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 Julia set Jλ of the complex exponential function Eλ: z → λez for a real parameter λ(0 < λ < 1/e) is known to be a Cantor bouquet of rays extending from the set Aλ of endpoints of Jλ to ∞. Since Aλ contains all the repelling periodic points of Eλ, it follows that Jλ = Cl (Aλ). We show that Aλ is a totally disconnected subspace of the complex plane ℂ, but if the point at ∞ is added, then is a connected subspace of the Riemann sphere . As a corollary, Aλ has topological dimension 1. Thus, ∞ is an explosion point in the topological sense for Âλ. It is remarkable that a space with an explosion point occurs ‘naturally’ in this way.

We prove the decidability of shift equivalence of sofic systems and discuss algebraic invariants.

We consider a dynamical system consisting of a compact subset of RN or CN with several contracting maps chosen with prescribed probabilities, which may depend on position. We show that if the maps and the probabilities are Cl+α functions of the spatial variable and an external parameter, then the average value of a Cl+α function is a differentiate function of the parameter. One implication of this theorem is that for certain families of complex functions dependent on a parameter the reciprocal of the dimension of an invariant measure on the Julia set is a harmonic function of the parameter.

We study the behavior of the twist coefficient near an elliptic fixed point for a one-parameter family of area-preserving diffeomorphisms. By looking at the singularities near resonance we can explain the sign changes which are typically found in such a family.

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.

Let p and q be relatively prime natural numbers. Define T0 and S0 to be multiplication by p and q (mod 1) respectively, endomorphisms of [0,1).

Let μ be a borel measure invariant for both T0 and S0 and ergodic for the semigroup they generate. We show that if μ is not Lebesgue measure, then with respect to μ both T0 and S0 have entropy zero. Equivalently, both T0 and S0 are μ-almost surely invertible.

We consider the crossed product or transformation group C*-algebras arising from actions of the group of integers on a totally disconnected compact metrizable space. Under a mild hypothesis, we give a necessary and sufficient dynamical condition for the invertibles in such a C*-algebra to be dense. We also examine the property of residual finiteness for such C*-algebras.

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