We investigate the dynamics of substitution subshifts and their associated tiling spaces. For a given subshift, the associated tiling spaces are all homeomorphic, but their dynamical properties may differ. We give criteria for such a tiling space to be weakly mixing, and for the dynamics of two such spaces to be topologically conjugate.

We prove that homoclinic classes for a residual set of C^1 vector fields X on closed n-manifolds are maximal transitive, and depend continuously on periodic orbit data. In addition, X does not exhibit cycles formed by homoclinic classes. We also prove that a homoclinic class of X is isolated if and only if it is \Omega-isolated, and it is the intersection of its stable set with its unstable set. All these properties are well known for structurally stable Axiom A vector fields.

For a compact Riemannian manifold of negative curvature, the geodesic foliation of its unit tangent bundle is independent of the negatively curved metric, up to Hölder bicontinuous homeomorphism. However, the Riemannian metric defines a natural transverse measure to this foliation, the Liouville transverse measure, which does depend on the metric. For a surface S, we show that the map which to a hyperbolic metric on S associates its Liouville transverse measure is differentiable, in an appropriate sense. Its tangent map is valued in the space of transverse Hölder distributions for the geodesic foliation.

This paper shows that any symplectic diffeomorphism $\mathbf{F} :(\Sigma^{2n},\eta) \to (\Sigma^{2n},\eta)$ can be embedded as a subsystem of a Liouville-integrable Hamiltonian flow on some symplectic manifold. If F is real-analytic, then the flow can be chosen to be real-analytic, but it is Liouville integrable with smooth first integrals. Examples are constructed of integrable, volume-preserving Hamiltonian flows on Poisson manifolds whose metric entropy with respect to the volume form is positive. Completely integrable Hamiltonian flows on a symplectic manifold are constructed which have positive metric entropy with respect to an invariant probability measure that is absolutely continuous with respect to the canonical volume form.

We study the convergence of pointwise ergodic means for random subsequences, in a universal framework, together with ergodic means which are modulated by random weights. The methods used in this work mainly involve Gaussian tools, transference principles and new results on oscillation functions.

We investigate a basic system T of two rotations with a common angle of rotation. Using a new idea, we prove that, except for an invertible T, the system T must have a global attractor and a global repeller. We give a specific condition for both cases.

In this paper we consider an analog of the regions of instability for twist maps in the context of area preserving diffeomorphisms which are not twist maps. Several properties analogous to those of classical regions of instability are proved.

Let X(x,y) and Y(x,y) be real analytic functions without constant and linear terms defined in a neighborhood of the origin. Assume that the analytic differential system \dot{x}=y+ X(x,y), \dot{y}=Y(x,y) has a nilpotent center at the origin. The first integrals, formal or analytic, will be real except if we say explicitly the converse. We prove the following.

1. If X= y f(x,y^2) and Y= g(x,y^2), then the systemhas a local analytic first integral of the form H=y^2+F(x,y),where F starts with terms of order higher than two.

2. If the system has a formal first integral, then it hasa formal first integral of the form H=y^2+F(x,y), where Fstarts with terms of order higher than two. In particular, if thesystem has a local analytic first integral defined at the origin,then it has a local analytic first integral of the formH=y^2+F(x,y), where F starts with terms of order higher than two.

3. As an application we characterize the nilpotent centersfor the differential systems \dot{x}=y+P_3(x,y),\dot{y}=Q_3(x,y), which have a local analytic first integral,where P_3 and Q_3 are homogeneous polynomials of degree three.

We consider quadratic skew-products over angle-doubling of the circle and prove that they admit positive Lyapunov exponents almost everywhere and an absolutely continuous invariant probability measure. This extends corresponding results of M. Viana and J. F. Alvès for skew-products over the linear strongly expanding map of the circle.

The purpose of this article is two fold. First, we show how quasi-periodic solutions for the N-body problem can be constructed by variational methods. We illustrate this by constructing uncountably many quasi-periodic solutions for the four- and six-body problems with equal masses. Second, we show by examples that a system of N masses can possess infinitely many simple or multiple choreographic solutions. In particular, it is shown that the four-body problem with equal masses has infinitely many double choreographic solutions and the six-body problem with equal masses has infinitely many simple and double choreographic solutions. Our approach is based on the technique of binary decomposition and some variational properties of Keplerian orbits.

For a closed irreducible non-positively curved manifold M, we show that if at almost every point one of the positive Lyapunov exponents for the geodesic flow achieves the minimum allowed by the curvature, then M is locally symmetric of non-compact type. Among the applications of this result, we show that rank one symmetric spaces may be characterized among negatively curved Hadamard manifolds admitting a cocompact lattice solely by the quasiconformal structure and Hausdorff dimension of their ideal boundary. We also prove a rigidity result for semiconjugacies.

This paper has four main results: (i) it shows that left-invariant geodesic flows on a broad class of two-step nilmanifolds—which are dubbed almost non-singular—are integrable in the non-commutative sense of Nehoros˘ev; (ii) the left-invariant geodesic flows on all Heisenberg–Reiter nilmanifolds are Liouville integrable; (iii) the topological entropy of a left-invariant geodesic flow on a two-step nilmanifold vanishes; (iv) there exist two-step nilmanifolds with non-integrable left-invariant geodesic flows. It is also shown that for each of the integrable Hamiltonians investigated here, there is a C^2-open neighbourhood in C^2(T^* M) such that every integrable Hamiltonian vector field in this neighbourhood must have wild first integrals.

Let $\mathcal{N}$ be a 2n-dimensional manifold equipped with a symplectic structure $\omega$ and $\Lambda(\mathcal{N})$ be the Lagrangian Grassmann bundle over $\mathcal{N}$. Consider a flow $\phi^t$ on $\mathcal{N}$ that preserves the symplectic structure and a $\phi^t$-invariant connected submanifold $\Sigma$. Given a continuous section $\Sigma\to\Lambda(\mathcal{N})$, we can associate to any finite $\phi^t$-invariant measure with support in $\Sigma$, a quantity, The asymptotic Maslov index, which describes the way Lagrangian planes are asymptotically wrapped in average around the Lagrangian Grassmann bundle. We pay particular attention to the case when the flow is derived from an optical Hamiltonian and when the invariant measure is the Liouville measure on compact energy levels. The situation when the energy levels are not compact is discussed.

A dynamical 2-complex is a 2-complex equipped with a set of combinatorial properties which allow one to define non-singular semi-flows on the complex. After giving a combinatorial characterization of the dynamical 2-complexes which define hyperbolic attractors when embedded in compact 3-manifolds, we give an effective criterion for the existence of cross sections to the semi-flows on these 2-complexes. In the embedded case, this gives an effective criterion of existence of cross sections to the associated hyperbolic attractors. We present a similar criterion for boundary-tangent flows on compact 3-manifolds which are constructed by means of our dynamical 2-complexes.

Suppose that S is a transitive locally compact Markov shift with Gurevic entropy less than log p, where p is an integer. If the inequality is strict, then S admits a graph presentation with degree p. Additionally, we realize any pre-assigned values for the Salama entropies. This result is best possible, since there are examples of Gurevic entropy log p which do not admit a graph presentation with degree p. We also show that every non-compact locally compact Markov shift has presentations without synchronizing blocks.

We give a partial answer to a question posed by C. Pugh and M. Shub in [PS2]. A partially hyperbolic diffeomorphism $f\in \mathit{Diff}^{2}(M)$ is accessible if every two points of M can be connected by a path which is piecewise in a stable or an unstable leaf. We prove that accessibility is open in $\mathit{Diff}^{2}(M)$ among the partially hyperbolic diffeomorphisms with one-dimensional center bundle.

We study Ruelle's transfer operator \mathcal{L} induced by a C^{\mathbf{r}+1}-smooth expanding map \varphi of a smooth manifold and a C^{\mathbf{r}}-smooth bundle automorphism \Phi of a real vector bundle \mathcal{E}. We prove the following exact formula for the essential spectral radius of \mathcal{L} on the space C^{\mathbf{r},\alpha} of \mathbf{r}-times continuously differentiable sections of \mathcal{E} with \alpha-Hölder \mathbf{r}th derivative:

{\rm r}_{\rm ess}(\mathcal{L};C^{\mathbf{r},\alpha})=\exp\Big(\sup_{\nu\in{\rm Erg}} \{h_\nu+\lambda_\nu-(\mathbf{r}+\alpha)\chi_\nu\}\Big),

where Erg is the set of \varphi-ergodic measures, h_\nu the entropy of \varphi with respect to \nu, \lambda_\nu the largest Lyapunov exponent of the cocycle induced by \Phi, and \chi_\nu the smallest Lyapunov exponent for the differential D\varphi.

This paper addresses the question of genericity of existence of elliptic islands for the billiard map associated to strictly convex closed curves. More precisely, we study 2-periodic orbits of billiards associated to C^5 closed and strictly convex curves and show that the existence of elliptic islands is a dense property on the subset of those billiards having an elliptic 2-periodic point.

Our main tools are normal perturbations, the Birkhoff Normal Form for elliptic fixed points and Moser's Twist Theorem. In order to perform some of the long computations involved, Maple^{\circledR} software was employed.

We classify all factors of the Cartesian product of any two non-singular type III$_\lambda$, $0<\lambda \leq 1$, or type II1 Chacón transformations, as well as the centralizer of finite Cartesian products of such transformations.

We prove a multidimensional version of the theorem that every shift of finite type has a power that can be realized as the same power of a tiling system. We also show that the set of entropies of tiling systems equals the set of entropies of shifts of finite type.