Invariance Principles for Interval Maps with an Indifferent Fixed Point

In this note we establish an almost sure invariance principle for a large class of interval maps with indifferent fixed points, including the Manneville-Pomeau map. This implies a number of well-known corollaries, including the Weak Invariance Principle and the Law of the Interated Logarithm. 0. Introduction It is a classical problem in ergodic theory to understand the statistical properties of typical orbits. For example, the Birkhoff ergodic theorem describes the average behaviour of such orbits and the Central Limit Theorem describes the deviation from this average. These results are subsumed by more general invariance principles. The situation for uniformly hyperbolic systems is reasonably well understood. In this note, we shall study a particular class of non-uniformly hyperbolic systems. Let T : X → X be a continuous transformation of the interval X = [0, 1] preserving an absolutely continuous probability measure μ. Assume that T is expanding, except at an indifferent fixed point. More precisely, for 0 < α < 1, we consider the class Iα of C interval maps T : X → X, with a fixed point T (0) = 0, such that: (i) T ′(0) = 1; (ii) T ′(x) > 1 for 0 < x ≤ 1; (iii) there exists c 6= 0 such that limx↘0 T ′′(x)x1−α = c. Any transformation T ∈ Iα has an absolutely continuous invariant probability measure μ. A simple example is provided by the Manneville-Pomeau map Tα : [0, 1]→ [0, 1] defined by Tα(x) = x+ x (mod 1), for 0 < α < 1. Let φ : X → R be a Hölder continuous function with ∫ φdμ = 0. We say that φ is a coboundary if there exists u ∈ C(X,R) such that φ = u ◦T −u. We introduce the sequence φ(x) = φ(x) + φ(Tx) + . . .+ φ(Tn−1x), for each n ≥ 1. Under the hypothesis that 0 < α < 12 , Young [19] and Liverani, Saussol and Vaienti [14] established the Central Limit Theorem, i.e., provided φ is not a coboundary then lim n→∞ μ { x : φ(x) < √ nσt } = 1 √ 2π ∫ t −∞ e−u 2/2du, We are grateful to Carlangelo Liverani and Manfred Denker for useful comments. Typeset by AMS-TEX 1