In this paper, we introduce a new entropy-like invariant, named Hausdorff metric entropy, for finitely generated semigroups acting on compact metric spaces from a set-valued view and study its properties. We establish the relation between Hausdorff metric entropy and topological entropy of a semigroup defined by Bis. Some examples with positive or zero Hausdorff metric entropy are given. Moreov...

We consider locally symmetric manifolds with a fixed universal covering, and construct for each such manifold M a simplicial complex R whose size is proportional to the volume of M . When M is non-compact, R is homotopically equivalent to M , while when M is compact, R is homotopically equivalent to M \ N , where N is a finite union of submanifolds of fairly smaller dimensions. This expresses h...

A theorem of V. L. Klee, Jr. [9] asserts that any homeomorphism between two compact sets in a separable, infinite-dimensional Hilbert space can be extended to a homeomorphism of the entire space onto itself. Since it has recently been shown ( [ l ] , [5] and [6]) that all separable, infinite-dimensional Frechet spaces are homeomorphic (a Frechet space being a metrizable, complete, locally conve...

In this article we lay out the details of Fukaya’s $$A_\infty $$ -structure Morse complexes manifold possibly with non-empty boundary. We show that these -structures are homotopically independent made choices. emphasize transversality arguments make some fiber products smooth.