The natural lexicographic semigroupoids associated with Cantor product spaces indexed by countable linear orders are classified. Applications are given to the classification of triangular operator algebras which are direct limits of upper-triangular matrix algebras.

We construct an example of a C∞ diffeomorphism on a 7-manifold which has an invariant set with an uncountable number of pseudocircle components. Furthermore, any diffeomorphism which is sufficiently close (in the C1 metric) to the constructed map has a similar invariant set. We also discuss the topological nature of the invariant set.

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.

We develop a new type of backward continued fractions that can be associated to each Hecke-type group. We study its symbolic dynamics, and the corresponding interval maps and their invariant measures. These measures are infinite if and only if the corresponding groups are discrete. For the discrete Hecke groups the invariant measure is computed explicitly by studying the geodesic flow on the associated Riemann surface.

For all masses, there are at least n − 2, O2-orbits of non-collinear planar central configurations. In particular, this estimate is valid even if the potential function is not a Morse function. If the potential function is a Morse function, then an improved lower bound, of the order of n! ln(n + 1/3)/2, can be given.

We investigate invariant ergodic measures for certain partially hyperbolic and Anosov actions of ℝk, ℤk and We show that they are either Haar measures or that every element of the action has zero metric entropy.

We construct two real Gaussian dynamical systems of zero entropy; the first one is not loosely Bernoulli, and the second is a loosely Bernoulli Gaussian-Kronecker system. To get loose-Bernoullicity for the second system, we prove and use a property of planar Brownian motion on [0, 1]: we can recover the whole trajectory knowing only some angles formed by the motion.

We consider a C2 non-renormalizable polymodal map with finitely many non-flat critical points of turning type and we prove that any minimal set of f has zero Lebesgue measure.

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.

Consider the partial differential equation f(x, y(x), dy(x)) = 0, where f is a smooth real function on ℝn × ℝ × (ℝn)*. Near each singularity of the characteristic foliation, a Liouville field is associated to the equation; we classify hyperbolic germs of Liouville fields up to symplectic transformations, hence we deduce normal forms for partial differential equations up to transformations which preserve the standard contact form of ℝ2n+1. For n = 1, a theorem of Davydov enables us to deduce normal forms for such equations up to transformations of the x, y plane.

We study the unfolding of a smooth vector-field X on ℝ3 having a homoclinic orbit to a hyperbolic equilibrium point with three real eigenvalues satisfying − λss < λs < 0 < λu We say that Γ is an inclination-flip homoclinic orbit if the extended unstable manifold at the equilibrium point is, along Γ, non-transverse to the stable manifold and that Γ is of weak type if the unstable manifold has a non-trivial intersection with a special C2 weak stable manifold of dimension one. In this paper, we show the existence of a strange attractor in the unfolding of an inclination-flip homoclinic orbit (of weak type) in the case where the divergence at the equilibrium point is negative. The crucial idea is to compare the Poincaré return map with the Hénon family: being close to 0.

We give a definition for a Julia set J(f) for generic classes of polynomial endomorphisms f: ℂn→ ℂn. For n = 1, our definition is equivalent to the usual one, which gives the points where the iterates of f do not form a normal family. Moreover, the Julia set J(f1 × … × fn) ⊂ ℂn for a product of one-dimensional polynomials fi: ℂ → ℂ turns out to be the product J(f1) × … × J(fn) of the associated Julia sets J(fi) ⊂ ℂ. For a special class of mappings f: ℂ2 → ℂ2 which is not of this simple type, the so-called Cantor skews, we investigate topological structure as well as measure theoretic aspects of the Julia sets obtained using our definition.

We give some topological ergodic theorems inspired by the Wiener-Wintner ergodic theorem. These theorems are used to give results for uniquely ergodic transformations and to study unique equilibrium states for shift maps. These latter results give random L2 ergodic theorems for a finite set of commuting measure-preserving transformations.

For several types of stable flows φ and representations ρ of the fundamental group of the underlying manifold, R-torsion for ρ can be computed from the periodic orbits of φ. However, there are counterexamples and the purpose of this paper is to describe the role of heteroclinic orbits in such counterexamples.

We describe several generic properties of homeomorphisms on compact manifolds. These properties concern the attractors of a homeomorphism, its chain recurrent set, and the periodic points.

Necessary and sufficient analytical conditions are given for homogeneous multiple Wiener-Itô integral processes (MWIs) to be mixing, and sufficient conditions are given for mixing of general square-integrable Gaussian-subordinated processes. It is shown that every finite or infinite sum Y of MWIs (i.e. every real square-integrable stationary polynomial form in the variables of an underlying weakly mixing Gaussian process) is mixing if the process defined separately by each homogeneous-order term is mixing, and that this condition is necessary for a large class of Gaussian-subordinated processes. Moreover, for homogeneous MWIs Y1, for sums of MWIs of order ≤ 3, and for a large class of square-integrable infinite sums Y1, of MWIs, mixing holds if and only if Y2 has correlation-function decaying to zero for large lags. Several examples of the criteria for mixing are given, including a second-order homogeneous MWI, i.e. a degree two polynomial form, orthogonal to all linear forms, which has auto-correlations tending to zero for large lags but is not mixing.

In this paper we estimate fractal dimensions of almost periodic orbits in terms of two kinds of exponents: the exponent in the inclusion lengths for ε-almost period and the exponent in Hölder conditions. Further, we estimate the inclusion lengths for ε-almost period of quasi-periodic functions by using Diophantine approximations. In the n-frequency quasi-periodic case we can show that the fractal dimension of its orbit is majorized by the value n/δ when it is Hölder continuous with exponent δ, 0 < δ ≤ 1.