Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

   
 
 

 

Stability of statistical properties in two-dimensional piecewise hyperbolic maps


Authors: Mark F. Demers and Carlangelo Liverani
Journal: Trans. Amer. Math. Soc. 360 (2008), 4777-4814
MSC (2000): Primary 37D50, 37D20, 37C30
DOI: https://doi.org/10.1090/S0002-9947-08-04464-4
Published electronically: April 8, 2008
MathSciNet review: 2403704
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the statistical properties of a piecewise smooth dynamical system by directly studying the action of the transfer operator on appropriate spaces of distributions. We accomplish such a program in the case of two-dimensional maps with uniformly bounded second derivative. For the class of systems at hand, we obtain a complete description of the SRB measures, their statistical properties and their stability with respect to many types of perturbations, including deterministic and random perturbations and holes.


References [Enhancements On Off] (What's this?)

  • [Ba] V.I. Bakhtin, A direct method for constructing an invariant measure on a hyperbolic attractor. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), 934-957; English transl., Russian Acad. Sci. Izv. Math. 41:2 (1993), 207-227. MR 1209028 (94d:58083)
  • [B1] V. Baladi, Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics, 16, World Scientific (2000). MR 1793194 (2001k:37035)
  • [B2] V. Baladi, Anisotropic Sobolev spaces and dynamical transfer operators: $ \mathcal{C}^\infty$ foliations, Algebraic and Topological Dynamics, Sergiy Kolyada, Yuri Manin and Tom Ward, eds. Contemporary Mathematics, Amer. Math. Society, (2005) 123-136. MR 2180233 (2007c:37022)
  • [BT] V. Baladi, M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier 57 (2007), 127-154. MR 2313087 (2008d:37034)
  • [BY] V. Baladi, L.-S. Young, On the spectra of randomly perturbed expanding maps, Comm. Math. Phys. 156:2 (1993), 355-385; 166:1 (1994), 219-220. MR 1233850 (94g:58172)
  • [BC] H. van den Bedem and N. Chernov, Expanding maps of an interval with holes, Ergod. Th. and Dynam. Sys. 22 (2002), 637-654. MR 1908547 (2003c:37046)
  • [BKL] M. Blank, G. Keller, C. Liverani, Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity 15:6 (2001), 1905-1973. MR 1938476 (2003m:37033)
  • [Bu] J. Buzzi, Absolutely continuous invariant probability measures for arbitrary expanding piecewise $ \mathbb{R}$-analytic mappings of the plane, Ergod. Th. and Dynam. Sys. 20:3 (2000), 697-708. MR 1764923 (2001h:37070)
  • [BK] J. Buzzi and G. Keller, Zeta functions and transfer operators for multidimensional piecewise affine and expanding maps, Ergod. Th. and Dynam. Sys. 21:3 (2001), 689-716. MR 1836427 (2002d:37036)
  • [C] N. N. Čencova, A natural invariant measure on Smale's horseshoe, Soviet Math. Dokl. 23 (1981), 87-91.
  • [CG] J.-R. Chazottes and S. Gouezel, On almost-sure versions of classical theorems for dynamical systems, Probability Theory and Related Fields 138 (2007), 195-234. MR 2288069 (2008a:60066)
  • [Ch] N. Chernov, Advanced statistical properties of dispersing billiards, J. Stat. Phys. 122 (2006), 1061-1094. MR 2219528 (2007h:37047)
  • [CD] N. Chernov, D.Dolgopyat, Brownian Brownian Motion - I, to appear in Memoirs of AMS.
  • [CM1] N. Chernov and R. Markarian, Ergodic properties of Anosov maps with rectangular holes, Bol. Soc. Bras. Mat. 28 (1997), 271-314. MR 1479505 (98k:58140)
  • [CM2] N. Chernov and R. Markarian, Anosov maps with rectangular holes. Nonergodic cases, Bol. Soc. Bras. Mat. 28 (1997), 315-342. MR 1479506 (99b:58142)
  • [CMT1] N. Chernov, R. Markarian and S. Troubetskoy, Conditionally invariant measures for Anosov maps with small holes, Ergod. Th. and Dynam. Sys. 18 (1998), 1049-1073. MR 1653291 (99k:58111)
  • [CMT2] N. Chernov, R. Markarian and S. Troubetskoy, Invariant measures for Anosov maps with small holes, Ergod. Th. and Dynam. Sys. 20 (2000), 1007-1044. MR 1779391 (2001f:37042)
  • [CY] N. Chernov and L.-S. Young, Decay of correlations for Lorentz gases and hard balls, in Hard Ball Systems and the Lorentz Gas, D. Szasz, ed., Enclyclopaedia of Mathematical Sciences 101, Springer-Verlag, Berlin, 2000, 89-120. MR 1805327 (2001k:37046)
  • [D1] M. Demers, Markov Extensions for Dynamical Systems with Holes: An Application to Expanding Maps of the Interval, Israel J. of Math. 146 (2005), 189-221. MR 2151600 (2006d:37062)
  • [D2] M. Demers, Markov Extensions and Conditionally Invariant Measures for Certain Logistic Maps with Small Holes, Ergod. Th. and Dynam. Sys. 25:4 (2005), 1139-1171. MR 2158400 (2007d:37057)
  • [DY] M. Demers and L.-S. Young, Escape rates and natural conditionally invariant measures, Nonlinearity 19 (2006), 377-397. MR 2199394 (2006i:37051)
  • [GL] S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergod. Th. and Dynam. Sys., 26, 1, 189-217 (2006). MR 2201945 (2007h:37037)
  • [HH] H. Hennion and L.Hevré, Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness, 1766, Lectures Notes in Mathematics, Springer-Verlag, Berlin, 2001. MR 1862393 (2002h:60146)
  • [K] G. Keller On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys. 96 (1984), no. 2, 181-193. MR 768254 (86k:58071)
  • [KL] G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Annali della Scuola Normale Superiore di Pisa, Scienze Fisiche e Matematiche (4) XXVIII (1999), 141-152. MR 1679080 (2000b:47030)
  • [LY] A. Lasota and J.A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1963), 481-488. MR 0335758 (49:538)
  • [L1] C. Liverani, Decay of Correlations, Annals of Mathematics 142 (1995), 239-301. MR 1343323 (96e:58090)
  • [L2] C. Liverani, Invariant measures and their properties. A functional analytic point of view, Dynamical Systems. Part II: Topological Geometrical and Ergodic Properties of Dynamics. Pubblicazioni della Classe di Scienze, Scuola Normale Superiore, Pisa. Centro di Ricerca Matematica ``Ennio De Giorgi'': Proceedings. Published by the Scuola Normale Superiore in Pisa (2004). MR 2071241 (2005d:37045)
  • [L3] C. Liverani, Fredholm determinants, Anosov maps and Ruelle resonances, Discrete and Continuous Dynamical Systems 13:5 (2005), 1203-1215. MR 2166265 (2006k:37047)
  • [LiM] C. Liverani and V. Maume-Deschamps, Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set, Annales de l'Institut Henri Poincaré Probability and Statistics 39 (2003), 385-412. MR 1978986 (2004c:37009)
  • [LW] C. Liverani and M. Wojtkowski, Ergodicity in Hamiltonian Systems, Dynamics Reported 4 (1995), 130-202. MR 1346498 (96g:58144)
  • [LM] A. Lopes and R. Markarian, Open billiards: cantor sets, invariant and conditionally invariant probabilities, SIAM J. Appl. Math. 56 (1996), 651-680. MR 1381665 (97a:58108)
  • [PP1] W. Parry and M. Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flows, Annals of Math. 118:3 (1983), 573-591. MR 727704 (85i:58105)
  • [PP2] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque No. 187-188 (1990), 268 pp. MR 1085356 (92f:58141)
  • [P] Ya.B.Pesin, Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties, Ergod. Th. and Dynam. Sys. 12 (1992), 123-151. MR 1162404 (93b:58095)
  • [Ru1] D. Ruelle, Locating resonances for Axiom A dynamical systems, J. Stat. Phys. 44:3-4 (1986), 281-292. MR 857060 (87k:58214)
  • [Ru2] D. Ruelle, Resonances for Axiom $ A$ flows, J. Differential Geom. 25:1 (1987), 99-116. MR 873457 (88j:58098)
  • [R1] H.H. Rugh, The correlation spectrum for hyperbolic analytic maps, Nonlinearity 5:6 (1992), 1237-1263. MR 1192517 (93i:58121)
  • [R2] H.H. Rugh, Fredholm determinants for real-analytic hyperbolic diffeomorphisms of surfaces. XIth International Congress of Mathematical Physics (Paris, 1994), 297-303, Internat. Press, Cambridge, MA, 1995. MR 1370685 (96k:58182)
  • [R3] H.H. Rugh, Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems. Ergod. Th. and Dynam. Sys. 16:4 (1996), 805-819. MR 1406435 (97j:58125)
  • [S] B. Saussol, Absolutely continuous invariant measures for multidimensional expanding maps, Israel J. Math. 116 (2000), 223-248. MR 1759406 (2001e:37037)
  • [T1] M. Tsujii, Absolutely continuous invariant measures for piecewise real-analytic expanding maps on the plane, Comm. Math. Phys. 208:3 (2000), 605-622. MR 1736328 (2000i:37034)
  • [T2] M. Tsujii, Absolutely continuous invariant measures for expanding piecewise linear maps, Invent. Math. 143:2 (2001), 349-373. MR 1835391 (2002g:37026)
  • [Y] L.-S. Young, Statistical properties of dynamical systems with some hyperbolicity, Annals of Math. 147:3 (1998), 585-650. MR 1637655 (99h:58140)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37D50, 37D20, 37C30

Retrieve articles in all journals with MSC (2000): 37D50, 37D20, 37C30


Additional Information

Mark F. Demers
Affiliation: Department of Mathematics, Fairfield University, Fairfield, Connecticut 06824
Email: mdemers@mail.fairfield.edu

Carlangelo Liverani
Affiliation: Dipartimento di Matematica, Università di Roma \sl Tor Vergata, Via della Ricerca Scientifica, 00133 Roma, Italy
Email: liverani@mat.uniroma2.it

DOI: https://doi.org/10.1090/S0002-9947-08-04464-4
Keywords: Hyperbolic, piecewise discontinuous, transfer operator, decay of correlations, open systems
Received by editor(s): July 17, 2006
Published electronically: April 8, 2008
Additional Notes: The authors would like to thank the Institut Henri Poincaré where part of this work was done (during the trimester Time at Work). Also the authors enjoyed partial support from M.I.U.R. (Cofin 05-06 PRIN 2004028108). The first author was partially supported by NSF VIGRE Grant DMS-0135290 and by the School of Mathematics of the Georgia Institute of Technology. Finally, the second author would like to warmly thank G. Keller with whom, several years ago, he had uncountably many discussions on these types of problems. Although we were not able to solve the problem at the time, as the technology was not ripe yet, the groundwork we did has been precious for the present work.
Article copyright: © Copyright 2008 by the authors

American Mathematical Society