Zeta functions and topological entropy of periodic nonautonomousdynamical systems

João Ferreira Alves; Michal Málek

doi:10.3934/dcds.2013.33.465

Article Contents

2013, Volume 33, Issue 2: 465-482. Doi: 10.3934/dcds.2013.33.465

This issue Previous Article Brownian-time Brownian motion SIEs on $\mathbb{R}_{+}$ × $\mathbb{R}^d$: Ultra regular direct and lattice-limits solutions and fourth order SPDEs links Next Article Inertial manifolds for a class of non-autonomous semilinear parabolic equations with finite delay

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
2.
Mathematical Institute, Silesian University in Opava

Received: August 2011

Revised: September 2011

Published: February 2013

Abstract / Introduction Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 37B55, 37C30; Secondary: 37B40, 37C25, 37E05.

Citation:

Related Papers

Cited by

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.