Entropy of a semigroup of maps from a set-valued view

Document Type: Research Paper

Authors

College of Mathematics‎, ‎Jilin university‎, ‎130012‎, ‎Changchun‎, ‎P‎. ‎R‎. ‎China.

Abstract

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. Moreover, some notions of chaos are also well generalized for finitely generated semigroups from a set-valued view.

Keywords


R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965) 303--319.

J.-P. Aubin, H. Frankowska, Set Valued Analysis, Birkhauser, Basel, 1990.

F. Balibrea, J. Smítal and M. Stefankova, The three versions of distributional chaos, Chaos Solitons Fractals 23 (2005) 1581--1583.

F. Beguin, S. Crovisier and F. Le Roux, Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique, Ann. Sci. Ec. Norm. Super (4) 40 (2007) 251--308.

A. Bis, Entropies of a semigroup of maps, Discrete Contin. Dyn. Syst. 11 (2004) 639--648.

A. Bis and M. Urbanski, Some remarks on topological entropy of a semigroup of continuous maps, Cubo 8 (2006) 63--71.

F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math. 547 (2002) 51--68.

R. Bowen, Entropy for group endomorphisms and homogenous spaces, Trans. Amer. Math. Soc. 153 (1971) 401--414.

A. Bufetov, Topological entropy of free semigroup actions and skew-product transformations, J. Dyn. Control Syst. 5 (1999) 137--143.

E.I. Dinaburg, The relation between topological entropy and metric entropy, Soviet Math. Dokl. 11 (1970) 13--16.

T. Downarowicz, Positive topological entropy implies chaos DC2, Proc. Amer. Math Soc. 142 (2014) 137--149.

S. Friedland, Entropy of graphs, semigroups and groups, in: Ergodic Theory of Zd Actions (Warwick, 1993-1994), pp. 319--343, London Math. Soc Lecure Note Ser. Cambridge Univ. Press, Cambridge, 1996.

E. Ghys, R. Langevin and P.Walczak, Entropie geometrique des feuilletages, Acta Math. 160 (1988) 105--142.

C. Grillenberger, Construction of strictly ergodic systems I. Given entropy, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 (1972/1973) 323--334.

F. Hahn and Y. Katznelson, On t