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Transience in dynamical systems

Published online by Cambridge University Press:  06 August 2012

GODOFREDO IOMMI  and
MIKE TODD
GODOFREDO IOMMI
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile (PUC), Avenida Vicuña Mackenna 4860, Santiago, Chile (email: giommi@mat.puc.cl)
MIKE TODD
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, KY16 9SS, Scotland (email: mjt20@st-andrews.ac.uk)
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Abstract

We extend the theory of transience to general dynamical systems with no Markov structure assumed. This is linked to the theory of phase transitions. We also provide new examples to illustrate different kinds of transient behaviour.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 5 , October 2013 , pp. 1450 - 1476
Copyright
Copyright © 2012 Cambridge University Press 

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References

[Aa]Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.Google Scholar
[A]Abramov, L. M.. On the entropy of a flow. Dokl. Akad. Nauk SSSR 128 (1959), 873–875.Google Scholar
[Ba]Barreira, L.. Dimension and Recurrence in Hyperbolic Dynamics (Progress in Mathematics, 272). Birkhäuser, Basel, 2008.Google Scholar
[Bo1]Bowen, R.. Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429–460.Google Scholar
[Bo2]Bowen, R.. Some systems with unique equilibrium states. Math. Syst. Theory 8 (1974), 193–202.CrossRefGoogle Scholar
[Bo3]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, 2nd revised edn.(Lecture Notes in Mathematics, 470). Ed. Chazottes, J.-R.. Springer, Berlin, 2008, with a preface by David Ruelle.Google Scholar
[Br]Bruin, H.. Minimal Cantor systems and unimodal maps. J. Difference Equ. Appl. 9 (2003), 305–318.Google Scholar
[BT1]Bruin, H. and Todd, M.. Equilibrium states for potentials with $\sup \phi - \inf \phi \lt h_{top}(f)$. Comm. Math. Phys. 283 (2008), 579–611.CrossRefGoogle Scholar
[BT2]Bruin, H. and Todd, M.. Equilibrium states for interval maps: the potential $ -t \log |Df|$. Ann. Sci. Éc. Norm. Supér.(4) 42 (2009), 559–600.Google Scholar
[ChH]Chazottes, J. R. and Hochman, M.. On the zero-temperature limit of Gibbs states. Comm. Math. Phys. 297 (2010), 265–281.Google Scholar
[CRL]Cortez, M. I. and Rivera-Letelier, J.. Invariant measures of minimal post-critical sets of logistic maps. Israel J. Math. 176 (2010), 157–193.Google Scholar
[C1]Cyr, V.. Transient Markov shifts. PhD Thesis, The Pennsylvania State University, 2010.Google Scholar
[C2]Cyr, V.. Countable Markov shifts with transient potentials. Proc. Lond. Math. Soc. (3) 103 (2011), 923–949.Google Scholar
[CS]Cyr, V. and Sarig, O.. Spectral gap and transience for Ruelle operators on countable Markov shifts. Comm. Math. Phys. 292 (2009), 637–666.Google Scholar
[D]Dobbs, N.. Renormalisation induced phase transitions for unimodal maps. Comm. Math. Phys. 286 (2009), 377–387.Google Scholar
[Fe]Feller, W.. An Introduction to Probability Theory and Its Applications, 3rd edn. Vol. I. John Wiley and Sons, New York, 1968.Google Scholar
[FL]Fisher, A. M. and Lopes, A.. Exact bounds for the polynomial decay of correlation, $1/f$ noise and the CLT for the equilibrium state of a non-Hölder potential. Nonlinearity 14 (2001), 1071–1104.Google Scholar
[Gu1]Gurevič, B. M.. Topological entropy for denumerable Markov chains. Dokl. Akad. Nauk SSSR 10 (1969), 911–915.Google Scholar
[Gu2]Gurevič, B. M.. Shift entropy and Markov measures in the path space of a denumerable graph. Dokl. Akad. Nauk SSSR 11 (1970), 744–747.Google Scholar
[H]Hofbauer, F.. Examples for the nonuniqueness of the equilibrium state. Trans. Amer. Math. Soc. 228 (1977), 223–241.Google Scholar
[HK]Hofbauer, F. and Keller, G.. Equilibrium states for piecewise monotonic transformations. Ergod. Th. & Dynam. Sys. 2 (1982), 23–43.CrossRefGoogle Scholar
[IT1]Iommi, G. and Todd, M.. Natural equilibrium states for multimodal maps. Comm. Math. Phys. 300 (2010), 65–94.Google Scholar
[LSV]Liverani, C., Saussol, B. and Vaienti, S.. Probabilistic approach to intermittency. Ergod. Th. & Dynam. Sys. 19 (1999), 671–685.CrossRefGoogle Scholar
[Lo]Lopes, A. O.. The zeta function, nondifferentiability of pressure, and the critical exponent of transition. Adv. Math. 101 (1993), 133–165.Google Scholar
[MP]Manneville, P. and Pomeau, Y.. Intermittent transition to turbulence in dissipative dynamical systems. Comm. Math. Phys. 74 (1980), 189–197.Google Scholar
[MPa]Markley, N. G. and Paul, M. E.. Equilibrium states of grid functions. Trans. Amer. Math. Soc. 274 (1982), 169–191.Google Scholar
[MU]Mauldin, R. and Urbański, M.. Dimensions and measures in infinite iterated function systems. Proc. Lond. Math. Soc. (3) 73 (1996), 105–154.Google Scholar
[O]Olivier, E.. Multifractal analysis in symbolic dynamics and distribution of pointwise dimension for $g$-measures. Nonlinearity 12 (1999), 1571–1585.CrossRefGoogle Scholar
[PP]Parry, W. and Pollicott, M.. Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990).Google Scholar
[PS]Pesin, Y. and Senti, S.. Equilibrium measures for maps with inducing schemes. J. Mod. Dyn. 2 (2008), 1–31.Google Scholar
[Pi]Pinheiro, V.. Expanding Measures. Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), 889–939.CrossRefGoogle Scholar
[PU]Przytycki, F. and Urbański, M.. Conformal Fractals: Ergodic Theory Methods. Cambridge University Press, Cambridge, 2010.Google Scholar
[Ra]Ratner, M.. Markov partitions for Anosov flows on $n$-dimensional manifolds. Israel J. Math. 15 (1973), 92–114.Google Scholar
[Ru]Ruelle, D.. Statistical mechanics on a compact set with $Z^{v}$ action satisfying expansiveness and specification. Trans. Amer. Math. Soc. 187 (1973), 237–251.Google Scholar
[S1]Sarig, O.. Thermodynamic formalism for countable Markov shifts. Ergod. Th. & Dynam. Sys. 19 (1999), 1565–1593.CrossRefGoogle Scholar
[S2]Sarig, O.. Thermodynamic formalism for null recurrent potentials. Israel J. Math. 121 (2001), 285–311.CrossRefGoogle Scholar
[S3]Sarig, O.. Phase transitions for countable Markov shifts. Comm. Math. Phys. 217 (2001), 555–577.Google Scholar
[S4]Sarig, O.. Lectures Notes on Thermodynamic Formalism for Topological Markov Shifts, 2009, http://www.wisdom.weizmann.ac.il//∼sarigo/TDFnotes.pdf.Google Scholar
[S5]Sarig, O.. Symbolic dynamics for surface diffeomorphisms with positive topological entropy. Preprint, 2011, arXiv:1105.1650v3.Google Scholar
[T]Todd, M.. Multifractal analysis for multimodal maps. Preprint.Google Scholar
[V1]Vere-Jones, D.. Geometric ergodicity in denumerable Markov chains. Q. J. Math. 13 (1962), 7–28.Google Scholar
[V2]Vere-Jones, D.. Ergodic properties of nonnegative matrices I. Pacific J. Math. 22 (1967), 361–386.Google Scholar
[W1]Walters, P.. Ruelle’s operator theorem and $g$-measures. Trans. Amer. Math. Soc. 214 (1975), 375–387.Google Scholar
[W2]Walters, P.. Equilibrium states for $\beta $-transformations and related transformations. Math. Z. 159 (1978), 65–88.Google Scholar
[W3]Walters, P.. Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1978), 121–153.CrossRefGoogle Scholar
[W4]Walters, P.. Convergence of the Ruelle operator for a function satisfying Bowen’s condition. Trans. Amer. Math. Soc. 353 (2001), 327–347.CrossRefGoogle Scholar
[W5]Walters, P.. A natural space of functions for the Ruelle operator theorem. Ergod. Th. & Dynam. Sys. 27 (2007), 1323–1348.Google Scholar
[Z]Zweimüller, R.. Invariant measures for general(ized) induced transformations. Proc. Amer. Math. Soc. 133 (2005), 2283–2295.Google Scholar