Assume that x∈[0,1) admits its continued fraction expansion x=[a1(x),a2(x),…]. The Khintchine exponent γ(x) of x is defined by when the limit exists. The Khintchine spectrum dim Eξ is studied in detail, where Eξ:={x∈[0,1):γ(x)=ξ}(ξ≥0) and dim denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum dim Eξ, as a function of , is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by are also studied, where φ(n) tends to infinity faster than n does. Under some regular conditions on φ, it is proved that the fast Khintchine spectrum dim ({x∈[0,1]:γφ(x)=ξ}) is a constant function. Our method also works for other spectra such as the Lyapunov spectrum and the fast Lyapunov spectrum.

We consider open sets of transformations in a manifold M, exhibiting non-uniformly expanding behavior in some forward invariant domain U\subset M. Assuming that each transformation has a unique Sinai–Ruelle–Bowen (SRB) measure in U, and some general uniformity conditions, we prove that the SRB measure varies continuously with the dynamics in the L1-norm. We also describe a concrete application.

A Riemannian metric is adapted to a hyperbolic set of a diffeomorphism if, in this metric, the expansion/contraction of the unstable/stable directions is seen after only one iteration. A dominated splitting is a notion of weak hyperbolicity where the tangent bundle of the manifold splits in invariant subbundles such that the vector expansion on one bundle is uniformly smaller than that on the next bundle. The existence of an adapted metric for a dominated splitting has been considered by Hirsch, Pugh and Shub (M. Hisch, C. Pugh and M. Shub. Invariant Manifolds(Lecture Notes in Mathematics, 583). Springer, Berlin, 1977). This paper gives a complete answer to this problem, building adapted metrics for dominated splittings and partially hyperbolic splittings in arbitrarily many subbundles of arbitrary dimensions. These results stand for diffeomorphisms and for flows.

We consider a Ruelle—Perron—Frobenius type of selection procedure for probability measures that are invariant under random subshifts of finite type. In particular we prove that for a class of random functions this method leads to a unique probability exhibiting properties that justify the names Gibbs measure and equilibrium states. In order to do this we introduce the notion of bundle random dynamical systems and provide a theory for their invariant measures as well as give a precise definition of Gibbs measures.

We study the asymptotic behavior of iterates of rational functions with small perturbations. In presence of attractive cycles we show that almost surely, in the parameter space, the iterates converge to a given neighborhood of the attractive cycles. When there is no attractive cycle, we prove an ergodic theorem with respect to Lebesgue measure.

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.

We extend results of Bowen and Manning on systems with good symbolic dynamics. In particular we identify the class of dynamical systems that admit Markov partitions. For these systems the Manning-Bowen method of counting periodic points is explained in terms of topological coincidence numbers. We show, in particular, that an expansive system with a finite cover by rectangles has a rational zeta function.

We prove that if f: M2→ M2 is pseudo-Anosov and if g ⋍ f then there is a closed subset X ⊂ M2 and a continuous surjection π: X→ M that is homotopic to inclusion such that f ° π = π ° g|X.

We prove that in a compact manifold of dimension n≥2, C1+α volume-preserving diffeomorphisms that are robustly transitive in the C1-topology have a dominated splitting. Also we prove that for three-dimensional compact manifolds, an isolated robustly transitive invariant set for a divergence-free vector field cannot have a singularity. In particular, we prove that robustly transitive divergence-free vector fields in three-dimensional manifolds are Anosov. For this, we prove a ‘pasting’ lemma, which allows us to make perturbations in conservative systems.

If X is a compact, zero-dimensional group and T is an expansive, transitive automorphism then (X, T) is shown to be topologically conjugate to a full shift on finitely many symbols.

The problem of classifying such automorphisms up to simultaneous algebraic isomorphism and topological conjugacy is discussed but not solved. It is proved that for any entropy there are only finitely many such equivalence classes. When the entropy is log p for a prime p, there is only one equivalence class. All are then equivalent to

The phase space of the geodesic flow on an almost integrable polyhedral surface is foliated into a one-parameter family of invariant surfaces. The flow on a typical invariant surface is minimal. We associate with an almost integrable polyhedral surface its holonomy group which is a subgroup of the group of motions of the Euclidean plane. We show that if the holonomy group is discrete then the flow on an invariant surface is ergodic if and only if it is minimal.

A mixing subshift of finite type T is a factor of a sofic shift S of greater entropy if and only if the period of any periodic point of S is divisible by the period of some periodic point of T. Mixing sofic shifts T satisfying this theorem are characterized, as are those mixing sofic shifts for which Krieger's Embedding Theorem holds. These and other results rest on a general method for extending shift-commuting continuous maps into mixing subshifts of finite type.

For S-unimodal maps $f$, we study equilibrium states maximizing the free energies $F_t(\mu) := h(\mu) - t\int \log|f'|\,d\mu$ and the pressure function $P(t):=\sup_\mu F_t(\mu)$. It is shown that if $f$ is uniformly hyperbolic on periodic orbits, then $P(t)$ is analytic for $t\approx 1$. On the other hand, examples are given where no equilibrium states exist, where equilibrium states are not unique and where the notions of equilibrium state for $t=1$ and of observable measure do not coincide.

This paper investigates the dynamical properties of a class of urn processes and recursive stochastic algorithms with constant gain which arise frequently in control, pattern recognition, learning theory, and elsewhere.

It is shown that, under suitable conditions, invariant measures of the process tend to concentrate on the Birkhoff center of irreducible (i.e. chain transitive) attractors of some vector field $F: {\Bbb R}^d \rightarrow {\Bbb R}^d$ obtained by averaging. Applications are given to simple situations including the cases where $F$ is Axiom A or Morse–Smale, $F$ is gradient-like, $F$ is a planar vector field, $F$ has finitely many alpha and omega limit sets.

A dynamical version of the Bourgain–Fremlin–Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of $\beta{\mathbb{N}}$, or it is a ‘tame’ topological space whose topology is determined by the convergence of sequences. In the latter case, Glasner (On tame enveloping semigroups. Colloq. Math.105 (2006), 283–395) calls the system tame. In this paper, we study the tame system under an Abelian group action. We introduce the notion of scrambled pairs for a dynamical system under an Abelian group action and show that a tame system has no scrambled pair. At the same time, we give some sufficient conditions such that a pair is a scrambled one. Moreover, using these sufficient conditions we prove that a minimal tame system under an Abelian group action is almost automorphic and uniquely ergodic. This gives a positive answer to Problem 2.5 in the paper cited above. Finally, we prove that for ${\mathbb Z}$-actions a tame system is $M$-null, that is, the metric sequence entropy is zero for any invariant measure and any increasing sequence of natural numbers. We also give examples to show that tameness is strictly weaker than nullness for ${\mathbb Z}$-actions.

For functions meromorphic in the plane, apart from an exceptional case, the Julia set J is the closure of the set of all preimages of poles. The repelling periodic cycles are dense in J. In contrast with the case of transcendental entire functions, J may be a subset of a straight line and general classes of functions for which this is the case can be determined. J may also lie on a quasicircle through infinity which is not a straight line.

We investigate the relation between the complexity function of a sequence, that is the number p(n) of its factors of length n, and the rank of the associated dynamical system, that is the number of Rokhlin towers required to approximate it. We prove that if the rank is one, then lim , but give examples with lim for any prescribed function G with G (n) = 0(an) for every a > 1. We give exact computations for examples of the ‘staircase’ type, which are strongly mixing systems with quadratic complexity. Conversely, for minimal sequences, if p(n) < an + b for some a ≥ 1, the rank is at most 2[a], with bounded strings of spacers, and the system is generated by a finite number of substitutions.

The purpose of this paper is to present some simple examples that are hyperbolic everywhere except at one point, but which do not admit SBR measures. Each example has a fixed point at which the larger eigenvalue is equal to one and the smaller eigenvalue is less than one.

Semi-invertible multiplicative ergodic theorems establish the existence of an Oseledets splitting for cocycles of non-invertible linear operators (such as transfer operators) over an invertible base. Using a constructive approach, we establish a semi-invertible multiplicative ergodic theorem that for the first time can be applied to the study of transfer operators associated to the composition of piecewise expanding interval maps randomly chosen from a set of cardinality of the continuum. We also give an application of the theorem to random compositions of perturbations of an expanding map in higher dimensions.

We survey the theory of quasi-periodic Schrödinger-type operators, focusing on the advances made since the early 2000s by adopting a dynamical systems point of view.