Ergodic theory for inner functions of the upper half plane

The real restriction of an inner function of the upper half plane leaves Lebesgue measure quasi-invariant. It may have a finite or infinite invariant measure. We give conditions for the rational ergodicity and exactness of such restrictions. ABSTRAIT. 2014 La restriction a la droite réelle d’une fonction intérieure du demi-plan supérieur laisse la mesure de Lebesgue quasi-invariante, et peut avoir une mesure invariante finie ou infinie. Nous donnons les conditions pour l’ergodicité rationnelle et l’exactitude de telles transformations. § 0. INTRODUCTION In this paper, we consider the ergodic properties of the real restrictions of inner functions on the open upper half plane : Let f : f~2 + ~ l~2 + be an analytic function. We say that f is an inner function on f~2 + if for ~,-a. e. x the limit li m f(x + iy) exists, and is y~o real. (Here, and throughout the paper, /). denotes Lebesgue measure on R). Consider the limit lim f(x + i y) = Tx. This is defined 03BB-a. e. on R. We call this limit the (real) restriction of f, and will sometimes write this as T = T( f ). Annales de l’Institut Henri Poincaré Section B Vol. XIV, n° 3 1978. 234 J. AARONSON We will denote the class of inner functions on (~2 + by I((~2 + ) = I, and their real restrictions by M(R). We note that f E iff ~-1 f ~(z) is an inner function of the unit disc, according to the definition on p. 370 of [9] (where ~(z) = L 1 -z The following characterisation of I{1R2+) appears in [6] and [17]. f E I(~2 + ) iff where a > 0, ~3 E f~ and ,u is a bounded, positive Borel measure, singular w. r. t. ~,. Since we shall be refering to (o .1) rather a lot, we shall denote the class of bounded, positive, singular measures on f~ by G. Letac ([6]) has shown that a measurable transformation T of (~ preserves the class of Cauchy distributions iff either T E or T E In particular, if dPa + ib(x) = b 03C0 dx 03C0(x a)2 + b2 for a + ib E and T = T( ) E then: ( ) + This equation shows that M(R) is a class of non-singular transformations of the measure space (R, B, 2), and is therefore an object of ergodic theory. Let f E I((~2 +) have a fixed point c~o E f~2 +. By (0.2), T( f ) preserves the Cauchy distribution It was shown in [16], that if f is 1 1, then T( f ) is conjugate to a rotation of the circle, and shown in that otherwise, T( f ) is mixing. We show in § 1 that if f is not 1 1 then T( f ) is exact. In § 2 we recall some well known facts about inner functions of f~2 +-. The Denjoy-Wolff theorem (see [13], [l4] and [18]) adapted to 1R2+ shows that when studying the ergodic properties of T( f ), for f E I(f1~2 +) with no fixed points in I~2 ±, we may assume that a( f ) >1. In case a( f ) > 1, T( f ) is dissipative, and when a( f ) = 1, T( f ) preserves Lebesgue measure. In § 3, we consider the case a( f ) = 1. Here, the conservativity of a restriction T( f ) is sufficient for its rational ergodicity ([l]) (ergodicity was established in [15] ). We also give sufficient conditions for exactness, and discuss the similarity classes ( [I] ) of restrictions. The ergodic theory of certain restrictions has been considered in [2], [5], [7], [IO], [11], and [l6]. The author would like to thank B. Weiss for helpful conversations, and G. Letac, J. Neuwirth and F. Schweiger for making preprints of their works available. Annales de l’Institut Henri Poincaré Section B 235 INNER FUNCTIONS OF THE UPPER HALF PLANE § 1. MIXING RESTRICTIONS PRESERVING FINITE MEASURES The purpose of this section is to prove THEOREM 1.1. Let f E +) and assume that f is not 1 1. If f has a fixed point then (R, T( f )) is an exact measure preserving transformation. Before proving theorem 1.1, we shall need some auxiliary results. The first of these is Lin’s criterion for exactness of Markov operators (theorem 4.4 in [8]) as applied to our case. To state this, we shall need some extra notation : Let T E then (IR, B, ~,, T) is a non-singular transformation, and so ~, ~,) iff g o T E ~, ~,). We define the dual operator of T, T : L 1 ( I~, B, À) --~ B, À) by If we write, for cc~ = a + ib E 1R2 + then equation (0.2) translates to : Clearly, t is a positive linear operator, R hd03BB = R hd03BB for Lin’s Criterion (for restrictions). 2014 Let T = T(/)eM(tR). T is exact iH’ (1.2) ~ 0 for every R ud03BB = 0. Here, and throughout, ~u~1 = R|u|d03BB. We shall also need the following (elementary) lemma. Vol. XIV, n~ 3-1978. 236 J. AARONSON LEMMA 1. 2. If cv" E f~2 + and cvn -~ cv E (1~2 + then : Proof of theorem 1.1. 2014 We first show that f "(cv) ~ a~o where = f(w) and = Let § : U = [ ~ Z ~ 1] R2 + be a conformal map. Then g = ~ lf ~ : U --> U is analytic, and g(~(cc~o)) _ ~(a~o). By the Schwartz lemma ([9]): I g’(~(a~o)) ~ 1 as g is not 1 1. It is now not hard to see that and hence that -~ mo Hence, by lemma 1.2 We will now establish that with = 0 which, by Lin’s criterion, will ensure the exactness of T. Let u E L 1 with = 0 and let E > 0. By Wiener’s Tauberian theo. rem (see [l2], p. 357), there exist al . aN, al . such that N Clearly, this implies that a~ E/2 and so : i= 1 Since 8 > 0 was arbitrary : [ 1 -~ 0. D Annales de l’Institut Henri Poincaré Section B 237 INNER FUNCTIONS OF THE UPPER HALF PLANE § 2. BASIC CLASSIFICATION PROPOSITION 2.1. [l7]. Let f E I(1R2+). Then Moreover Proof From the representation 0.1, we immediately calculate that : It follows from elementary integration theory that To check the limit as b 0, we « flip » f to get : Since f E I((1~2 + ), we have that but this decodes to : Now, if y( f ) oo then, by 2.1 : Hence y( f ) > a( f ) with equality iff --_ 0. D _ PROPOSITION 2.2. Let f E I((1~2 +) and T = T( f ). If a( f ) > 1 then T is dissipative. Proof 2014 Write + Vol. XIV, n° 3 1978. 238 J. AARONSON From the representation (0.1), we have : Hence v"(i ) >_ oc" for n > 1, and Clearly