Zeta functions and topological entropy of periodic nonautonomous dynamical systems

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February 2013, 33(2): 465-482. doi: 10.3934/dcds.2013.33.465

Zeta functions and topological entropy of periodic nonautonomous dynamical systems

João Ferreira Alves ^1, and Michal Málek ^2,

1.	Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, TUL, Lisboa, Portugal
2.	Mathematical Institute, Silesian University in Opava, Czech Republic

Received August 2011 Revised September 2011 Published September 2012

Abstract
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In the setting of continuous piecewise monotone interval maps, we study the analytic properties of the zeta function of a periodic nonautonomous dynamical system. Based upon these properties we discuss the relationship between topological entropy and growth number of periodic points of a periodic nonautonomous dynamical system.

Keywords: topological entropy, piecewise monotone maps, Nonautonomous dynamical systems, periodic points, zeta functions, kneading determinants..

Mathematics Subject Classification: Primary: 37B55, 37C30; Secondary: 37B40, 37C25, 37E0.

Citation: João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465

References:

[1]	R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy,, Trans. Amer. Math. Soc., 114 (1965), 309. doi: 10.1090/S0002-9947-1965-0175106-9. Google Scholar
[2]	Z. AlSharawi, J. Angelos, S. Elaydi and L. Rakesh, An extension of Sharkovsky's theorem to periodic difference equations,, J. Math. Anal. Appl., 316 (2006), 128. doi: 10.1016/j.jmaa.2005.04.059. Google Scholar
[3]	J. F. Alves and Sousa J. Ramos, Kneading theory for tree maps,, Ergodic Theory Dynam. Systems, 24 (2004), 957. doi: 10.1017/S014338570400015X. Google Scholar
[4]	J. F. Alves, R. Hric and J. S. Ramos, Topological entropy, homological growth and zeta functions on graphs,, Nonlinearity, 18 (2005), 591. doi: 10.1088/0951-7715/18/2/008. Google Scholar
[5]	J. F. Alves, What we need to find out the periods of a periodic difference equation,, J. Difference Equ. Appl., 15 (2009), 833. doi: 10.1080/10236190802357701. Google Scholar
[6]	M. Artin and B. Mazur, On periodic points,, Ann. of Math, 81 (1965), 82. doi: 10.2307/1970384. Google Scholar
[7]	M. Baillif, Dynamical zeta functions for tree maps,, Nonlinearity, 12 (1999), 1511. doi: 10.1088/0951-7715/12/6/305. Google Scholar
[8]	V. Baladi and D. Ruelle, An extension of the theorem of Milnor and Thurston on the zeta functions of interval maps,, Ergodic Theory Dynam. Systems, 14 (1994), 621. doi: 10.1017/S0143385700008087. Google Scholar
[9]	V. Baladi, Correlation spectrum of quenched and annealed equilibrium states for random expanding maps,, Comm. Math. Phys., 186 (1997), 671. doi: 10.1007/s002200050124. Google Scholar
[10]	V. Baladi, Periodic orbits and dynamical spectra,, Ergodic Theory Dynam. Systems, 18 (1998), 255. doi: 10.1017/S0143385798113925. Google Scholar
[11]	J. Buzzi, Some remarks on random zeta functions,, Ergodic Theory Dynam. Systems, 22 (2002), 1031. doi: 10.1017/S0143385702000524. Google Scholar
[12]	J. S. Cánovas and A. Linero, Periodic structure of alternating continuous interval maps,, J. Difference Equ. Appl., 12 (2006), 847. doi: 10.1080/10236190600772515. Google Scholar
[13]	S. Kolyada, M. Misiurewicz and L. Snoha, Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval,, Fund. Math, 160 (1999), 161. Google Scholar
[14]	S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems,, Random Comput. Dynam., 4 (1996), 205. Google Scholar
[15]	J. Milnor and W. Thurston, "On Iterated Maps of the Interval,", Dynamical systems (College Park, (1342), 1986. Google Scholar
[16]	M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings,, Studia Math, 67 (1980), 45. Google Scholar
[17]	D. Ruelle, An extension of the theory of Fredholm determinants,, Inst. Hautes Études Sci. Publ. Math, 72 (1990), 175. doi: 10.1007/BF02699133. Google Scholar

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References:

[1]	R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy,, Trans. Amer. Math. Soc., 114 (1965), 309. doi: 10.1090/S0002-9947-1965-0175106-9. Google Scholar
[2]	Z. AlSharawi, J. Angelos, S. Elaydi and L. Rakesh, An extension of Sharkovsky's theorem to periodic difference equations,, J. Math. Anal. Appl., 316 (2006), 128. doi: 10.1016/j.jmaa.2005.04.059. Google Scholar
[3]	J. F. Alves and Sousa J. Ramos, Kneading theory for tree maps,, Ergodic Theory Dynam. Systems, 24 (2004), 957. doi: 10.1017/S014338570400015X. Google Scholar
[4]	J. F. Alves, R. Hric and J. S. Ramos, Topological entropy, homological growth and zeta functions on graphs,, Nonlinearity, 18 (2005), 591. doi: 10.1088/0951-7715/18/2/008. Google Scholar
[5]	J. F. Alves, What we need to find out the periods of a periodic difference equation,, J. Difference Equ. Appl., 15 (2009), 833. doi: 10.1080/10236190802357701. Google Scholar
[6]	M. Artin and B. Mazur, On periodic points,, Ann. of Math, 81 (1965), 82. doi: 10.2307/1970384. Google Scholar
[7]	M. Baillif, Dynamical zeta functions for tree maps,, Nonlinearity, 12 (1999), 1511. doi: 10.1088/0951-7715/12/6/305. Google Scholar
[8]	V. Baladi and D. Ruelle, An extension of the theorem of Milnor and Thurston on the zeta functions of interval maps,, Ergodic Theory Dynam. Systems, 14 (1994), 621. doi: 10.1017/S0143385700008087. Google Scholar
[9]	V. Baladi, Correlation spectrum of quenched and annealed equilibrium states for random expanding maps,, Comm. Math. Phys., 186 (1997), 671. doi: 10.1007/s002200050124. Google Scholar
[10]	V. Baladi, Periodic orbits and dynamical spectra,, Ergodic Theory Dynam. Systems, 18 (1998), 255. doi: 10.1017/S0143385798113925. Google Scholar
[11]	J. Buzzi, Some remarks on random zeta functions,, Ergodic Theory Dynam. Systems, 22 (2002), 1031. doi: 10.1017/S0143385702000524. Google Scholar
[12]	J. S. Cánovas and A. Linero, Periodic structure of alternating continuous interval maps,, J. Difference Equ. Appl., 12 (2006), 847. doi: 10.1080/10236190600772515. Google Scholar
[13]	S. Kolyada, M. Misiurewicz and L. Snoha, Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval,, Fund. Math, 160 (1999), 161. Google Scholar
[14]	S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems,, Random Comput. Dynam., 4 (1996), 205. Google Scholar
[15]	J. Milnor and W. Thurston, "On Iterated Maps of the Interval,", Dynamical systems (College Park, (1342), 1986. Google Scholar
[16]	M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings,, Studia Math, 67 (1980), 45. Google Scholar
[17]	D. Ruelle, An extension of the theory of Fredholm determinants,, Inst. Hautes Études Sci. Publ. Math, 72 (1990), 175. doi: 10.1007/BF02699133. Google Scholar

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