Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Projective metrics and mixing properties on towers
HTML articles powered by AMS MathViewer

by Véronique Maume-Deschamps PDF
Trans. Amer. Math. Soc. 353 (2001), 3371-3389 Request permission

Abstract:

We study the decay of correlations for towers. Using Birkhoff’s projective metrics, we obtain a rate of mixing of the form: \[ c_n (f,g) \leq \operatorname {Ct} \alpha (n) \Vert f \Vert \, \Vert g \Vert _1\] where $\alpha (n)$ goes to zero in a way related to the asymptotic mass of upper floors, $\Vert f\Vert$ is some Lipschitz norm and $\Vert g \Vert _1$ is some $L^1$ norm. The fact that the dependence on $g$ is given by an $L^1$ norm is useful to study asymptotic laws of successive entrance times.
References
  • Jon Aaronson, Manfred Denker, and Mariusz Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), no. 2, 495–548. MR 1107025, DOI 10.1090/S0002-9947-1993-1107025-2
  • Viviane Baladi and Marcelo Viana, Strong stochastic stability and rate of mixing for unimodal maps, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 4, 483–517. MR 1386223
  • V. Baladi and L.-S. Young, On the spectra of randomly perturbed expanding maps, Comm. Math. Phys. 156 (1993), no. 2, 355–385. MR 1233850
  • M. BENEDICKS and L.-S. YOUNG Decay of correlations for certain Henon maps. (1996) preprint.
  • Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
  • P. Erdős and András Hajnal, On chromatic graphs, Mat. Lapok 18 (1967), 1–4 (Hungarian, with English summary). MR 227050
  • Jérôme Buzzi, Markov extensions for multi-dimensional dynamical systems, Israel J. Math. 112 (1999), 357–380. MR 1714974, DOI 10.1007/BF02773488
  • A.A. CASTRO Backward inducing and exponential decay of correlations for partially hyperbolic attractors with mostly contracting central direction. PhD Thesis (1998).
  • P. COLLET Statistics of closest return for some non uniformly hyperbolic systems. Preprint (1999)
  • P. Collet, A. Galves, and B. Schmitt, Unpredictability of the occurrence time of a long laminar period in a model of temporal intermittency, Ann. Inst. H. Poincaré Phys. Théor. 57 (1992), no. 3, 319–331 (English, with English and French summaries). MR 1185337
  • N. Chernov, Statistical properties of piecewise smooth hyperbolic systems in high dimensions, Discrete Contin. Dynam. Systems 5 (1999), no. 2, 425–448. MR 1665752, DOI 10.3934/dcds.1999.5.425
  • D. DOLGOPYAT On dynamics of mostly contracting diffeomorphisms. Preprint (1998).
  • P. FERRERO, B. SCHMITT Ruelle Perron Frobenius theorems and projective metrics. Colloque Math. Soc. J. Bolyai Random Fields. Estergom (Hungary) (1979).
  • P. FERRERO, B. SCHMITT On the rate of convergence for some limit ratio theorems related to endomorphisms with a non regular invariant density. Preprint (1994).
  • A. Galves and B. Schmitt, Inequalities for hitting times in mixing dynamical systems, Random Comput. Dynam. 5 (1997), no. 4, 337–347. MR 1483874
  • F. HOFBAUER On intrinsic ergodicity of piecewise monotonic transformations with positive entropy. Israel J. Math. (1979), 34, 1, 213-237; (1981), 38, 11, 107-115.
  • Gerhard Keller and Tomasz Nowicki, Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps, Comm. Math. Phys. 149 (1992), no. 1, 31–69. MR 1182410
  • Abdelaziz Kondah, Véronique Maume, and Bernard Schmitt, Vitesse de convergence vers l’état d’équilibre pour des dynamiques markoviennes non höldériennes, Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), no. 6, 675–695 (French, with English and French summaries). MR 1484537, DOI 10.1016/S0246-0203(97)80109-4
  • Carlangelo Liverani, Decay of correlations, Ann. of Math. (2) 142 (1995), no. 2, 239–301. MR 1343323, DOI 10.2307/2118636
  • Carlangelo Liverani, Central limit theorem for deterministic systems, International Conference on Dynamical Systems (Montevideo, 1995) Pitman Res. Notes Math. Ser., vol. 362, Longman, Harlow, 1996, pp. 56–75. MR 1460797
  • V. MAUME-DESCHAMPS Propriétés de mélange pour des systèmes dynamiques markoviens. PhD Thesis, Université de Bourgogne (1998), http://www.u-bourgogne.fr/monge/v.maume/accueil.html.
  • V. MAUME-DESCHAMPS Correlation decay for Markov maps on a countable state space. To appear in Erg. Th Dyn. Syst.
  • F. PACCAUT Statistics of return times for weighted maps of the interval Preprint (1999).
  • Omri M. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1565–1593. MR 1738951, DOI 10.1017/S0143385799146820
  • B. SAUSSOL Étude statistique de systèmes dynamiques dilatants. PhD. Thesis, Université de Toulon.
  • E. Seneta, Nonnegative matrices and Markov chains, 2nd ed., Springer Series in Statistics, Springer-Verlag, New York, 1981. MR 719544, DOI 10.1007/0-387-32792-4
  • M. VIANA Stochastic dynamics of deterministic systems. (1997).
  • Lai-Sang Young, Statistical properties of dynamical systems with some hyperbolicity, Ann. of Math. (2) 147 (1998), no. 3, 585–650. MR 1637655, DOI 10.2307/120960
  • L.-S. YOUNG Recurrence times and rates of mixing. Israel J. Math. 110 (1999), 153-188.
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37A25, 37C30, 37C40
  • Retrieve articles in all journals with MSC (2000): 37A25, 37C30, 37C40
Additional Information
  • Véronique Maume-Deschamps
  • Affiliation: Département de Mathématiques, Université de Genève, Geneva, Switzerland
  • Address at time of publication: Université de Bourgogne, B.P. 47870, 21078 Dijon Cedex, France
  • Email: vmaume@topolog.u-bourgogne.fr
  • Received by editor(s): May 23, 1999
  • Received by editor(s) in revised form: January 13, 2000
  • Published electronically: April 9, 2001
  • © Copyright 2001 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 3371-3389
  • MSC (2000): Primary 37A25, 37C30, 37C40
  • DOI: https://doi.org/10.1090/S0002-9947-01-02786-6
  • MathSciNet review: 1828610