Regularity of solutions to the measurable Livsic equation

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) 0-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 H6lder continuous function c: X -* IR then any solution to the equation (1) c(x) u(Tx) u(x) with u: X ---1iR measurable and essentially bounded (with respect to an equilibrium measure for a H6lder continuous function) must have a continuous version i.e. eu': 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 3-transformations (section 1), rational maps (section 2) and certain multi-dimensional maps (section 3). 1. }5-TRANSFORMATION We begin by recalling the definition of the well-known 3-transformation on the unit interval. Let / > 1 and define T: [0,1) -* [0,1) by T(x) = fx (mod 1). Let v denote Lebesgue measure on the interval [0,1] and Ll ([0, 1], 13, v) denotes the space of integrable functions on [0,1]. Received by the editors August 5, 1996. 1991 Mathematics Subject Classification. Primary 58Fxx. (?)1999 American Mathematical Society 559 This content downloaded from 157.55.39.255 on Wed, 25 May 2016 04:56:50 UTC All use subject to http://about.jstor.org/terms 560 M. POLLICOTT AND M. YURI Definition. We define a Perron-Frobenius operator Li : Ll ([O, 1], 6,B v) --Ll ([O, 1], 6,B v) by Ty= 13 where the summation is over the [,3] or [a1] + 1 pre-images of x. We denote by BBV the space of functions of bounded variation i.e. if f E BBV then we have that var(f) :=sup{ZEif(Xi+1)-f(xi)I: O xoX 0 for x E [0,1). We shall also need the following more general operator. Definition. Given a Lipschitz function c: [0,11 --? R. We define a c-weighted Perron-Frobenius operator L2: L' ([O, 1], B, v) -* L1 ([O, 1], 13, v) by L2 f (x) = E ecp f (y) TY=X (When c= 0 then L2 clearly reduces to Lp.) It is convenient to denote cn (x) = c(x) + c(Tx) +.. + c(Tn lx) for n > 1. The following result is important in proving Theorem 1. Lemma 2. (i) Vf E L1([0, 1], B, v) we have I2nf h(f fdv) in the L1 norm, (ii) Let c: [0,11 -? IR be a Lipschitz function. There exists w E BBV with w > 0 and A > 0 such that the sequence of functions An (T-y=x BBV. The convergence Ijf -* h(J f dv), as n -4 +oo is proved in [9, Theorem 10] when f is uniformly continuous. Since any function in L1 ([O,1], B, v) can be approximated from above and below by such functions, this is sufficient. This content downloaded from 157.55.39.255 on Wed, 25 May 2016 04:56:50 UTC All use subject to http://about.jstor.org/terms REGULARITY OF SOLUTIONS l'O THE MEASURABLE LIVSIC EQUATION 561 (ii) The operator L2: BBV -BBV has a simple maximal eigenvalue A, say, with positive eigenvector w E BBV ([9, Lemma 9]). Moreover, there exists a measure ,t on [0, 1] such that fn =~ --> w f dti, as n +oo [9, Theorem 10]. 1] Our main result on the measurable Livsic equation for fl-transformations is the following. Theorem 1. If c: [0, 1t -* IR is Lipschitz then for any measurable and essentially bounded solution u to (1') c(x) = u(Tx) u(x) there exists uo(x) = v(x) log h(x), where v(x) E BBV such that u == uo a.e. (nt). Proof. We can make the choice f 1 and then iterating the operator 12 introduced above we can write (L n1)(X)= ASn fn (X) Tny3/X In particular, substituting from the identity (1') we get that f Ix eCn(y) fn (x) An E , n