Horizontal systems of rays arise in the study of integral curves of Hamiltonian systems $v_H$ on T*X, which are tangent to a given distribution V of hyperplanes on X. We investigate the local properties of systems of rays for general pairs (H,V) as well as for Hamiltonians H such that the corresponding Hamiltonian vector fields $v_H$ are horizontal with respect to V. As an example we explicitly calculate the space of horizontal geodesics and the corresponding systems of rays for the canonical distribution...

It is shown that the limit in an abstract version of Szegő's limit theorem can be expressed in terms of the antistable dynamics of the system. When the system dynamics are regular, it is shown that the limit equals the difference between the antistable Lyapunov exponents of the system and those of its inverse. In the general case, the elements of the dichotomy spectrum give lower and upper bounds.

Let X be a locally compact, separable metric space. We prove that $di{m}_{T}X=infdi{m}_L{X}^{\text{'}}:X^{\text{'}}ishomeomorphictoX$, where $di{m}_{L}X$ and $di{m}_{T}X$ stand for the concentration dimension and the topological dimension of X, respectively.

A finite-state stationary process is called (one- or two-sided) super-K if its (one- or two-sided) super-tail field-generated by keeping track of (initial or central) symbol counts as well as of arbitrarily remote names-is trivial. We prove that for every process (α,T) which has a direct Bernoulli factor there is a generating partition β whose one-sided super-tail equals the usual one-sided tail of β. Consequently, every K-process with a direct Bernoulli factor has a one-sided super-K generator....

Let $ℰ$ be a disjoint decomposition of ${ℝ}^n$ and let $X$ be a vector field on ${ℝ}^n$, defined to be linear on each cell of the decomposition $ℰ$. Under some natural assumptions, we show how to associate a semiflow to $X$ and prove that such semiflow belongs to the o-minimal structure ${ℝ}_{\mathrm{an},exp}$. In particular, when $X$ is a continuous vector field and $\Gamma$ is an invariant subset of $X$, our result implies that if $\Gamma$ is non-spiralling then the Poincaré first return map associated $\Gamma$ is also in ${ℝ}_{\mathrm{an},exp}$.

After giving an introduction to the procedure dubbed slow polynomial mating and quickly recalling known results about more classical notions of polynomial mating, we show conformally correct pictures of the slow mating of two degree $3$ post critically finite polynomials introduced by Shishikura and Tan Lei as an example of a non matable pair of polynomials without a Levy cycle. The pictures show a limit for the Julia sets, which seems to be related to the Julia set of a degree $6$ rational map. We...

Soit $F_0$ un difféomorphisme d’une surface possédant deux fers à cheval $\Lambda ,{\Lambda }^{\text{'}}$ tels que $W^s\Lambda$ et $W^u{\Lambda }^{\text{'}}$ aient en un point $q$ une tangence quadratique isolée. Nous montrons que, si la somme des dimensions transverses de $W^s\Lambda$ et $W^u{\Lambda }^{\text{'}}$ est strictement plus grande que 1, les difféomorphismes voisins de $F_0$ tels que $W^s\Lambda$ et $W^u{\Lambda }^{\text{'}}$ soient stablement tangents au voisinage de $q$ forment une partie de densité inférieure strictement positive en $F_0$.

We show that any transversally complete Riemannian foliation $ℱ$ of dimension one on any possibly non-compact manifold $M$ is tense; namely, $M$ admits a Riemannian metric such that the mean curvature form of $ℱ$ is basic. This is a partial generalization of a result of Domínguez, which says that any Riemannian foliation on any compact manifold is tense. Our proof is based on some results of Molino and Sergiescu, and it is simpler than the original proof by Domínguez. As an application, we generalize some...