This paper establishes a version of the Chebotarev density theorem in which number fields are replaced by 3-manifolds.

Given a 4-manifold, we build a non-empty ${C}^{1} $-open set of vector fields having a (chain transitive) attractor containing singularities of different indices. Then, we begin the study of the hyperbolic properties of such a robust singular attractor.

For impulsive differential equations, we construct topological conjugacies between linear and nonlinear perturbations of non-uniform exponential dichotomies. In the case of linear perturbations, the topological conjugacies are constructed in a more or less explicit manner. In the nonlinear case, we obtain an appropriate version of the Grobman–Hartman Theorem for impulsive equations, with a simple and direct proof that involves no discretization of the dynamics.

In [Kelmer, Scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms, Comm. Math. Phys. 276 (2007), 381–395] we introduced a family of symplectic maps of the torus whose quantization exhibits scarring on invariant co-isotropic submanifolds. The purpose of this note is to show that in contrast to other examples, where failure of quantum unique ergodicity is attributed to high multiplicities in the spectrum, for these examples the spectrum is (generically) simple.

We obtain an asymptotic formula for the number of pairs of closed orbits of a weak-mixing transitive Anosov flow whose homology classes have a fixed difference.

We prove that if f is a C1-generic symplectic diffeomorphism then the Oseledets splitting along almost every orbit is either trivial or partially hyperbolic. In addition, if f is not Anosov then all the exponents in the centre bundle vanish. This establishes in full a result announced by Mañé at the International Congress of Mathematicians in 1983. The main technical novelty is a probabilistic method for the construction of perturbations, using random walks.

We consider a simple model to describe the widths of the mode-locked intervals for the critical circle map. By using two different partitions of the rational numbers based on Farey series and Farey tree levels, respectively, we calculate the free energy analytically at selected points for each partition. It emerges that the result of the calculation depends on the method of partition. An implication of this finding is that the generalized dimensions Dq are different for the two types of partition except when q=0; that is, only the Hausdorff dimension is the same in both cases.

We prove that if f:I=[0,1]→I is a C3-map with negative Schwarzian derivative, nonflat critical points and without wild attractors, then exactly one of the following alternatives must occur: (i) R(f) has full Lebesgue measure λ; (ii) both S(f) and Scramb(f) have positive measure. Here R(f), S(f), and Scramb(f) respectively stand for the set of approximately periodic points of f, the set of sensitive points to the initial conditions of f, and the two-dimensional set of points (x,y) such that {x,y} is a scrambled set for f.Also, we show that if f is piecewise monotone and has no wandering intervals, then either λ(R(f))=1 or λ(S(f))>0, and provide examples of maps g,h of this type satisfying S(g)=S(h)=I such that, on the one hand, λ(R(g))=0and λ2 (Scramb (g))=0 , and, on the other hand, λ(R(h))=1 .