Entropy-expansive maps

Abstract

Let f : X → X f:X \to X be a uniformly continuous map of a metric space. f is called h-expansive if there is an ε > 0 \varepsilon > 0 so that the set Φ ε ( x ) = { y : d ( f n ( x ) , f n ( y ) ) ≦ ε {\Phi _\varepsilon }(x) = \{ y:d({f^n}(x),{f^n}(y)) \leqq \varepsilon for all n ≧ 0 n \geqq 0 } has zero topological entropy for each x ∈ X x \in X . For X compact, the topological entropy of such an f is equal to its estimate using ε : h ( f ) = h ( f , ε ) \varepsilon :h(f) = h(f,\varepsilon ) . If X is compact finite dimensional and μ \mu an invariant Borel measure, then h μ ( f ) = h μ ( f , A ) {h_\mu }(f) = {h_\mu }(f,A) for any finite measurable partition A of X into sets of diameter at most ε \varepsilon . A number of examples are given. No diffeomorphism of a compact manifold is known to be not h-expansive.