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.

show all references

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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