Entropy-Expansive Maps

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