For non-uniformly hyperbolic maps of the interval with exponential decay of correlations we prove that the law of closest return to a given point when suitably normalized is almost surely asymptotically exponential. A similar result holds when the reference point is the initial point of the trajectory. We use the framework for non-uniformly hyperbolic dynamical systems developed by L. S. Young.

Given a piecewise invertible map T:X\to X and a weight g:X\rightarrow\ ]0,\infty[, a conformal measure \nu is a probability measure on X such that, for all measurable A\subset X with T:A\to TA invertible, \nu(TA)= \lambda \int_{A}\frac{1}{g}\ d\nu with a constant \lambda>0. Such a measure is an essential tool for the study of equilibrium states. Assuming that the topological pressure of the boundary is small, that \log g has bounded distortion and an irreducibility condition, we build such a conformal measure.

Let X\subset\mathbb{R}^2 be a finite union of bounded polytopes and let T:X\to X be piecewise affine and eventually expanding. Then the Perron–Frobenius operator \mathcal{L} of T is quasicompact as an operator on the space of functions of bounded variation on \mathbb{R}^2 and its isolated eigenvalues (including multiplicities) are just the reciprocals of the poles of the dynamical zeta function of T. In higher dimensions the result remains true under an additional generically satisfied transversality assumption.

A homeomorphism on a metric space (X,d) is completely scrambled if for each x\not= y\in X, \lim sup_{n\longrightarrow +\infty} d(f^n(x),f^n(y))>0 and \lim inf_{n\ longrightarrow +\infty}d(f^n(x),f^n(y))=0. We study the basic properties of completely scrambled homeomorphisms on compacta and show that there are ‘many’ compacta admitting completely scrambled homeomorphisms, which include some countable compacta (we give a characterization), the Cantor set and continua of arbitrary dimension.

Intuitively, a Liouvillian function on \mathbb{C} P(n) is one which is obtained from rational functions by a finite process of integrations, exponentiations and algebraic operations. This paper is devoted to the study of foliations determined by polynomial 1-forms which have a Liouvillian first integral. Our main result states that, under some mild restrictions on the singularities of the foliation, such a foliation must be either a linear foliation or an exponent two Bernoulli foliation after some rational pull-back. This proves that the highest level of transcendence for the ordinary differential equations which can be integrated by the use of elementary functions is reached at the Riccati equations.

We show that piecewise isometries, i.e. non-necessarily invertible maps defined on a finite union of polytopes and coinciding with an isometry on the interior of each polytope, have zero topological entropy in any dimension. This had been conjectured by a number of authors. The proof is by an induction on the dimension and uses a device (the differential of a piecewise linear map) introduced by M. Tsujii.

Certain classes of automorphisms of reduced amalgamated free products of C^*-algebras are shown to have Brown–Voiculescu topological entropy zero. Also, for automorphisms of exact C^*-algebras, the Connes–Narnhofer–Thirring entropy is shown to be bounded above by the Brown–Voiculescu entropy. These facts are applied to generalize Størmer's result about the entropy of automorphisms of the {\rm II}_1-factor of a free group.

Let ({\cal T}^1 S, m, (T^t){t \in \mathbb{R}}) be the geodesic flow on the unit tangent bundle of a surface S of negative constant curvature and finite volume. We show that every Hölder function on {\cal T}^1 S is, for the discrete time action of the geodesic flow, homologous to a martingale increment. From this representation follow the central limit theorem and its improvements, and a characterization of Hölder functions which are coboundaries in the class of measurable functions.

Soit ({\cal T}^1 S, m, (T^t)_{t \in \mathbb{R}}) le flot géodésique sur le fibré unitaire d'une surface S de courbure négative constante de volume fini. Nous montrons que toute fonction höldérienne sur {\cal T}^1 S est, pour l'action du flot géodésique à temps discret, homologue à un accroissement de martingale. Cette représentation permet d'obtenir le théorème de la limite centrale et ses extensions, et de caractériser les fonctions höldériennes qui sont des cobords dans la classe des fonctions mesurables.

Residual entropy of a topological system is defined as the infimum increase of entropy necessary to build a symbolic extension of this system. If no symbolic extension exists then residual entropy is set at infinity. In this paper we provide a direct formula for the residual entropy of a system on a totally disconnected compact space in terms of basic notions of conditional entropies viewed as functions of invariant measures. This formula allows us to evaluate residual entropy in many examples as well as to construct new examples with arbitrarily preset topological and residual entropies. The appendix contains a condition equivalent to asymptotic h-expansiveness.

In this paper we investigate the rigidity of symmetric spaces of non-compact type using ergodic theory such as Patterson–Sullivan measure and the marked length spectrum along with the cross ratio on the limit set. In particular, we prove that the marked length spectrum determines the Zariski dense subgroup up to conjugacy in the isometry group of the product of rank-one symmetric spaces. As an application, we show that two convex cocompact, negatively curved, locally symmetric manifolds are isometric if the Thurston distance is zero and the critical exponents of the Poincaré series are the same, and the same is true if the geodesic stretch is equal to one.

Pointwise convergence a.e. of \mu^k f has been studied by Bellow, Jones and Rosenblatt. We answer two of their questions: the strong sweeping out property holds (in this case) iff the measure \mu has unbounded angular ratio and the stability is equivalent to the bounded angular ratio.

We consider the set of maps f\in\mathcal{F}_{\alpha+} = \cup_{\beta>\alpha} \mathcal{C}^{1+\beta} of the circle which are covering maps of degree D, expanding, \min_{x\in S^1}f'(x) >1 and orientation preserving. We are interested in characterizing the set of such maps f which admit a unique f-invariant probability measure \mu minimizing \int \ln f'\,d\mu over all f-invariant probability measures. We show there exists a set \mathcal{G}_+\subset\mathcal{F}_{\alpha+}, open and dense in the \mathcal{C}^{1+\alpha}-topology, admitting a unique minimizing measure supported on a periodic orbit. We also show that, if f admits a minimizing measure not supported on a finite set of periodic points, then f is a limit in the \mathcal{C}^{1+\alpha}-topology of maps admitting a unique minimizing measure supported on a strictly ergodic set of positive topological entropy.

We use in an essential way a sub-cohomological equation to produce the perturbation. In the context of Lagrangian systems, the analogous quation was introduced by R. Mañé and A. Fathi extended it to the all configuration space in [8].

We will also present some results on the set of f-invariant measures \mu maximizing \int A\,d\mu for a fixed \mathcal{C}^1-expanding map f and a general potential A, not necessarily equal to -\ln f'.

A general framework for investigating topological actions of \mathbb{Z}^d on compact metric spaces was proposed by Boyle and Lind in terms of expansive behavior along lower-dimensional subspaces of \mathbb{R}^d. Here we completely describe this expansive behavior for the class of algebraic \mathbb{Z}^d-actions given by commuting automorphisms of compact abelian groups. The description uses the logarithmic image of an algebraic variety together with a directional version of Noetherian modules over the ring of Laurent polynomials in several commuting variables.

We introduce two notions of rank for topological \mathbb{Z}^d-actions, and for algebraic \mathbb{Z}^d-actions describe how they are related to each other and to Krull dimension. For a linear subspace of \mathbb{R}^d we define the group of points homoclinic to zero along the subspace, and prove that this group is constant within an expansive component.

We prove that the Jouanolou foliation of degree k ≤ 5 admits no non-trivial minimal sets. The proof uses reliable computations based on interval arithmetic.

Let G be a semisimple Lie group with no compact factors, K a maximal compact subgroup of G, and \Gamma a lattice in G. We study automorphic forms for \Gamma if G is of real rank one with some additional assumptions, using a dynamical approach based on properties of the homogeneous flow on \Gamma\backslash G and a Livshitz type theorem we prove for such a flow. In the Hermitian case G=SU(n,1) we construct relative Poincaré series associated to closed geodesics on \Gamma\backslash G/K for one-dimensional representations of K, and prove that they span the corresponding spaces of holomorphic cusp forms.

We find asymptotics for the number of cylinders and saddle connections on flat surfaces. These results extend previous results of Veech.

To obtain connections between grammars of languages and subshifts, Krieger introduced the idea of associating to a subshift X a certain increasing sequence of shifts of finite type X_{n} sitting inside X. Their union is the set of presynchronizing points. A system is called presynchronized if there is a sequence of irreducible components C_{n} \subset X_{n}, n \in \mathbb{N}, increasing with n and with dense union (a characterization of coded systems shows that presynchronized systems are coded). We investigate the uniqueness (up to eventual equality) of such sequences. For that an equivalence relation on the periodic presynchronizing points captures all the possible choices of increasing sequences of irreducible components. Dense equivalence classes correspond to dense unions. There is a one-sided and a two-sided theory. In the one-sided setting we show the uniqueness of a dense equivalence class and identify the right presynchronized systems as the well-known half-synchronized systems. The two-sided theory turns out to be more complicated (and interesting). There can be more than one dense class. Many examples are discussed in detail. There is a natural ordering on the set of (dense) equivalence classes. For each finite-order structure we find a representative in the countable class of systems which are intersections of Dyck shifts with shifts of finite type. Finally we discuss the relation to other known classes of subshifts, which is more subtle than in the one-sided setting. Almost all systems considered in this work will be coded.

For L equal to the unilateral or bilateral shift on a weighted sequence space, we characterize, in terms of f and the weight function, those f holomorphic on the spectrum of L for which f(L) is a chaotic operator. For B equal to d/dx, the generator of left translation, on weighted L^p spaces on [0, \infty) or \mathbb{R}, we similarly characterize those polynomials Q for which the differential operator Q(B) generates a chaotic semigroup.

Under some weak hyperbolicity conditions, we establish C^0- and C^\infty-local rigidity theorems for two classes of standard algebraic actions: (1) left translation actions of higher real rank semisimple Lie groups and their lattices on quotients of Lie groups by uniform lattices; (2) higher rank lattice actions on nilmanifolds by affine diffeomorphisms. The proof relies on an observation that local rigidity of the standard actions is a consequence of the local rigidity of some constant cocycles. The C^0-local rigidity for weakly hyperbolic standard actions follows from a cocycle C^0-local rigidity result proved in the paper. The main ingredients in the proof of the latter are Zimmer's cocycle superrigidity theorem and stability properties of partially hyperbolic vector bundle maps. The C^\infty-local rigidity is deduced from the C^0-local rigidity following a procedure outlined by Katok and Spatzier.

Using similar considerations, we also establish C^0-global rigidity of volume preserving, higher rank lattice Anosov actions on nilmanifolds with a finite orbit.

We show that for a smooth contact Anosov flow on a closed three manifold the measure of maximal entropy is in the Lebesgue class if and only if the flow is, up to finite covers, conjugate to the geodesic flow of a metric of constant negative curvature on a closed surface. This shows that the ratio between the measure theoretic entropy and the topological entropy of a contact Anosov flow is strictly smaller than one on any closed three manifold which is not a Seifert bundle.