ar X iv : m at h / 04 08 20 5 v 1 [ m at h . D S ] 1 6 A ug 2 00 4 ON THE MIXING COEFFICIENTS OF PIECEWISE MONOTONIC MAPS

We investigate the mixing coefficients of interval maps satisfying Rychlik’s conditions. A mixing Lasota-Yorke map is reverse φ-mixing. If its invariant density is uniformly bounded away from 0, it is φ-mixing iff all images of all orders are big in which case it is ψ-mixing. Among β-transformations, non-φ-mixing is generic. In this sense, the asymmetry of φ-mixing is natural. §0 Introduction Piecewise monotonic maps. A non-singular, piecewise monotonic map (PM map) of the interval X := [0, 1] is denoted (X,T, α) where α is a finite or countable collection of open subintervals of X which is a partition in the sense that ⋃ a∈α a = X mod m (where m is Lebesgue measure) and T : X → X is a map such that T |A is absolutely continuous, strictly monotonic for each A ∈ α. Let (X,T, α) be a PM map. For each n ≥ 1, (X,T , αn) is also an PM map, where αn := {[a0, . . . , an−1] := n−1 ⋂ k=0 Tak : a0, . . . , an−1 ∈ α}. A PM map (X,T, α) satisfies m ◦ T−1 ≪ m, whence f ∈ L∞(m) ⇒ f ◦ T ∈ L∞(m) . Let T̂ : L(m) → L(m) be the predual of f 7→ f ◦ T , then T̂ g = ∑ a∈αn v′ a1Tnag ◦ va where va : T a → a is given by va := (T n|a)−1. Under certain additional assumptions (see below), ∃ h ∈ L(m), h ≥ 0, ∫ X hdm = 1, so that T̂ h = h, i.e. dP = hdm is an absolutely continuous, T -invariant probability (a.c.i.p.). Mixing and measures of dependence between σ-algebras. A mixing property of a stationary stochastic process (. . . , X−1, X0, X1, . . . ) reflects a decay of the statistical dependence between the past σ-algebra σ({Xk : k ≤ 0}) and the asymptotic future σ-algebra σ({Xk : k ≥ n}) as n → ∞ and the various mixing properties are described by corresponding measures of dependence between σ-algebras (see [Br]). 1991 Mathematics Subject Classification. 37A25, 37E05 (37D20, 37C30, 37C40, 60G10).