We prove that a Riemannian metric on the 2-sphere or the projective plane can be C2 approximated by a $C^\infty$ metric whose geodesic flow has an elliptic closed geodesic.

We elaborate thermodynamic and multifractal formalisms for general classes of potential functions and their average growth rates. We then apply these formalisms to certain geometrically finite Kleinian groups which may have parabolic elements of different ranks. We show that for these groups our revised formalisms give access to a description of the spectrum of ‘homological growth rates’ in terms of Hausdorff dimension. Furthermore, we derive necessary and sufficient conditions for the existence of ‘thermodynamic phase transitions’.

We consider systems that have some hyperbolicity behavior and which preserve conformal structures on the stable and unstable bundles. We show that two such systems that are topologically conjugate are smoothly conjugate. This is somewhat more general than a conjecture of the author in 2002. Related results have also been obtained by Kalinin and Sadovskaia.

Let V be a finite-dimensional vector space over $\mathbb{R}$and let $\Gamma \subset \mathrm{GL}(V)$ be a semigroup. We study the closed $\Gamma$-invariant subsets of V-\{0\} under the condition that the Zariski closure of $\Gamma$ is semi-simple. We use the results to show that, if $\Gamma\subset\mathrm{SL}(\mathbb{R}^d)$ acts on $\mathbb{T}^d$ by automorphisms, then the orbits of $\Gamma$ are finite or dense.

Let f be a diffeomorphism of a manifold preserving a Borel probability measure $\mu$ having no zero Lyapunov exponents and $\mathcal{W}^s$ a set with asymptotically attracting properties. An upper bound on the Hausdorff dimension of $\mathcal{W}^s$ is given in terms of the Lyapunov exponents and the measure-theoretic entropy. For a surface diffeomorphism, in particular, this upper bound relates, roughly speaking, to the dimension of $\mu$ in the direction of the unstable subspace. Furthermore, we apply this result to establish the equivalence of several properties of $\mu$: its statistical properties, its physical observability and its dimension-theoretical properties (Corollary 2.7).

One of the corollaries of Ornstein's isomorphism theorem is that if $(Y,S,\nu)$ is an invertible measure-preserving transformation and $(Y,S^2,\nu)$ is isomorphic to a Bernoulli shift, then $(Y,S,\nu)$ is isomorphic to a Bernoulli shift. In this paper we show that non-invertible transformations do not share this property. We do this by exhibiting a uniformly two-to-one endomorphism $(X,\sigma,\mu)$ which is not isomorphic to the one-sided Bernoulli two shift. However, $(X,T^2,\mu)$ is isomorphic to the one-sided Bernoulli four shift.

We study the growth of fibers of coverings of pinched negatively curved Riemannian manifolds. The applications include counting estimates for horoballs in the universal cover of geometrically finite manifolds with cusps. Continuing our work on diophantine approximation in negatively curved manifolds started in an earlier paper (Math. Zeit.241 (2002), 181–226), we prove a Khintchine–Sullivan-type theorem giving the Hausdorff measure of the geodesic lines starting from a cusp that are well approximated by the cusp returning ones.

We consider a piecewise smooth expanding map f on the unit interval that has the form $f(x)=x+x^{1+\gamma}+o(x^{1+\gamma})$ near 0, where $0<\gamma < 1$. We prove by showing both lower and upper bounds that the rate of decay of correlations with respect to the absolutely continuous invariant probability measure $\mu$ is polynomial with the same degree $1/\gamma-1$ for Lipschitz functions. We also show that the density function h of $\mu$ has the order $x^{-\gamma}$ as $x\to 0$. Perron–Frobenius operators are the main tool used for proofs.

In this paper we consider a quasi-linear approximation for the 3D Euler system as well as for the Navier–Stokes system in our previous paper Moscow Math. J.1(3) (2001), 381–388). We define finite-dimensional versions of the approximation and study properties of their solutions.

Let (X, T) be a topological dynamical system and let $\mu$ be a T-invariant probability measure on X. In this paper, we study two properties of the notions of measure theoretical entropy for a measurable cover $\mathcal{U},\ h_\mu^+(\mathcal{U},T)$ and $h_\mu^-(\mathcal{U},T)$ introduced by P. P. Romagnoli (Ergod. Th. & Dynam. Sys. 23 (2003), 1601–1610). The main result of the paper states that entropy pairs for the measure $\mu$ can be defined using either $h_\mu^+$ or $h_\mu^-$. We also prove that both $h_\mu^+$ and $h_\mu^-$ have an ergodic decomposition and we use it to prove a local Abramov formula for $h_\mu^-$.

The volume entropy, h(g), of a compact Riemannian manifold (M,g) measures the growth rate of the volume of a ball of radius R in its universal cover. Under the Ricci flow, g evolves along a certain path $(g_t, t\geq0)$ that improves its curvature properties. For a compact surface of variable negative curvature we use a Katok–Knieper–Weiss formula to show that h(gt) is strictly decreasing.

In this paper we find conditions which guarantee that a given flow $\Phi$ on a compact metric space X admits a Lyapunov 1-form $\omega$ lying in a prescribed Čech cohomology class $\xi\in\check H^1(X;\mathbb{R})$. These conditions are formulated in terms of the restriction of $\xi$ to the chain recurrent set of $\Phi$. The result of the paper may be viewed as a generalization of a well-known theorem by Conley about the existence of Lyapunov functions.

We prove that the billiard map in a Bunimovich stadium has polynomial decay of correlations of the order of n-1. Ergodic Bunimovich-type billiards (two arcs not bigger than half a circle) that do not have trajectories with infinitely many consecutive bounces on straight lines satisfy the same property. We also prove that a very particular class of these billiards have polynomial decay of correlations of the order of n-2.

In Blanchard et al (Topological complexity. Ergod. Th. & Dynam. Sys.20 (2000), 641–662), the authors introduced the notion of scattering and a weaker notion of 2-scattering. It is an open question whether the two notions are equivalent. The question is answered affirmatively in this paper. Using the complexity function of an open cover along some sequences of natural numbers, we characterize mild mixing, strong scattering and scattering. We show that mildly mixing (respectively strongly mixing) systems are disjoint from minimal uniformly rigid (respectively minimal rigid) systems.

Moreover, assuming minimality we show that a dynamical system is full-scattering (respectively mildly mixing or weakly mixing) if and only if it is strongly mixing (respectively IP*-transitive or $\mathcal{D}$-transitive), where $\mathcal{D}$ is the collection of subsets of $\mathbb{Z}_+$ with the lower Banach density 1.

The $\lambda$-entropy of a right-resolving $\lambda$-graph system is introduced as the upper asymptotic growth rate of the number of vertices at level n. It is shown that shift equivalent right-resolving $\lambda$-graph systems have the same $\lambda$-entropy. The $\lambda$-entropies of right-resolving $\lambda$-graph systems that are invariantly associated to subshifts are invariants of topological conjugacy. Finite directed graphs are used in the construction of compact forward separated Shannon graphs that have the structure of a pushdown automaton and that are invariantly associated to the subshift they present. To these compact forward separated Shannon graphs there correspond $\lambda$-graph systems whose $\lambda$-entropy is equal to the topological entropy of the edge shift of the finite directed graph. Examples are given.

We consider one-dimensional dynamical systems with indifferent fixed points (fixed points with derivative one). Many such maps have absolutely continuous ergodic infinite invariant measures. We study the limit of the ratio of the ergodic sum of fA to that of fB, where the integrals of fA and fB are infinite with respect to the absolutely continuous ergodic infinite invariant measure. If fA and fB are analytic functions on [0, 1], the result in this paper makes it clear whether the ratio of the ergodic sum of fA to that of fB converges in the Lebesgue measure or not.

Consider a wave equation (WE) with constant coefficients in $\mathbb{R}^n$ for even $n\ge 2$ and with variable coefficients for even $n\ge4$. We study the distribution $\mu_t$ of the random solution at time $t\in\mathbb{R}$. The initial probability measure $\mu_0$ has a translation-invariant covariance, zero mean and finite mean density for the energy. It also satisfies a Rosenblatt- or Ibragimov–Linnik-type mixing condition. The main result is the convergence of $\mu_t$ to a Gaussian probability measure as $t\to\infty$ which gives a Central Limit Theorem (CLT) for the WE. The proof for the case of constant coefficients is based on stationary phase asymptotics of the solution in the Fourier representation and Bernstein's ‘room–corridor’ argument. The case of variable coefficients is reduced to that of constant ones by a version of the scattering theory, based on Vainberg's results on local energy decay.

Cocyclic subshifts arise as the supports of matrix cocycles over a full shift and generalize topological Markov chains and sofic systems. We compute the topological entropy of a cocyclic subshift as the logarithm of the spectral radius of an appropriate transfer operator and give a concrete description of the measure of maximal entropy in terms of the eigenvectors. Unlike in the Markov or sofic case, the operator is infinite-dimensional and the entropy may be a logarithm of a transcendental number.

In the study of systems which combine slow and fast motions, the averaging principle suggests that a good approximation of the slow motion on long time intervals can be obtained by averaging its parameters over the fast variables. When the slow and fast motions depend on each other (fully coupled), as is usually the case, for instance, in perturbations of Hamiltonian systems, the averaging prescription cannot always be applied, and when it does work, this is usually only in some averaged with respect to initial conditions sense. In this paper we first give necessary and sufficient conditions for the averaging principle to hold (in the above sense) and then, relying on some large deviations arguments, verify them in the case when the fast motions are hyperbolic (Axiom A) flows for each freezed slow variable. It turns out that in this situation the Lebesgue measure of initial conditions with bad averaging approximation tends to zero exponentially fast as the parameter tends to zero.

Symbolic matrix systems are generalizations of finite symbolic matrices for sofic systems to subshifts. We prove that if two symbolic matrix systems are strong shift equivalent, then the gauge actions of the associated C*-algebras are stably outer conjugate. The proof given here is based on the construction of an imprimitivity bimodule from a bipartite $\lambda$-graph system, so that an equivariant version of the Brown–Green–Rieffel Theorem proved by Combes is used, together with its proof. As a corollary, if two subshifts are topologically conjugate, then the gauge actions of the associated C*-algebras are stably outer conjugate.