On a surface of negative curvature, we study the distribution of closed geodesics which are not far from minimizing length in their homology classes. These geodesics do not have the statistical behaviour obtained by Margulis for ‘generic’ geodesics and later extended by Lalley to geodesics satisfying certain homological constraints. We cannot use the usual techniques of ‘thermodynamic formalism for Anosov flows’, but we study the problem using ideas from Aubry–Mather theory, as well as some facts of low-dimensional topology.

Let r, s be natural numbers, r, s \geq 2, such that {\rm log}\,r/{\rm log}\,s is irrational. Let \mu_j = \sum_{k=0}^{s-1} p^{(j)}_{k}\delta_{k/s^j}, where p^{(j)}_k \in [0,1), \sum_{k=0}^{s-1} p^{(j)}_k =1, j = 1, 2, \dotsc and {\delta}_{x} denotes the probability atom at x. We prove that if there exists k_0\in \{0,\dotsc , s-2\}, such that \inf \{ p^{(j)}_{k_0}, p^{(j)}_{k_0+1} : j\in \mathbb N \} > 0 and \mu is the infinite convolution measure \mu = *^{\infty}_{j=1} {\mu_{j} }, where the convergence is in the weak* sense, then \mu-almost all numbers are normal to base r. This generalizes some earlier results of Feldman, Smorodinsky and Pushkin.

A partially hyperbolic diffeomorphism is dynamically coherent if its center, center-stable and center-unstable invariant distributions are integrable, i.e. tangent to foliations. Dynamical coherence is a key assumption in the theory of stable ergodicity. The main result: a partially hyperbolic diffeomorphism f\colon M\to M is dynamically coherent if the strong stable and unstable foliations are quasi-isometric in the universal cover \widetilde{M}, i.e. for any two points in the same leaf, the distance between them in \widetilde{M} is bounded from below by a linear function of the distance along the leaf.

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.

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.

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.

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.

In this paper, random expansions to non-integer bases \beta >1 are studied. For \beta's satisfying \beta^2=n\beta +k (with 1\leq k\leq n) and \beta^n=\beta^{n-1}+\dotsb + \beta +1 the ergodic properties of such expansions are described.

Recently, the stretched exponential decay of multiple correlations in a periodic Lorentz gas has been used to show the convergence of a series of correlations which has a physical interpretation as the fourth-order Burnett coefficient, a generalization of the diffusion coefficient. Here the result is extended to include all higher-order Burnett coefficients and a plausible argument is given that the expansion constructed from the Burnett coefficients has a finite radius of convergence.

We consider the standard family (or Arnold family) of circle maps given by f_{\alpha, \beta}(x)=x + \alpha + \beta \sin(x) \pmod{2\pi}, for x,\alpha\in [0,2\pi), \beta \in (0,1) and its complexification F_{\alpha,\beta}(z)=z e^{i\alpha} \exp [\frac12\beta(z-\frac{1}{z})]. If f_{\alpha,\beta} is analytically linearizable, there is a Herman ring around the unit circle in the dynamical plane of F_{\alpha,\beta}. Given an irrational rotation number \theta, the parameters (\alpha,\beta) such that f_{\alpha, \beta} has rotation number \theta form a curve T_\theta in the parameter plane. Using quasi-conformal surgery of the simplest type, we show that if \theta is a Brjuno number, the curve T_\theta can be parametrized real-analytically by the modulus of the Herman ring, from \beta=0 up to a point (\alpha_0,\beta_0) with \beta_0 \leq 1, for which the Herman ring collapses. Using a result of Herman and a construction in I. N. Baker and P. Domínguez (Complex Variables37 (1998), 67–98) we show that for a certain set of angles \theta \in \mathcal{B} \setminus \mathcal{H}, the point \beta_0 is strictly less than 1 and, moreover, the boundary of the Herman rings with the corresponding rotation number have two connected components which are quasi-circles, and do not contain any critical point. For rotation numbers of constant type, the boundary consists of two quasi-circles, each containing one of the two critical points of F_{\alpha, \beta}.

For diffeomorphisms on surfaces with basic sets, we show the following type of rigidity result: if a topological conjugacy between them is differentiable at a point in the basic set then the conjugacy has a smooth extension to the surface. These results generalize the similar ones of D. Sullivan, E. de Faria and ours for one-dimensional expanding dynamics.

We construct d-dimensional substitution sequences for which the continuous part of the spectrum is generated by measures equal to Lebesgue measure. A special case is the Rudin–Shapiro substitution sequence. The construction uses Hadamard matrices in an essential way, so the dimension and size of a substitution is restricted by the size of the Hadamard matrix defining it. Each such substitution automatically has a dual substitution, which is defined by the same Hadamard matrix, and which retains a Lebesgue spectral component. We also see that random application of our substitutions produces sequences with a Lebesgue component in their spectrum. Finally, we see that any d-dimensional substitution with d > 1 can be ‘unraveled’ into lower-dimensional substitutions which still have Lebesgue spectral components.

We show that any Pisot substitution dynamical system on two symbols has pure discrete spectrum. This follows from an analysis of a combinatorial ‘balanced pair algorithm’, in view of recent progress on the Strong Coincidence Conjecture made by Barge and Diamond.

We generalize the average asymptotic linking number of a pair of divergence-free vector fields on homology three-spheres by considering the linking of a divergence-free vector field on a manifold of arbitrary dimension with a codimension-two foliation endowed with an invariant transverse measure. We prove that the average asymptotic linking number is given by an integral of Hopf type. Considering appropriate vector fields and measured foliations, we obtain an ergodic interpretation of the Godbillon–Vey invariant of a family of codimension-one foliations discussed by Kotschick (Symplectic and Contact Topology: Interactions and Perspectives. Eds. Y. Eliashberg, B. Khesin and F. Lalonde. American Mathematical Society, Providence, RI).

We construct dynamical partitions of the torus for a diffeomorphism that is isotopic to the identity. The existence and the combinatorics of the partitions is solely determined by the rotation set of the diffeomorphism. When the rotation set consists of a single non-resonant vector, there is a whole hierarchy of partitions analogous to the partitions of the circle into the closest return intervals under an irrational circle rotation. In particular, all such torus maps are infinitely renormalizable in a natural sense.

In this paper we introduce foundational techniques and prove the following: if X is a \mathbb{Z}^d subshift without periodic points, if Y is a \mathbb{Z}^d square mixing subshift of finite type containing a finite orbit and if there exists a homomorphism X\rightarrow Y, then X embeds into Y if and only if h(X)<h(Y). For the proof, clopen markers are used to generate Voronoi tiles whose thickened boundaries are coded using the homomorphism. The entropy gap and the square mixing permit the construction of an injective code on the tile interiors. A second paper will show that \mathbb{Z}^2 square filling mixing shifts of finite type are square mixing and that homomorphisms exist, resulting in an extension of Krieger's Embedding Theorem to \mathbb{Z}^2 subshifts.

The topological transitivity of non-compact group extensions of topologically mixing subshifts of finite type has been studied recently by Niţică. We build on these methods, and give the first examples of stably transitive non-compact group extensions of hyperbolic dynamical systems. Our examples include extensions of hyperbolic basic sets by the Euclidean group SE(n) for n even, n\ge4.

We show that if f is a C^2 diffeomorphism with positive entropy on an n-dimensional closed manifold (n \geq 2) then f is chaotic in the sense of Li–Yorke, and moreover f is \ast-chaotic on the closure of the stable manifold for some point. The notion of ‘\ast-chaos’ was introduced by Kato in relation to Li–Yorke's chaos in the context of topological dynamics.

This paper investigates the set of angles of the parameter rays which land on the real slice [-2,1/4] of the Mandelbrot set. We prove that this set has zero length but Hausdorff dimension 1. We obtain the corresponding results for tuned images of the real slice. Applications of these estimates in the study of critically non-recurrent real quadratics as well as bi-accessible points of quadratic Julia sets are given.

In the second paragraph on page 30 [1], we wrote ‘Making a critical point precritical decreases the number of points about which we have to worry’. Indeed, we stopped worrying about these points so much, that we forgot to mention them in several statements. This includes Lemma 3.13, Theorem 3.14 and the Main Theorem. Thus, the possibility of critical points being precritical (not only sinking or recurrent) has to be added to those statements. In particular, the Main Theorem should read as follows.