Entropy-Expansive Maps

Let /: X -> X be a uniformly continuous map of a metric space./ is called /(-expansive if there is an oO so that the set $>c(x) = {y : d(Jn(x),fn(y))^e for all nO) has zero topological entropy for each xe X.For Xcompact, the topological entropy of such an /is equal to its estimate using e: h(f) = h(f, e).If X is compact finite dimensional and y. an invariant Borel measure, then /V"(/) = A"(/, A) for any finite measurable partition A of X into sets of diameter at most e.A number of examples are given.No diffeomorphism of a compact manifold is known to be not A-expansive.Let/: X -> X be a homeomorphism of a metric space.For e > 0 and xe X define Ys(x) = {yeX: d(f(y),f"(x)) i for all n e Z).fis called expansive if for some e these sets are as small as possible, i.e. if Ye(x)=x for all x.We are concerned with entropy and shall call /h-expansive provided that for some e > 0 the Fs(x) are negligible in terms of entropy, i.e. if the topological entropy (/, TE(x)) = 0 for all x.We have two main results for -expansive maps with X compact.First, the topological entropy satisfies (/) = h(fi e).Second, assuming X is finite dimensional, "(/) = "(/> A) when p. is an/-invariant normalized Borel measure on Zand A is a finite measurable partition of X into sets of diameter at most e.Both these results are well known in case/is expansive (see [11] and [14] respectively).Arov [2] noted that the second statement was true for/an endomorphism of a torus and fi Haar measure when he calculated hu(f) for this case (see Example 1.2).1. Definitions and examples.We now review the definition of topological entropy given in [4].For X compact this definition was given independently by Dinaburg [7]; is related to the e-entropy of Kolmogorov [12].Topological entropy was defined first in [1].Let /: X -> X be uniformly continuous on the metric space X.For E, F<= X we say that E(n, 8)-spans F (with respect to fi), if for each y e FthereisanxeFso that d(fk(x),fkiy)) i 8 for all 0 i k < n.We let rn(F, 8) = rn(F, 8,f) denote the minimum cardinality of a set which (, 8)-spans F. If K is compact, then the continuity off guarantees rn(K, 8) < oo.