Regularity of Solutions to the Measurable Livsic Equation

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) β-transformations; (b) rational maps; and (c) 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. 0. Introduction In 1972 Livsic showed that given a mixing subshift of finite type T : X → X and a Hölder continuous function c : X → R then any solution to the equation c(x) = u(Tx)− u(x) (1) with u : X → R measurable and essentially bounded (with respect to an equilibrium measure for a Hölder continuous function) must have a continuous version i.e. ∃u′ : X → R continuous such that u(x) = u′(x) a.e. [2], [3] This is an elementary type of “rigidity result”. Livsic also applied this result to Anosov systems. Another approach to this result for subshifts of finite type and using Perron-Frobenius type operators appeared in [6]. In this note we shall discuss some simple generalisations of Livsic’s result to other examples of dynamical systems. We shall be interested in systems with discontinuities or which are not uniformly hyperbolic and thus are not covered by Livsic’s original results. In this context the Perron-Frobenius type operator method adapts easily to prove the generalisations of Livsic’s original results. To illustrate this approach we shall apply this method to β-transformations (section 1), rational maps (section 2) and certain multi-dimensional maps (section 3). 1. β-transformation We begin by recalling the definition of the well-known β-transformation on the unit interval. Let β > 1 and define T : [0, 1) → [0, 1) by T (x) = βx (mod 1). Let ν denote Lebesgue measure on the interval [0, 1] and L([0, 1],B, ν) denotes the space of integrable functions on [0, 1]. Received by the editors August 5, 1996. 1991 Mathematics Subject Classification. Primary 58Fxx. c ©1999 American Mathematical Society 559 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 560 M. POLLICOTT AND M. YURI Definition. We define a Perron-Frobenius operator L1 : L([0, 1],B, ν) → L([0, 1],B, ν) by L1f(x) = ∑