A component U of the complement of the Julia set of an entire function ƒ is a wandering domain if the sets ƒn(U) are mutually disjoint, where n ∈ℕ and ƒn is the n-th iterate of ƒ. Examples are given of entire ƒ of order , which have multiply-connected wandering domains. An example is given where the connectivity is infinite.

Let δ be a closed *-derivation on a commutative C*-algebra , suppose that is dense in for some n = 1, 2, …, ∞, and let be a linear operator satisfying the locality condition

It is shown that , for some finite integer p≤n and functions lm on X. Estimates on the coefficients lm are obtained and applied to flows and local flows.

Let λ > 1 be a real eigenvalue of an automorphism of the two dimensional torus. We prove that for a dense, open subset of intervals the sequence where {x} denotes the fractional part of x and χ[a, b] the characteristic function of [a, b], satisfies the local limit theorem with respect to Lebesgue measure on [0, 1].

We define a new property of a Borel group action on a Lebesgue measure space, which we call approximate transitivity. Our main results are (i) a type III0 hyperfinite factor is ITPFI if and only if its flow of weights is approximately transitive, and (ii) for ergodic transformations preserving a finite measure, approximate transitivity implies zero entropy.

Periodic points of piecewise monotonic transformations are investigated using an oriented graph, whose closed paths represent the periodic orbits of the transformation. In the first part it is shown that the inverse of the ζ-function is a kind of characteristic polynomial of this graph, which is a power series if the graph is infinite. In the second part, the sets are determined, which can occur as{ with and where T is a monotonic mod one transformation.

The spaces of interval exchange maps and measured foliations are considered and an alternative proof that almost all interval exchange maps and measured foliations are uniquely ergodic is given. These spaces are endowed with a refinement process, called a simplicial system, which is studied abstractly and is shown to be normal under a simple assumption. The results follow and thus are a corollary of a more general theorem in a broader setting.

In this paper we prove that if the entropy of an ergodic measure preserved by a C2 surface diffeomorphism is positive then it is equal to the product of the Hausdorff dimension of the quotient measure defined by the family of stable manifolds and the positive Lyapunov exponent.

Two Cr, r ≥ 2, expanding maps of the circle which are absolutely continuously conjugate are Cr conjugate. Here ƒ and g: S1 → S1 are expanding if they stretch tangent vectors in some metric, and a conjugacy is an isomorphism h: S1 → S1 such that fh = hg.

The aim of this paper is to introduce and study the class of boundedly non-additive processes. The main result is the decomposition in theorem (2.1) and theorem (3.1), which says that a boundedly non-additive process is the sum of a non-positive subadditive, a non-negative superadditive and an additive process. By this decomposition we can extend the mean and local ergodic theorems for superadditive processes of M. A. Akcoglu and U. Krengel to boundedly non-additive processes. At the end of this paper some examples are given.

In this paper we study actions of lattices in semisimple groups preserving a G-structure of finite type.

This is an extended report on a problem session about geodesic flows held on May 30, 1984 at MSRI, Berkeley. The list of six authors reflects the principal participants in the discussion. Contributions of other participants are mentioned in the text. The session was chaired by A. Katok; this report was prepared by K. Burns in collaboration with A. Katok.

In our paper ‘Hausdorff dimension for horseshoes’ (H. McCluskey and A. Manning, Ergod. Th. & Dynam. Sys. (1983) 3, 251–260), § 3 entitled ‘Continuity across a bifurcation’ should be deleted since the proof of theorem 3 there is wrong.

The mistake is that does not in fact imply that δs → 1. This is because, for q = q0, the function is is only non-positive rather than negative and so is not strictly decreasing function of t.(Thus the pressure curve in figure 3 may, for q = q0, have become horizontal before t = l.) A similar mistake occurs in the second half of the same proof. Thus the claim of theorem 3 that the Hausdorff dimension of the basic set changes continuously in the bifurcation from Anosov to DA remains unproved. The rest of the paper is not affected.