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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

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, 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), 309-319. 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), 128-141. 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), 957-985. 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), 591-607. 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), 833-847. doi: 10.1080/10236190802357701.  Google Scholar

[6]

M. Artin and B. Mazur, On periodic points, Ann. of Math, 81 (1965), 82-99. doi: 10.2307/1970384.  Google Scholar

[7]

M. Baillif, Dynamical zeta functions for tree maps, Nonlinearity, 12 (1999), 1511-1529. 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), 621-632. 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), 671-700. doi: 10.1007/s002200050124.  Google Scholar

[10]

V. Baladi, Periodic orbits and dynamical spectra, Ergodic Theory Dynam. Systems, 18 (1998), 255-292. doi: 10.1017/S0143385798113925.  Google Scholar

[11]

J. Buzzi, Some remarks on random zeta functions, Ergodic Theory Dynam. Systems, 22 (2002), 1031-1040. 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), 847-858. 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), 161-181.  Google Scholar

[14]

S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., 4 (1996), 205-233.  Google Scholar

[15]

J. Milnor and W. Thurston, "On Iterated Maps of the Interval," Dynamical systems (College Park, MD, 1986-87), 465-563, Lecture Notes in Math, 1342, Springer, Berlin, 1988.  Google Scholar

[16]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math, 67 (1980), 45-63.  Google Scholar

[17]

D. Ruelle, An extension of the theory of Fredholm determinants, Inst. Hautes Études Sci. Publ. Math, 72 (1990), 175-193. doi: 10.1007/BF02699133.  Google Scholar

show all references

References:
[1]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. 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), 128-141. 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), 957-985. 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), 591-607. 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), 833-847. doi: 10.1080/10236190802357701.  Google Scholar

[6]

M. Artin and B. Mazur, On periodic points, Ann. of Math, 81 (1965), 82-99. doi: 10.2307/1970384.  Google Scholar

[7]

M. Baillif, Dynamical zeta functions for tree maps, Nonlinearity, 12 (1999), 1511-1529. 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), 621-632. 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), 671-700. doi: 10.1007/s002200050124.  Google Scholar

[10]

V. Baladi, Periodic orbits and dynamical spectra, Ergodic Theory Dynam. Systems, 18 (1998), 255-292. doi: 10.1017/S0143385798113925.  Google Scholar

[11]

J. Buzzi, Some remarks on random zeta functions, Ergodic Theory Dynam. Systems, 22 (2002), 1031-1040. 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), 847-858. 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), 161-181.  Google Scholar

[14]

S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random Comput. Dynam., 4 (1996), 205-233.  Google Scholar

[15]

J. Milnor and W. Thurston, "On Iterated Maps of the Interval," Dynamical systems (College Park, MD, 1986-87), 465-563, Lecture Notes in Math, 1342, Springer, Berlin, 1988.  Google Scholar

[16]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math, 67 (1980), 45-63.  Google Scholar

[17]

D. Ruelle, An extension of the theory of Fredholm determinants, Inst. Hautes Études Sci. Publ. Math, 72 (1990), 175-193. doi: 10.1007/BF02699133.  Google Scholar

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