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

DOI:
https://doi.org/10.1090/S0002-9947-99-02383-1

MathSciNet review:
1621702

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Abstract | References | Similar Articles | Additional Information

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:

- (a)
- -transformations;
- (b)
- rational maps; and
- (c)
- planar maps with indifferent periodic points.

**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 -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 -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

Email:
mp@ma.man.ac.uk

**M. Yuri**

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

Email:
yuri@math.sci.hokudai.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-99-02383-1

Received by editor(s):
August 5, 1996

Article copyright:
© Copyright 1999
American Mathematical Society