Ergodic Theory and Dynamical Systems



Topological entropy for divergence points


CHEN ERCAI a1, TASSILO KÜPPER a2 and SHU LIN a1
a1 Department of Mathematics, Nanjing Normal University, Nanjing 210097, People's Republic of China (e-mail: ecchen@njnu.edu.cn, shulin003@sohu.com)
a2 Mathematical Institute, University of Cologne, D-50931 Cologne, Germany (e-mail: kuepper@mi.uni-koeln.de)

Article author query
ercai c   [Google Scholar] 
kupper t   [Google Scholar] 
lin s   [Google Scholar] 
 

Abstract

Let X be a compact metric space, f a continuous transformation on X, and Y a vector space with linear compatible metric. Denote by M(X) the collection of all the probability measures on X. For a positive integer n, define the nth empirical measure $L_{n}:X\mapsto M(X)$ as

\[ L_{n}x=\frac{1}{n}\sum _{k=0}^{n-1}\delta_{f^{k}x}, \]

where $\delta_{x}$ denotes the Dirac measure at x. Suppose $\Xi:M(X)\mapsto Y$ is continuous and affine with respect to the weak topology on M(X). We think of the composite

\[ \Xi\circ L_{n}:X \xrightarrow{L_n} M(X)\xrightarrow{\Xi} Y \]

as a continuous and affine deformation of the empirical measure Ln. The set of divergence points of such a deformation is defined as

\[ D(f,\Xi)=\{x\in X\mid \mbox{the limit of }\Xi L_n x \mbox{ does not exist}\}. \]

In this paper we show that for a continuous transformation satisfying the specification property, if $\Xi(M(X))$ is a singleton, then set of divergence points is empty, i.e. <formula form="inline" disc="math" id="ffm008"><formtex notation="AMSTeX">$D(f,\Xi)=\emptyset$, and if $\Xi(M(X))$ is not a singleton, then the set of divergence points has full topological entropy, i.e.

\[ h_{\rm top}(D(f,\Xi))=h_{\rm top}(f). \]

(Received May 1 2004)
(Revised September 20 2004)