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Thermodynamic Formalism for Random Weighted Covering Systems

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Abstract

We develop for the first time a quenched thermodynamic formalism for random dynamical systems generated by countably branched, piecewise-monotone mappings of the interval that satisfy a random covering condition. Given a random contracting potential \(\varphi \) (in the sense of Liverani–Saussol–Vaienti), we prove there exists a unique random conformal measure \(\nu _\varphi \) and unique random equilibrium state \(\mu _\varphi \). Further, we prove quasi-compactness of the associated transfer operator cocycle and exponential decay of correlations for \(\mu _\varphi \). Our random driving is generated by an invertible, ergodic, measure-preserving transformation \(\sigma \) on a probability space \((\Omega ,{\mathscr {F}},m)\); for each \(\omega \in \Omega \) we associate a piecewise-monotone, surjective map \(T_\omega :I\rightarrow I\). We consider general potentials \(\varphi _\omega :I\rightarrow {\mathbb {R}}\cup \{-\infty \}\) such that the weight function \(g_\omega =e^{\varphi _\omega }\) is of bounded variation. We provide several examples of our general theory. In particular, our results apply to new examples of linear and non-linear systems including random \(\beta \)-transformations, randomly translated random \(\beta \)-transformations, countably branched random Gauss–Renyi maps, random non-uniformly expanding maps (such as intermittent maps and maps with contracting branches) composed with expanding maps, and a large class of random Lasota–Yorke maps.

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Notes

  1. Our theory works equally well if I is taken to be an uncountable, totally ordered, order-complete set as in [33]. In this setting the order topology makes I compact. We choose to work under the assumption that I is a compact interval in \({\mathbb {R}}\) for ease of exposition.

  2. In [33] they consider the case \({\hat{\alpha }}=4\) and \({\hat{\gamma }}=2\).

  3. Note that these limits exist by the subadditive ergodic theorem due to the submultiplicativity and supermultiplicativity of the respective sequences \(\{\omega \mapsto \Vert g_\omega ^{(n)}\Vert _\infty \}_{n\in {\mathbb {N}}}\) and \( \left\{ \omega \mapsto \inf {\mathcal {L}}_\omega ^n \mathbb {1}_\omega \right\} _{n\in {\mathbb {N}}}\). Furthermore, the limits are constant m-a.e. and the left-hand limit may be equal to \(-\infty \).

  4. If \(a\in X_\omega \) we may take \(\alpha _a=a=\beta _a\).

  5. One could prove the existence of such a function in a similar manner to Urysohn’s Lemma to show that for \(x<y\) there exists a continuous, increasing function (hence BV) from [x, y] onto [0, 1].

  6. 55 is chosen because \({1}/{\log 55}<{1}/{4}\), which will be useful in Sect. 7.

  7. Our final estimate of Lemma 7.5 differs from Buzzi’s [10, Lemma 3.4] because of a term in the sum that was overlooked. The problem can be corrected by carefully choosing \(R_*=R_{a_0}\) and then by defining \(a_*\ge a_0\) in terms of \(R_*\) as we have done here.

  8. Here the number \(\Sigma \) is playing the same role as the number \(K_{n-{\hat{y}}_*}\) from Lemma 7.6.

  9. Note that if \(\tau _0\in \Omega _G\), in which case we have \(\tau _0=\tau _0^*\), then \(\Sigma _I=0\) and \(\Sigma =\Sigma _+{\hat{r}}_{\tau _0}(n)\).

  10. Indeed, one can see this by considering the continuous linear functional \(\Lambda _\omega \) which is equivalent to \(\nu _\omega \) on \({\mathrm{BV}}(I)\). As \(f_{\omega ,n}\rightarrow q_{\omega ,f}\) we must have \(1=\Lambda _{\omega }(f_{\omega ,n})\rightarrow \Lambda (q_{\omega ,f})=\nu _\omega (q_{\omega ,f})=1\).

  11. Without loss of generality, we assume the partitions on which \(\beta \) and \(\Phi \) are constant coincide.

  12. Note that this condition is equivalent with the assumption that \(\int _\Omega \log \left\lceil \beta _\omega \right\rceil \, dm<\infty \).

  13. The maps are still required to be onto, however some branches may be contracting.

  14. The same argument given in Remark 13.14 allows us to replace \(\beta \ge 5\) with \(\beta \ge 2\) at the expense of taking \(N_*=3\).

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Acknowledgements

J.A., G.F., and C.G.-T. thank the Centro de Giorgi in Pisa and CIRM in Luminy for their support and hospitality. J.A. is supported by an ARC Discovery project and thanks the School of Mathematics and Physics at the University of Queensland for their hospitality. G.F., C.G.-T., and S.V. are partially supported by an ARC Discovery Project. S.V. thanks the Laboratoire International Associé LIA LYSM, the INdAM (Italy), the UMI-CNRS 3483, Laboratoire Fibonacci (Pisa) where this work has been completed under a CNRS delegation and the Centro de Giorgi in Pisa for various supports.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    Jason Atnip & Gary Froyland

  2. School of Mathematics and Physics, The University of Queensland, St Lucia, QLD, 4072, Australia

    Cecilia González-Tokman

  3. Aix Marseille Université, Université de Toulon, CNRS, CPT, 13009, Marseille, France

    Sandro Vaienti

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  1. Jason Atnip
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Correspondence to Jason Atnip.

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Communicated by C. Liverani.

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Atnip, J., Froyland, G., González-Tokman, C. et al. Thermodynamic Formalism for Random Weighted Covering Systems. Commun. Math. Phys. 386, 819–902 (2021). https://doi.org/10.1007/s00220-021-04156-1

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