We show that if π: ΣG → ΣH is a bounded-to-1 factor map from an irreducible shift of finite type ΣG with period pG to a shift of finite type ΣH with period pH, then there is a factor map that is (pG/pH)-to-1 almost everywhere. Moreover, if π is right closing, then may be taken to be right closing also.

Under a natural assumption the Hausdorff dimension of a measure μ canonically associated with a given self-affine set is computed. A simplified proof of Bowen's formula for the box dimension of self-affine sets proved earlier is given. A condition for the box dimension and Hausdorff dimension to be equal is proven, and a collection of examples in which this condition can be checked is discussed.

Let be an irreducible 0–1 matrix such that the non-zero entries in each row are consecutive. Let be the class of piecewise linear Markov transformations τ on [0, 1] into [0, 1] induced by for which the absolutely continuous invariant measure has maximal entropy. The main result presents necessary and sufficient slope conditions on τ which guarantee that τ ∈ .

We consider Anosov flows on a 5-dimensional smooth manifold V that possesses an invariant symplectic form (transverse to the flow) and a smooth invariant probability measure λ. Our main technical result is the following: If the Anosov foliations are C∞, then either (1) the manifold is a transversely locally symmetric space, i.e. there is a flow-invariant C∞ affine connection ∇ on V such that ∇R ≡ 0, where R is the curvature tensor of ∇, and the torsion tensor T only has nonzero component along the flow direction, or (2) its Oseledec decomposition extends to a C∞ splitting of TV (defined everywhere on V) and for any invariant ergodic measure μ, there exists χμ > 0 such that the Lyapunov exponents are −2χμ, −χμ, 0, χμ, and 2χμ, μ-almost everywhere.

As an application, we prove: Given a closed three-dimensional manifold of negative curvature, assume the horospheric foliations of its geodesic flow are C∞. Then, this flow is C∞ conjugate to the geodesic flow on a manifold of constant negative curvature.

This paper contains a study of attractors in cellular automata, particularly the minimal attractors as defined by J. Milnor. Milnor's definition of attractor uses a measure on the state space; the measures that we consider are Bernoulli product measures. Given a Bernoulli measure it is shown that a cellular automaton has at most one minimal attractor; when there is one, it is the omega-limit set of almost all points. Examples are given to show that the minimal attractor can change as the Bernoulli measure is varied. Other examples illustrate the difference between this result and the corresponding result that is obtained by replacing Milnor's definition of attractor by the purely topological definition used by C. Conley. The examples also show that certain invariant sets of cellular automata are less well-behaved than one might hope: for instance the periodic points are not necessarily dense in the nonwandering set.

Let p = {p1,  , pn} and q = {qi,  , qm} be finite probability vectors, each having at least three non-zero components, such that . Let C = {1,  , n}, D = {1,  , m} and let (Cℤ, pℤ, σ) and (Dℤ, qℤ, σ) be the corresponding Bernoulli shifts. Then there exists an isomorphism φ between these shifts such that for a.a. x ∈ Cℤ σ(x)(0) is determined by finitely many of the future co-ordinates x(0), x(1),   and for a. a.y ∈ Dℤφ−1(y)(0) is determined by finitely many of the co-ordinates   y(−1), y(0), y(1),  .

Our main results, specialized to unimodal interval maps T with negative Schwarzian derivative, are the following:

(1) There is a set CT such that the ω-limit of Lebesgue-a.e. point equals CT. CT is a finite union of closed intervals or it coincides with the closure of the critical orbit.

(2) There is a constant λT such that for Lebesgue-a.e. x.

(3) λT > 0 if and only if T has an absolutely continuous invariant measure of positive entropy.

(4) , i.e. uniform hyperbolicity on periodic points implies λT > 0, and λT < 0 implies the existence of a stable periodic orbit.

Andrés del Junco has proposed a definition of topological minimal self-joinings intended to parallel Dan Rudolph's measure-theoretic concept. By means of a rank-two ‘cutting and stacking’, this article constructs the first example of a system (a subshift) satisfying his proposed definition of 2-fold topological minimal self-joinings.

The second part of the article shows that 2-fold topological minimal self-joinings does not imply 3-fold and that no map has 4-fold topological minimal self-joinings. This latter result follows from a generalization of a theorem of Schwartzman.

For ergodic group extensions of transformations with discrete spectra it is proved that each invariant sub-σ-algebra is determined by a compact subgroup in the centralizer of a natural factor.

It is shown that the system of the form x + V′ (x) = p (t) with periodic V and p and with (p) = 0 is near-integrable for large energies. In particular, most (in the sense of Lebesgue measure) fast solutions are quasiperiodic, provided V ∈ C(5) and p ∈ L1; furthermore, for any solution x(t) there exists a velocity bound c for all time: |x(t)| < c for all t ∈ R. For any real number r there exists a solution with that average velocity, and when r is rational, this solution can be chosen to be periodic.

The statement and proof of the ℤ-spectral decomposition theorem for a pseudo-Anosov flow φ on a 3-manifold M are in error (see [1]). There are counter-examples which show that the theorem as stated is false. We remark that the results of § 9 of [1], concerning analogues of the ℤ-spectral decomposition theorem for basic sets of Axiom A flows, are unaffected by this error.

The main result of this paper is a construction of geometric Lorenz attractors (as axiomatically defined by J. Guckenheimer) by means of an Ω-explosion. The unperturbed vector field on ℝ3 is assumed to have a hyperbolic fixed point, whose eigenvalues satisfy the inequalities λ1 > 0, λ2 > 0, λ3 > 0 and |λ2|>|λ1|>|λ3|. Moreover, the unstable manifold of the fixed point is supposed to form a double loop. Under some other natural assumptions a generic two-parameter family containing the unperturbed vector field contains geometric Lorenz attractors.

A possible application of this result is a method of proving the existence of geometric Lorenz attractors in concrete families of differential equations. A detailed discussion of the method is in preparation and will be published as Part II.

In this note we answer a question raised by Friedland and Milnor in [FM] concerning the topological entropy of polynomial diffeomorphisms of C2.