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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

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.
Citation: João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete and 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.

[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.

[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.

[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.

[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.

[6]

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

[7]

M. Baillif, Dynamical zeta functions for tree maps, Nonlinearity, 12 (1999), 1511-1529. doi: 10.1088/0951-7715/12/6/305.

[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.

[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.

[10]

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

[11]

J. Buzzi, Some remarks on random zeta functions, Ergodic Theory Dynam. Systems, 22 (2002), 1031-1040. doi: 10.1017/S0143385702000524.

[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.

[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.

[14]

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

[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.

[16]

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

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[7]

M. Baillif, Dynamical zeta functions for tree maps, Nonlinearity, 12 (1999), 1511-1529. doi: 10.1088/0951-7715/12/6/305.

[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.

[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.

[10]

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

[11]

J. Buzzi, Some remarks on random zeta functions, Ergodic Theory Dynam. Systems, 22 (2002), 1031-1040. doi: 10.1017/S0143385702000524.

[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.

[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.

[14]

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

[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.

[16]

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

[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.

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