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Transactions of the American Mathematical Society

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



Regularity of solutions to
the measurable Livsic equation

Authors: M. Pollicott and M. Yuri
Journal: Trans. Amer. Math. Soc. 351 (1999), 559-568
MSC (1991): Primary 58Fxx
MathSciNet review: 1621702
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Abstract: In this note we give generalisations of Livsic's result that a priori measurable solutions to cocycle equations must in fact be more regular. We go beyond the original continuous hyperbolic examples of Livsic to consider examples of this phenomenon in the context of:

$\beta $-transformations;
rational maps; and
planar maps with indifferent periodic points.
Such examples are not immediately covered by Livsic's original approach either due to a lack of continuity or hyperbolicity.

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

  • 1. Sh.Ito and M.Yuri, Number theoretical transformations with finite range structure and their ergodic properties, Tokyo J.Math 10 (1987), 1-32. MR 88i:28031
  • 2. A. Livsic, Homology properties of $Y$-systems, Math. Zametki 10 (1971), 758-763.
  • 3. A. Livsic, Cohomology of dynamical systems, Math. USSR Izvestija 6 (1972), 1278-1301. MR 48:12606
  • 4. M. Lyubich, Ergodic properties of rationale endomorphisms of the Riemann sphere, Ergod. Theory and Dynam. Sys. 3 (1983), 351-385.
  • 5. W. Parry, Acta Math. Acad. Sci. Hungary 11 (1960), 401-416. MR 26:288
  • 6. W. Parry and M. Pollicott, Zeta functions and the closed orbit structure of hyperbolic systems, Asterisque 187-188 (1990), 1-267. MR 92f:58141
  • 7. F. Przytycki, On the Perron-Frobenius-Ruelle operator for rational maps on the Riemann sphere and for Hölder continuous functions, Bol. Bras. Math. Soc. 20 (1990), 95-125. MR 93b:58120
  • 8. S.Tanaka, A complex continued fraction and its ergodic properties, Tokyo J.Math. 8 (1985), 191-214. MR 87h:11073
  • 9. P. Walters, Equilibtium States for $\beta $-Transformations and Related Transformations, Math. Z. 159 (1978), 65-88. MR 57:6370
  • 10. M.Yuri, On a Bernoulli property for multi-dimensional maps with finite range structure, Tokyo J.Math 9 (1986), 457-485. MR 88d:28023
  • 11. M.Yuri, Invariant measures for certain multi-dimensional maps, Nonlinearity 7 (1994), 1093-1124. MR 95c:11101
  • 12. M.Yuri, Multi-dimensional maps with infinite invariant measures and countable state sofic shifts, Indagationes Mathematicae 6 (1995), 355-383. MR 96f:28021
  • 13. M.Yuri, On the convergence to equilibrium states for certain nonhyperbolic systems., Ergodic Theory and Dynam.Sys. 17 (1997), 977-1000. MR 98f:58155

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Additional Information

M. Pollicott
Affiliation: Department of Mathematics, Manchester University, Manchester, M13 9PL, England

M. Yuri
Affiliation: Department of Business Administration, Sapporo University, 3-jo, 7-chome, Nishioka, Toyohira-ku, Sapporo 062, Japan

Received by editor(s): August 5, 1996
Article copyright: © Copyright 1999 American Mathematical Society

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