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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
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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.


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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