Let (α, u) denote a cocycle-crossed action of a discrete group G on a unital C*-algebra A, and B = C*r (A, G, α, u) the associated reduced (twisted) C*-crossed product. We discuss the following problem: when is B a simple C*-algebra with a unique trace? To this aim, we introduce the concept of tracially properly outerness of actions.

A homoclinic orbit is considered for which the center-stable and center-unstable manifolds of a saddle-node equilibrium have a quadratic tangency. This bifurcation is of codimension two and leads generically to the creation of a bifurcation curve defining two independent transverse homoclinic orbits to a saddle-node. This latter case was shown by Shilnikov to imply shift dynamics. It is proved here that in a large open parameter region of the codimension-two singularity, the dynamics are completely described by a perturbation of the Hénon-map giving strange attractors, Newhouse sinks and the creation of the shift dynamics. In addition, an example system admitting this bifurcation is constructed and numerical computations are performed on it.

Linear mod one transformations are the maps of the unit interval given by fβα(x) = βx + α (mod 1), with β > 1 and 0 ≤ α < 1. The lap-counting function is the function where the lap number Ln essentially counts the number of monotonic pieces of the nth iterate . We derive an explicit factorization formula for Lβα(z) which directly shows that Lβα(z) is a function meromorphic in the open unit disk {z: |z| < 1} and analytic in the open disk {z: |z| < 1/β}, with a simple pole at z = 1/β.Comparison with a known formula for the Artin—Mazur—Ruelle zeta function ζβ,α(z) of fβα shows that Lβα(z) and ζβ,α(z) have identical sets of singularities in the disk {z: |z| < 1}. We derive two more factorization formulae for Lβ,α(z) valid for certain parameter ranges of (β, α). When 1 < α + β ≤ 2, there is sometimes a ‘renormalization’ structure of such maps present, which has previously been studied in connection with simplified models for the Lorenz attractor. In the case that fβα is non-trivially renormalizable, we obtain a factorization formula for Lβα(z). For (β, α) in a region contained in 2 < α + β ≤ 3 we obtain a factorization formula which relates Lβα(z) to a ‘rescaled’ lap-counting function from the region 1 < α + β ≤ 2. The various factorizations exhibit certain singularities of Lβα(z) on the circle |z| = 1/β. These singularities are related to topological dynamical properties of fβ,α. In parts II and III we show that these comprise the complete set of such singularities on the circle |z| = 1/β.

We construct linked twist maps of the two-dimensional torus which are Bernoulli and possess infinite entropy. Inparticular, we construct a Bernoulli toral linked twist map B* of infinite entropy which has smooth, absolutely continuous local (un)stable manifolds and positive Lyapunov exponents defined almost everywhere. This map is continuous at each point save those on two line segments.

B* is shown to be stochastically stable under the following random perturbation: apply the map to a point p and then jump (all points move the same distance and in the same direction) according to a B-process (not necessarily an independent process) such that the expected distance moved is equal to r. Stochastic stability means that given α > 0 if r > 0 is sufficiently small then the perturbed and unperturbed systems are α-congruent. We prove a similar stability result for B* under a perturbation in which the random jump described above is distributed according to a general stochastic process. These stability results are also shown to hold (in a slightly modified form) for a general class of finite-entropy toral linked twist maps under the same perturbations.

We consider a class of skew-products with base the one-sided two-shift equipped with the (½, ½)Bernoulli measure and fibre either or the circle. We give conditions for the first kind to be isomorphic to the base itself and use this result to establish an isomorphism with the base for the second kind when a certain irrationalis very well approximated by rationals. A consequence is that there is a circle action of automorphisms commuting with theone-sided two-shift.

We introduce a procedure for finding C1 Lebesgue measure-preserving maps of the circle isomorphic to one-sided shifts equipped with certain invariant probability measures. We use this to construct a C1 expanding map of the circle which preserves Lebesgue measure, but for which Lebesgue measure is non-ergodic (that is there is more than one absolutely continuous invariant measure). This is in contrast with results for C1+e maps. We also show that this example answers in the negative a question of Keane's on uniqueness of g-measures, which in turn is based on a question raised by an incomplete proof of Karlin's dating back to 1953.

We show that near the geodesic flow of a compact Riemannian manifold with no conjugate points which is expansive, every expansive geodesic flow has no conjugate points. We also prove that in the above hypotheses the geodesic flow istopologically stable.

We consider the one-parameter family of quadratic polynomials:

i.e. monic centered quadratic polynomials with an indifferent fixed point αt and prefixed point −αt. Let At, be any one of the sets {0, ±αt}, {±αt}, {0, αt}, or {0, −αt}. Then we prove that for quadratic Julia sets corresponding to a Gδ-dense subset of there is an explicitly given external ray accumulating on At. In the case At = {±αt} the theorem is known as the Douady accumulation theorem.

Corollaries are the non-local connectivity of these Julia sets and the fact that all such Julia sets contain a Cremer point. Existence of non-locally connected quadratic Julia sets of Hausdorff dimension two is derived by using a recent result of Shishikura. By tuning, the results hold on the boundary of any hyperbolic component of the Mandelbrot set.

Finally, we concentrate on quadratic Cremer point polynomials. Here we prove that any ray accumulating on two symmetrical points of the Julia set must accumulate the origin. As a consequence, the dense Gδsets arising from the first two possible choices of At are the same. We also prove that, if two distinct rays accumulate both to two distinct points, then the rays must accumulate on a common continuum joining the two points. This supports the conjecture that αt and –αt may be joined by an arc in the Julia set.

This paper is a contribution to the classification problem of expansive homeomorphisms. Let M be a compact connected oriented three dimensional topological manifold without boundary and f: M → M an expansive homeomorphism.We show that if the topologically hyperbolic period points of f are dense in M then M = , and f is conjugate to an Anosov diffeomorphism. This follows from our basic result: for such a homeomorphism, all stable and unstable sets are (tamely embedded) topological manifolds.