Smooth Livšic regularity for piecewise expanding maps
Authors:
Matthew Nicol and Tomas Persson
Journal:
Proc. Amer. Math. Soc. 140 (2012), 905914
MSC (2010):
Primary 37D50, 37A20; Secondary 37A25
Published electronically:
July 11, 2011
MathSciNet review:
2869074
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Abstract: We consider the regularity of measurable solutions to the cohomological equation where is a dynamical system and is a smooth realvalued cocycle in the setting in which is a piecewise GibbsMarkov map, an affine transformation of the unit interval or more generally a piecewise uniformly expanding map of an interval. We show that under mild assumptions, bounded solutions possess versions. In particular we show that if is a transformation, then has a version, thus improving a result of Pollicott and Yuri.
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 1.
 J. Aaronson and M. Denker. Local limit theorems for partial sums of stationary sequences generated by GibbsMarkov maps, Stochast. Dynam. 1 (2001), 193237. MR 1840194 (2002h:37014)
 2.
 J. Aaronson, M. Denker, O. Sarig and R. Zweimüller, Aperiodicity of cocycles and stochastic properties of nonMarkov maps, Stoch. Dyn. 4 (2004), 3162. MR 2069366 (2005e:37015)
 3.
 V. Baladi, Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics 16, World Scientific Publishing, River Edge, NJ, 2000. MR 1793194 (2001k:37035)
 4.
 H. Bruin, M. Holland and M. Nicol. Livšic regularity for Markov systems, Ergod. Th. and Dynam. Sys. 25 (2005), 17391765. MR 2183291 (2006j:37036)
 5.
 R. de la Llave, J. M. Marco and E. Moriyon, Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomological equation, Annals of Math. (2) 123 (1986), 537611. MR 840722 (88h:58091)
 6.
 D. Fried, The flattrace asymptotics of a uniform system of contractions, Ergodic Theory Dynam. Systems 15 (1995), no. 6, 10611073. MR 1366308 (97c:58124)
 7.
 S. Gouëzel, Regularity of coboundaries for nonuniformly expanding maps, Proceedings of the American Mathematical Society 134:2 (2006), 391401. MR 2176007 (2006g:37006)
 8.
 F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z. 180 (1982), 119140. MR 656227 (83h:28028)
 9.
 F. Hofbauer and G. Keller, Equilibrium states for piecewise monotonic transformations, Ergodic Theory and Dynamical Systems 2 (1982), 2343. MR 684242 (85f:58069)
 10.
 O. Jenkinson. Smooth cocycle rigidity for expanding maps and an application to Mostow rigidity, Math. Proc. Camb. Phil. Soc. 132 (2002), 439452. MR 1891682 (2002k:37027)
 11.
 H. Keynes and D. Newton. Ergodic measures for nonabelian compact group extensions. Compositio Math. 32 (1976), 5370. MR 0400188 (53:4023)
 12.
 C. Liverani, Decay of correlations for piecewise expanding maps, journal of Statistical Physics 78 (1995), 11111129. MR 1315241 (96d:58077)
 13.
 A. N. Livšic. Cohomology of dynamical systems, Mathematics of the USSR Izvestija 6(6) (1972), 12781301. MR 0334287 (48:12606)
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 A. N. Livšic. Homology properties of Ysystems, Math. Notes 10 (1971), 758763. MR 0293669 (45:2746)
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 M. Nicol and A. Scott, Livšic theorems and stable ergodicity for group extensions of hyperbolic systems with discontinuities, Ergodic Theory and Dynamical Systems 23 (2003), 18671889. MR 2032492 (2004k:37051)
 16.
 M. Nicol and M. Pollicott, Measurable cocycle rigidity for some noncompact groups, Bull. Lond. Math. Soc. 31 (1999), 592600. MR 1703845 (2000k:37004)
 17.
 M. Nicol and M. Pollicott, Livšic theorems for semisimple Lie groups, Erg. Th. and Dyn. Sys. 21 (2001), 15011509. MR 1855844 (2002h:37048)
 18.
 M. Nicol and A. Scott, Livšic theorem and stable ergodicity for group extensions of hyperbolic systems with discontinuities, Ergod. Th. and Dyn. Sys. 23 (2003), 18671889. MR 2032492 (2004k:37051)
 19.
 M. Noorani, Ergodicity and weakmixing of homogeneous extensions of measurepreserving transformations with applications to Markov shifts, Monatsh. Math. 123 (1997), 149170. MR 1430502 (98g:28023)
 20.
 W. Parry and M. Pollicott, Zeta functions and closed orbits for hyperbolic systems, Astérisque (Soc. Math. France), 187188 (1990), 1268. MR 1085356 (92f:58141)
 21.
 W. Parry and M. Pollicott. The Livsic cocycle equation for compact Lie group extensions of hyperbolic systems. J. London Math. Soc. (2) 56 (1997), 405416. MR 1489146 (99d:58109)
 22.
 M. Pollicott and C. Walkden, Livsic theorems for connected Lie groups, Trans. Amer. Math. Soc. 353 (2001), 28792895. MR 1828477 (2002c:37045)
 23.
 M. Pollicott and M. Yuri, Regularity of solutions to the measurable Livsic equation, Trans. Amer. Math. Soc. 351 (1999), 559568. MR 1621702 (99d:58110)
 24.
 A. Scott, Livšic theorems and the stable ergodicity of compact group extensions of systems with some hyperbolicity, thesis, University of Surrey (2003).
 25.
 C. Walkden, Livsic theorems for hyperbolic flows, Trans. Amer. Math. Soc. 352 (2000), 12991313. MR 1637106 (2000j:37036)
 26.
 C. Walkden, Livsic regularity theorems for twisted cocycle equations over hyperbolic systems, J. London Math. Soc. 61 (2000), 286300. MR 1745387 (2001i:37048)
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Additional Information
Matthew Nicol
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 772043008
Email:
nicol@math.uh.edu
Tomas Persson
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, ulica Śniadeckich 8, P.O. Box 21, 00956 Warszawa, Poland
Address at time of publication:
Centre for Mathematical Sciences, Lund University, Box 118, 22 100 Lund, Sweden
Email:
tomasp@maths.lth.se
DOI:
http://dx.doi.org/10.1090/S000299392011109493
Received by editor(s):
July 23, 2010
Received by editor(s) in revised form:
December 14, 2010
Published electronically:
July 11, 2011
Additional Notes:
The second author was supported by EC FP6 Marie Curie ToK programme CODY
Communicated by:
Bryna Kra
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
