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Exponential decay of correlations for surface semi-flows without finite Markov partitions. (English) Zbl 1055.37027
An important bound for the norms of iterates of the transfer operator associated to piecewise uniformly expanding $$C^2$$ interval maps due to D. Dolgopyat [Ann. Math. (2) 147, 357–390 (1998; Zbl 0911.58029)] is extended from finitely many intervals of monotonicity to infinitely many intervals. This may ultimately be useful for extending results on exponential decay of correlations in the setting of billiards.

##### MSC:
 37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. 37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
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##### References:
 [1] N. Anantharaman, Travaux de Dolgopyat sur le mélange des mesures de Gibbs, Chapter 1 of “Géodésiques fermées d’une surface sous contraintes homologiques,” unpublished, 2000. [2] V. Baladi and B. Vallée, Euclidean algorithms are Gaussian, Preprint (2003), www.arxiv.org. · Zbl 1114.11092 [3] Dmitry Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2) 147 (1998), no. 2, 357 – 390. · Zbl 0911.58029 · doi:10.2307/121012 · doi.org [4] D. Dolgopyat, Private communication (2003). [5] Mark Pollicott, On the mixing of Axiom A attracting flows and a conjecture of Ruelle, Ergodic Theory Dynam. Systems 19 (1999), no. 2, 535 – 548. · Zbl 0942.37013 · doi:10.1017/S0143385799120911 · doi.org [6] Mark Pollicott and Richard Sharp, Exponential error terms for growth functions on negatively curved surfaces, Amer. J. Math. 120 (1998), no. 5, 1019 – 1042. · Zbl 0999.37010 [7] Luchezar Stoyanov, Spectrum of the Ruelle operator and exponential decay of correlations for open billiard flows, Amer. J. Math. 123 (2001), no. 4, 715 – 759. · Zbl 0994.37018 [8] Lai-Sang Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2) 147 (1998), no. 3, 585 – 650. · Zbl 0945.37009 · doi:10.2307/120960 · doi.org
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