# American Institute of Mathematical Sciences

June  2011, 31(2): 545-557 . doi: 10.3934/dcds.2011.31.545

## Topological pressure and topological entropy of a semigroup of maps

 1 Department of Mathematics, South China University of Technology, Guangzhou, 510640, China 2 Department of Mathematics, South China University of Technology, Guangzhou, 510641, China

Received  April 2010 Revised  February 2011 Published  June 2011

By using the Carathéodory-Pesin structure(C-P structure), with respect to arbitrary subset, the topological pressure and topological entropy, introduced for a single continuous map, is generalized to the cases of semigroup of continuous maps. Several of their basic properties are provided.
Citation: Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 545-557 . doi: 10.3934/dcds.2011.31.545
##### References:
 [1] R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 303-319. doi: 10.1090/S0002-9947-1965-0175106-9. [2] L. Barreira, Y. Pesin and J. Schmeling, On a general concept of multifractality: Multifractal spectrum for dimensions, entropies and Lyapunv exponents, multifractal rigidity, Chaos, 7 (1997), 27-38. doi: 10.1063/1.166232. [3] A. Biś, Entropies of a semigroup of maps, Discrete Contin. Dyn. Systs. Series A., 11 (2004), 639-648. doi: 10.3934/dcds.2004.11.639. [4] A. Biś and M. Urbański, Some remarks on topological entropy of a semigroup of continuous maps, Cubo, 8 (2006), 63-71. [5] A. Biś and P. Walczak, Entropies of hyperbolic groups and some foliated spaces, in "Foliations-Geometry and Dynamics" (eds. P. Walczak et al. ), World Sci. Publ., Singapore, (2002), 197-211. [6] R. Bowen, Entropy for group endomorphisms and homogenous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414. doi: 10.1090/S0002-9947-1971-0274707-X. [7] R. Bowen, Topological entropy for non-compact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. [8] C. Carathéodory, Über das lineare mass, Göttingen Nachr, (1914), 406-426. [9] E. I. Dinaburg, The relation between topological entropy and metric entropy, Soviet Math. Dokl., 11 (1970), 13-16. [10] S. Friedland, Entropy of graphs,semigroups and groups, in "Ergodic Theory of $Z^d$ Actions" (eds. M. Policott and K. Schmidt), London Math. Soc, London, (1996), 319-343. [11] E. Ghys, R. Langevin and P. Walczak, Entropie geometrique des feuilletages, Acta Math., 160 (1988), 105-142. doi: 0.1007/BF02392274. [12] M. Hurley, On topological entropy of maps, Ergodic Th. and Dynam. Sys., 15 (1995), 557-568. [13] R. Langevin and F. Przytycki, Entropie de l'image inverse d'une application, Bull. Soc. Math. France., 120 (1992), 237-250. [14] R. Langevin and P. Walczak, Entropie d'une dynamique, C. R. Acad. Sci. Paris, Ser I, 312 (1991), 141-144. [15] Z. Nitecki and F. Przytycki, Preimage entropy for mapping, Internat. J. Bifur. Chaos Appl. Sci. Engrg, 9 (1999), 1815-1843. doi: 10.1142/S0218127499001309. [16] Y. Pesin, "Dimension Theory in Dynamical Systems," Chicago: The university of Chicago Press, 1997. [17] Y. Pesin and B. Pitskel, Topological pressure and the variational principle for non-compact sets, Functional Anal. and Its Applications, 18 (1984), 50-63. doi: 10.1007/BF01083692. [18] D. Ruelle, "Thermodynamic Formalism," Addison-Wesley, Reading, MA,1978. [19] P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971. doi: 10.2307/2373682.

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##### References:
 [1] R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 303-319. doi: 10.1090/S0002-9947-1965-0175106-9. [2] L. Barreira, Y. Pesin and J. Schmeling, On a general concept of multifractality: Multifractal spectrum for dimensions, entropies and Lyapunv exponents, multifractal rigidity, Chaos, 7 (1997), 27-38. doi: 10.1063/1.166232. [3] A. Biś, Entropies of a semigroup of maps, Discrete Contin. Dyn. Systs. Series A., 11 (2004), 639-648. doi: 10.3934/dcds.2004.11.639. [4] A. Biś and M. Urbański, Some remarks on topological entropy of a semigroup of continuous maps, Cubo, 8 (2006), 63-71. [5] A. Biś and P. Walczak, Entropies of hyperbolic groups and some foliated spaces, in "Foliations-Geometry and Dynamics" (eds. P. Walczak et al. ), World Sci. Publ., Singapore, (2002), 197-211. [6] R. Bowen, Entropy for group endomorphisms and homogenous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414. doi: 10.1090/S0002-9947-1971-0274707-X. [7] R. Bowen, Topological entropy for non-compact sets, Trans. Amer. Math. Soc., 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X. [8] C. Carathéodory, Über das lineare mass, Göttingen Nachr, (1914), 406-426. [9] E. I. Dinaburg, The relation between topological entropy and metric entropy, Soviet Math. Dokl., 11 (1970), 13-16. [10] S. Friedland, Entropy of graphs,semigroups and groups, in "Ergodic Theory of $Z^d$ Actions" (eds. M. Policott and K. Schmidt), London Math. Soc, London, (1996), 319-343. [11] E. Ghys, R. Langevin and P. Walczak, Entropie geometrique des feuilletages, Acta Math., 160 (1988), 105-142. doi: 0.1007/BF02392274. [12] M. Hurley, On topological entropy of maps, Ergodic Th. and Dynam. Sys., 15 (1995), 557-568. [13] R. Langevin and F. Przytycki, Entropie de l'image inverse d'une application, Bull. Soc. Math. France., 120 (1992), 237-250. [14] R. Langevin and P. Walczak, Entropie d'une dynamique, C. R. Acad. Sci. Paris, Ser I, 312 (1991), 141-144. [15] Z. Nitecki and F. Przytycki, Preimage entropy for mapping, Internat. J. Bifur. Chaos Appl. Sci. Engrg, 9 (1999), 1815-1843. doi: 10.1142/S0218127499001309. [16] Y. Pesin, "Dimension Theory in Dynamical Systems," Chicago: The university of Chicago Press, 1997. [17] Y. Pesin and B. Pitskel, Topological pressure and the variational principle for non-compact sets, Functional Anal. and Its Applications, 18 (1984), 50-63. doi: 10.1007/BF01083692. [18] D. Ruelle, "Thermodynamic Formalism," Addison-Wesley, Reading, MA,1978. [19] P. Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97 (1975), 937-971. doi: 10.2307/2373682.
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