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. For compact K we define rf (K, 8) = lim sup -log rn (K, 8) n-* to