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Decay of correlations for piecewise expanding maps. (English) Zbl 1080.37501
Summary: This paper investigates the decay of correlations in a large class of non-Markov one-dimensional expanding maps. The method employed is a special version of a general approach recently proposed by the author. Explicit bounds on the rate of decay of correlations are obtained.

MSC:
37A30 Ergodic theorems, spectral theory, Markov operators
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[1] V. Baladi and G. Keller, Zeta functions and transfer operators for piecewise monotone transformations,Commun. Math. Phys. 127:459–477 (1990). · Zbl 0703.58048 · doi:10.1007/BF02104498
[2] M. Benedicks and L.-S. Young, Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps,Ergodic Theory Dynam. Syst. 12:13–37 (1992). · Zbl 0769.58051 · doi:10.1017/S0143385700006556
[3] Garret Birkhoff, Extension of Jentzsch’s theorem,Trans. Am. Math. Soc. 85:219–227 (1957). · Zbl 0079.13502
[4] Garret Birkhoff,Lattice Theory, 3rd ed. (American Mathematical Society Colloquium Publications, Providence, Rhode Island, 1967).
[5] P. Diaconis and D. Stroock, Geometric bounds for eigenvalues of Markov chains,Ann. Appl. Prob. 1:36–61 (1991). · Zbl 0731.60061 · doi:10.1214/aoap/1177005980
[6] P. Collet, An estimate of the decay of correlations for mixing non Markov expanding maps of the interval, preprint (1994).
[7] P. Ferrero, Contribution à la théorie des états d’équilibre en méchanique statistique, Theses (1981).
[8] P. Ferrero and B. Schmitt, Produits aléatories d’opérateurs matrices de transfert,Prob. Theory Related Fields 79:227–248 (1988). · Zbl 0633.60014 · doi:10.1007/BF00320920
[9] P. Ferrero and B. Schmitt, On the rate of convergence for some limit ratio theorem related to endomorphisms with a nonregular invariant density, preprint.
[10] P. Ferrero and B. Schmitt, Ruelle’s Perron-Frobenius theorem and projective metrics, inRandom Fields, Esztergom (Hungary), Colloquia Mathematica Societas János Bolyai, Vol. 27 (1979).
[11] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotone transformations,Math. Z. 180:119–140 (1982). · Zbl 0485.28016 · doi:10.1007/BF01215004
[12] B. R. Hunt, Estimating invariant measures and Lyapunov exponents, preprint. · Zbl 0858.58029
[13] G. Keller, Un théorème de la limite centrale pour une classe de transformations monotones par morceaux,C. R. Acad. Sci. A 291:155–158 (1980). · Zbl 0446.60013
[14] C. Ionescu-Tulcea and Marinescu,Ann. Math. 52:140–147 (1950). · Zbl 0040.06502 · doi:10.2307/1969514
[15] C. Liverani, Decay of correlations, to appear inAnn. Math. · Zbl 0871.58059
[16] A. Lasota and J. Yorke, On existence of invariant measures for piecewise monotonic transformations,Trans. Am. Math. Soc. 186:481–487 (1973). · Zbl 0298.28015 · doi:10.1090/S0002-9947-1973-0335758-1
[17] D. Ruelle,Thermodynamic Formalism (Addison-Wesley, New York, 1978).
[18] D. Ruelle, The thermodynamic formalism for expanding maps,Commun. Math. Phys. 125:239–262 (1989). · Zbl 0702.58056 · doi:10.1007/BF01217908
[19] D. Ruelle, An extension of the theory of fredholm determinants,IHES 72:175–193 (1990). · Zbl 0732.47003
[20] Marek Rychlik, Regularity of the metric entropy for expanding maps,Trans. Am. Math. Soc. 315(2):833–847 (1989). · Zbl 0686.58023 · doi:10.1090/S0002-9947-1989-0958899-6
[21] H. Samelson, On the Perron-Frobenius theorem,Michigan Math. J. 4:57–59 (1956). · Zbl 0077.02303
[22] L.-S. Young, Decay of correlations for certain quadratic maps,Commun. Math. Phys. 146:123–138 (1992). · Zbl 0760.58030 · doi:10.1007/BF02099211
[23] K. Ziemian, On the Perron-Frobenius operator and limit theorems for some maps of an interval,Ergodic Theory and Related Topics II, Proceedings (Teubner, 1987), pp. 206–211.
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