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The pressure function for infinite equilibrium measures

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Abstract

Assume that (X, f) is a dynamical system and φ: X → [−∞, ∞) is a potential such that the f-invariant measure μφ equivalent to the φ-conformal measure is infinite, but that there is an inducing scheme F = fτ with a finite measure \(\mu_\phi^-\) and polynomial tails \(\mu_\phi^-\)(τ ≥ n) = O(n−β), β ∈ (0, 1). We give conditions under which the pressure of f for a perturbed potential φ + sψ relates to the pressure of the induced system as \(P(\phi + s\psi ) = (CP{(\overline {\phi + s\psi )} )^{1/\beta }}(1 + o(1)),\) together with estimates for the o(1)-error term. This extends results from Sarig [S06] to the setting of infinite equilibrium states. We give several examples of such systems, thus improving on the results of Lopes [L93] for the Pomeau-Manneville map with potential φt = −tlogf′, as well as on the results by Bruin and Todd [BTo09, BTo12] on countably piecewise linear unimodal Fibonacci maps. In addition, limit properties of the family of measures μφ+sψ as s → 0 are studied and statistical properties (correlation coefficients and arcsine laws) under the limit measure are derived.

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References

  • [A97] J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, Vol. 50, American Mathematical Society, providence, RI, 1997.

  • [AD01] J. Aaronson and M. Denker, Local limit theorems for partial sums of stationary sequences generated by Gibbs-Markov maps, Stochastics and Dynamics 1 (2001), 193–237.

    Article  MathSciNet  MATH  Google Scholar 

  • [BM89] M. Benedicks and M. Misiurewicz, Absolutely continuous invariant measures for maps with Hat tops, Institut des Hautes Etudes Scientifiques. Publications Mathématiques 69 (1989), 203–213.

    Article  MATH  Google Scholar 

  • [BGT] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, Vol. 27, Cambridge University Press, Cambridge, 1987.

  • [B75] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer, Berlin-New York, 1975.

  • [BKNS96] H. Bruin, G. Keller, T. Nowicki and S. van Strien, Wild Cantor attractors exist, Annals of Mathematics 143 (1996), 97–130.

    Article  MathSciNet  MATH  Google Scholar 

  • [BNT09] H. Bruin, M. Nicol and D. Terhesiu, On Young towers associated with infinite measure preserving transformations, Stochastics and Dynamics 9 (2009), 635–655.

    Article  MathSciNet  MATH  Google Scholar 

  • [BTe18] H. Bruin and D. Terhesiu, Upper and lower bounds for the correlation function via inducing with general return times, Ergodic Theory and Dynamical Systems 38 (2018), 34–62.

    Article  MathSciNet  MATH  Google Scholar 

  • [BTo09] H. Bruin and M. Todd, Df, Annales Scientifiques de l'École Normale Supérieure 42 (2009), 559–600.

    Article  MATH  Google Scholar 

  • [BTo12] H. Bruin and M. Todd, Transience and thermodynamic formalism for infinitely branched interval maps, Journal of the London Mathematical Society 86 (2012), 171–194.

    Article  MathSciNet  MATH  Google Scholar 

  • [BTo15] H. Bruin and M. Todd, Wild attractors and thermodynamic formalism, Monatshefte für Mathematik 178 (2015), 39–83.

    Article  MathSciNet  MATH  Google Scholar 

  • [DS15] N. Dobbs and M. Stenlund, Quasistatic dynamical systems, Ergodic Theory and Dynamical Systems 37 (2017), 2556–2596.

    Article  MathSciNet  MATH  Google Scholar 

  • [F66] W. Feller, An Introduction to Probability Theory and its Applications, II, Wiley, New York, 1966.

    MATH  Google Scholar 

  • [FF70] M. Fisher and B. Felderhof, Phase transitions in one-dimensional clusterinteraction fluids IA, Annals of Physics 58 (1970), 176–216.

    Article  Google Scholar 

  • [FS09] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, Cambridge, 2009.

    Book  MATH  Google Scholar 

  • [GW88] P. Gaspard and X. J. Wang, Sporadicity: between periodic and chaotic dynamical behaviours, Proceedings of the National Academy of Sciences of the United States of America 85 (1988), 4591–4595.

    Article  MathSciNet  MATH  Google Scholar 

  • [G04] S. Gouëzel, Sharp polynomial estimates for the decay of correlations, Israel Journal of Mathematics 139 (2004), 29–65.

    Article  MathSciNet  MATH  Google Scholar 

  • [G10] S. Gouëzel, Characterization of weak convergence of Birkhoff sums for Gibbs-Markov maps, Israel Journal of Mathematics 180 (2010), 1–41.

    Article  MathSciNet  MATH  Google Scholar 

  • [G11] S. Gouëzel, Correlations from large deviations in dynamical systems with infinite measure, Colloquium Mathematicum 125 (2011), 193–212

    Article  MathSciNet  MATH  Google Scholar 

  • [IT10] G. Iommi and M. Todd, Natural equilibrium states for multimodal maps, Communications in Mathematical Physics 300 (2010), 65–94.

    Article  MathSciNet  MATH  Google Scholar 

  • [IT13] G. Iommi and M. Todd, Thermodynamic formalism for interval maps: inducing schemes, Dynamical Systems 28 (2013), 354–380.

    Article  MathSciNet  MATH  Google Scholar 

  • [Ka76] T. Kato, Perturbation Theory for Linear Operators, Grundlehren de Mathematischen Wissenschaften, Vol. 132, Springer, New York, 1976.

  • [Kel89] G. Keller, Lifting measures to Markov extensions, Monatshefte für Mathematik 108 (1989), 183–200.

    Article  MathSciNet  MATH  Google Scholar 

  • [LSV99] C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic Theory and Dynamical Systems 19 (1999), 671–685.

    Article  MathSciNet  MATH  Google Scholar 

  • [L93] A. Lopes, The zeta function, non-differentiability of pressure, and the critical exponent of transition, Advances in Mathematics 101 (1993), 133–165.

    Article  MathSciNet  MATH  Google Scholar 

  • [MT12] I. Melbourne and D. Terhesiu, Operator renewal theory and mixing rates for dynamical systems with infinite measure, Inventiones Mathematicae 189 (2012), 61–110.

    Article  MathSciNet  MATH  Google Scholar 

  • [MT13] I. Melbourne and D. Terhesiu, First and higher order uniform ergodic theorems for dynamical systems with infinite measure, Israel Journal of Mathematics 194 (2013), 793–830.

    Article  MathSciNet  MATH  Google Scholar 

  • [MTo04] I. Melbourne and A. Török, Statistical limit theorems for suspension flows, Israel Journal of Mathematics 144 (2004), 191–209.

    Article  MathSciNet  MATH  Google Scholar 

  • [PP90] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188 (1990).

  • [S99] O. Sarig, Thermodynamic formalism for countable Markov shifts Ergodic Theory and Dynamical Systems 19 (1999), 1565–1593.

    Article  MathSciNet  MATH  Google Scholar 

  • [S01a] O. Sarig, Thermodynamic formalism for null recurrent potentials, Israel Journal of Mathematics 121 (2001), 285–311.

    Article  MathSciNet  MATH  Google Scholar 

  • [S01b] O. Sarig, Phase transitions for countable topological Markov shifts, Communications in Mathematical Physics 217 (2001), 555–577.

    Article  MathSciNet  MATH  Google Scholar 

  • [S02] O. Sarig, Subexponential decay of correlations, Inventiones Mathematicae 150 (2002), 629–653.

    Article  MathSciNet  MATH  Google Scholar 

  • [S06] O. Sarig, Continuous phase transitions for dynamical systems, Communications in Mathematical Physics 267 (2006), 631–667.

    Article  MathSciNet  MATH  Google Scholar 

  • [SV97] B. Stratmann and R. Vogt, Fractal dimension of dissipative sets, Nonlinearity 10 (1997), 565–577.

    Article  MathSciNet  MATH  Google Scholar 

  • [Te15] D. Terhesiu, Improved mixing rates for infinite measure preserving transformations, Ergodic Theory and Dynamical Systems 35 (2015), 585–614.

    Article  MathSciNet  MATH  Google Scholar 

  • [Tha00] M. Thaler, The asymptotics of the Perron-Frobenius operator of a class of interval maps preserving infinite measures, Studia Mathematica 143 (2000), 103–119.

    Article  MathSciNet  MATH  Google Scholar 

  • [Thu05] H. Thunberg, Positive exponent in families with flat critical point, Ergodic Theory and Dynamical Systems 19 (1999), 767–807.

    Article  MathSciNet  MATH  Google Scholar 

  • [Z98] R. Zweimüller, Ergodic structure and invariant densities of non-markovian interval maps with indifferent fixed points, Nonlinearity 11 (1998), 1263–1276.

    Article  MathSciNet  MATH  Google Scholar 

  • [Z04] R. Zweimüller, S-unimodal Misiurewicz maps with flat critical points, Fundamenta Mathematicae 181 (2004), 1–25.

    Article  MathSciNet  MATH  Google Scholar 

  • [Z05] R. Zweimüller, Invariant measures for generalized induced transformation, Proceedings of the American Mathematical Society 138 (2005), 2283–2295.

    Article  MATH  Google Scholar 

  • [Z07] R. Zweimüller, Mixing limit theorems for ergodic transformations, Journal of Theoretical Probability 20 (2007), 1059–1071.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Oskar Morgensternplatz 1, 1090, Vienna, Austria

    Henk Bruin

  2. Department of Mathematics, University of Exeter, Streatham Campus North Park Road, Exeter, EX4 4QF, UK

    Dalia Terhesiu

  3. Mathematical Institute, University of St Andrews North Haugh, St Andrews, KY16 9SS, Scotland

    Mike Todd

Authors
  1. Henk Bruin
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  2. Dalia Terhesiu
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  3. Mike Todd
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Correspondence to Henk Bruin.

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Bruin, H., Terhesiu, D. & Todd, M. The pressure function for infinite equilibrium measures. Isr. J. Math. 232, 775–826 (2019). https://doi.org/10.1007/s11856-019-1887-1

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  • DOI: https://doi.org/10.1007/s11856-019-1887-1

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