A selection of basic results in smooth ergodic theory and in the thermodynamic formalism of dynamical systems generated by compositions of random maps is reviewed in this article.

We show that Chacon's non-singular type III_\lambda transformation T_\lambda, 0<\lambda\leq 1, is power weakly mixing, i.e. for all sequences of non-zero integers \{k_{1},\dotsc,k_{r}\}, T_\lambda^{k_{1}}\times\dotsb\times T_\lambda^{k_{r}} is ergodic. We then show that in infinite measure, this condition is not implied by infinite ergodic index (having all finite Cartesian products ergodic), and that infinite ergodic index does not imply 2-recurrence.

Let \varphi be a primitive, non-periodic substitution. The tiling space \mathcal{T}_\varphi has a finite (non-zero) number of asymptotic composants. We describe the form and make use of these asymptotic composants to define a closely related substitution \varphi^* and prove that for primitive, non-periodic substitutions \varphi and \chi, \mathcal{T}_\varphi and \mathcal{T}_\chi are homeomorphic if and only if {\varphi^*} (or its reverse) and \chi^* are weakly equivalent. We also provide examples indicating that for substitution minimal systems, flow equivalence and orbit equivalence are independent.

Let \{U_{g}\}_{g\in G} be a weakly mixing unitary action of a finitely-generated nilpotent group on a Hilbert space \mathcal{H}. Extending a known result about abelian groups, we show that weakly wandering vectors are dense in \mathcal{H}.

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.

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'.

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.

We present a powerful approach to computing the Hausdorff dimension of certain conformally self-similar sets. We illustrate this method for the dimension \mathop{\rm dim}\nolimits_H(E_2) of the set E_2, consisting of those real numbers whose continued fraction expansions contain only the digits 1 or 2. A very striking feature of this method is that the successive approximations converge to \mathop{\rm dim}\nolimits_H(E_2) at a super-exponential rate

We give stronger versions and alternative simple proofs of some results of Beardon, [Be1] and [Be2]. These results concern contractions of locally compact metric spaces and generalize the theorems of Wolff and Denjoy about the iteration of a holomorphic map of the unit disk. In the case of unbounded orbits, there are two types of statements which can sometimes be proven; first, about invariant horoballs, and second, about the convergence of the iterates to a point on the boundary. A few further remarks of similar type are made concerning certain random products of semicontractions and also concerning semicontractions of Gromov hyperbolic spaces.

Consider the conjugacy action of the unitary group of an infinite-dimensional separable Hilbert space on the unitary operators. A strong generic ergodicity property of this action is established, by showing that any conjugacy invariants assigned in a definable way to unitary operators, and taking as values countable structures up to isomorphism, generically trivialize. Similar results are proved for conjugacy of self-adjoint operators and for measure equivalence. The proofs make use of the theory of turbulence for continuous actions of Polish groups, developed by Hjorth. These methods are also used to give a new solution to a problem of Mauldin in measure theory, by showing that any analytic set of pairwise orthogonal measures on the Cantor space is orthogonal to a product measure.

We consider the action of the (n-1)-dimensional group of diagonal matrices in SL(n,\mathbb{R}) on SL(n,\mathbb{R})/\Gamma, where \Gamma is a lattice and n\ge 3. Far-reaching conjectures of Furstenberg, Katok–Spatzier and Margulis suggest that there are very few closed invariant sets for this action. We examine the closed invariant sets containing compact orbits. For example, for \Gamma={\rm SL}(n,\mathbb{Z}) we describe all possible orbit-closures containing a compact orbit. In marked contrast to the case n=2, such orbit-closures are necessarily homogeneous submanifolds in the sense of Ratner.

In this paper we show that strong generalizations of the measurable Livšic theorem for cocycles taking values in connected non-compact linear semisimple Lie groups, a canonical example being SL(2, \mathbb{R}), can be deduced from an elegant approach of Brin and Pesin to the dynamics of partially hyperbolic systems.

The aim of this paper is to present a one-parameter family X_a of three-dimensional vector fields which has infinitely many persistent Hénon-like strange attractors. This family also has infinitely many periodic attractors for values of the parameter in an open set.

We consider measurable conjugacies between two non-solvable subgroups G_1, G_2 of the group of analytic diffeomorphisms of the circle which admit a finite generating set close to the identity. These measurable conjugacies are characterized by our main result which asserts that, up to finite coverings, they coincide almost everywhere with an analytic diffeomorphism of the circle.

The unstable manifolds of hyperbolic systems admitting a one-rectangle Markov partition are here characterized up to homeomorphism and up to conjugacy of the underlying dynamics in a very easy-to-compute way. We associate to each one-rectangle system a word over the alphabet \{+,-\} describing the bending of the image rectangle with respect to the initial rectangle. We endow the set \{+,-\}^{\ast} of such words with a product \wedge having a dynamical origin. The structure of the non-commutative semigroup (\{+,-\}^{\ast},\wedge) is made completely explicit. The topological and dynamical properties of the unstable manifolds of one-rectangle systems are translated in terms of the decomposition of the associated words in such a semigroup.