On chaos and topological entropy of tree maps
by
José S. Cánovas
Universidad Politécnica de Cartagena

Let (X, d) be a compact metric space and let f:X® X be a continuous map. For x, y Î X and t Î R⁺ define

Fx, y(t)= liminf n® ¥  1 n n-1 å i=0  c[0, t](d(fi(x), fi(y))

and

Fx, y*(t)= liminf n® ¥  1 n n-1 å i=0  c[0, t](d(fi(x), fi(y))).

F_{x, y} and F_{x, y}^* are called the lower and upper distribution functions of f at x and y (we identify functions that are equal in the L¹ metric). f is said to be distributionally chaotic if there are x, y Î X, x ¹ y, such that

Fx, y(t) < Fx, y*(t)

for some t Î R⁺. f is said to be chaotic in the sense of Li-Yorke if there is an uncountable set E Ì X such that

liminf i® ¥  d(fi(x), fi(y))=0

and

limsup i® ¥  d(fi(x), fi(y)) > 0

for any x, y Î E, x ¹ y (see [] for definitions). Denote by h_A(f) the topological sequence entropy of f relative to the increasing sequence of positive integers A (see []) and by h(f)=h_A(f) when A=(i)_i=0^¥ .

In the general case, if f is distributionally chaotic, then it is chaotic in the sense of Li-Yorke. When X=[0, 1], f is distributionally chaotic iff h(f) > 0 (see []) and f is chaotic in the sense of Li-Yorke iff h_A(f) > 0 for some A ([]). The same results hold when X=S¹ (see [] and []). However, they are false in general for two-dimensional maps (see [] and []).

Here we focus our attention on one-dimensional maps, and more precisely on finite trees. A finite tree (or simply a tree) T is a connected Hausdorff space which has a finite subspace V (points of V are called vertices) such that G\V is a disjoint union of finite number of open subsets e₁, e₂, ..., e_k (called edges), each of them homeomorphic to an open interval of the real line, and two vertices are attached at the boundary of each edge. For more information on trees see the survey paper [].

We consider the n-star X={z Î C:zⁿ Î [0, 1]} and prove the following results:

Let X be a tree and let f:X®X be a continuous map such that f(0)=0. Then

(a) f is distributionally chaotic iff h(f) > 0.
(b) f is chaotic in the sense of Li-Yorke iff h_A(f) > 0 for some A.

References

[]

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R. Hric, Topological sequence entropy for maps of the circle,  Comment. Math. Univ. Carolin. 41 (2000), 53-59.

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N. Franzová and J. Smítal,  Positive sequence topological entropy characterizes chaotic maps,  Proc. Amer.\ Math. Soc. 112 (1991), 1083-1086.

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L. Paganoni and P. Santambrogio, Chaos and sequence topological entropy for triangular maps, quaderno n. 59/1996, Universtá degli studi di Milano.

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B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344 (1994), 737-754.

Date received: February 23, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafw-24.