremes of Λ. For these extensions, we present here a computer verification of a result that compute its topological entropy. As a consequence, we can affirm: the longer the period of the periodic sequence is, then the lower complexity in the dynamics of the extension the associated map has.

Nonlinear Analysis: Theory, Methods & Applications, 2002

Ergodic Theory and Dynamical Systems, 2013

We extend the classical notion of block structure for periodic orbits of interval maps to the setting of tree maps and study the algebraic properties of the Markov matrix of a periodic tree pattern having a block structure. We also prove a formula which relates the topological entropy of a pattern having a block structure with that of the underlying periodic pattern obtained by collapsing each block to a point, and characterize the structure of the zero entropy patterns in terms of block structures. Finally, we prove that an $n$-periodic pattern has zero (positive) entropy if and only if all $n$-periodic patterns obtained by considering the $k\mathrm{th} $ iterate of the map on the invariant set have zero (respectively, positive) entropy, for each $k$ relatively prime to $n$.