In this article we study conditions to be a continuous or a measurable eigenvalue of finite rank minimal Cantor systems, that is, systems given by an ordered Bratteli diagram with a bounded number of vertices per level. We prove that continuous eigenvalues always come from the stable subspace associated with the incidence matrices of the Bratteli diagram and we study rationally independent generators of the additive group of continuous eigenvalues. Given an ergodic probability measure, we provide a general necessary condition for there to be a measurable eigenvalue. Then, we consider two families of examples, a first one to illustrate that measurable eigenvalues do not need to come from the stable space. Finally, we study Toeplitz-type Cantor minimal systems of finite rank. We recover classical results in the continuous case and we prove that measurable eigenvalues are always rational but not necessarily continuous.

In this article we prove that given a self-similar interval exchange transformation T(λ,π), whose associated matrix verifies a quite general algebraic condition, there exists an affine interval exchange transformation with wandering intervals that is semi-conjugated to it. That is, in this context the existence of Denjoy counterexamples occurs very often, generalizing the result of Cobo [Piece-wise affine maps conjugate to interval exchanges. Ergod. Th. & Dynam. Sys.22 (2002), 375–407].

We study two classes of dynamical systems with holes: expanding maps of the interval and Collet–Eckmann maps with singularities. In both cases, we prove that there is a natural absolutely continuous conditionally invariant measure μ (a.c.c.i.m.) with the physical property that strictly positive Hölder continuous functions converge to the density of μ under the renormalized dynamics of the system. In addition, we construct an invariant measure ν, supported on the Cantor set of points that never escape from the system, that is ergodic and enjoys exponential decay of correlations for Hölder observables. We show that ν satisfies an equilibrium principle which implies that the escape rate formula, familiar to the thermodynamic formalism, holds outside the usual setting. In particular, it holds for Collet–Eckmann maps with holes, which are not uniformly hyperbolic and do not admit a finite Markov partition. We use a general framework of Young towers with holes and first prove results about the a.c.c.i.m. and the invariant measure on the tower. Then we show how to transfer results to the original dynamical system. This approach can be expected to generalize to other dynamical systems than the two above classes.

We present an analysis of one-dimensional models of dynamical systems that possess ‘coherent structures’: global structures that disperse more slowly than local trajectory separation. We study cocycles generated by expanding interval maps and the rates of decay for functions of bounded variation under the action of the associated Perron–Frobenius cocycles. We prove that when the generators are piecewise affine and share a common Markov partition, the Lyapunov spectrum of the Perron–Frobenius cocycle has at most finitely many isolated points. Moreover, we develop a strengthened version of the Multiplicative Ergodic Theorem for non-invertible matrices and construct an invariant splitting into Oseledets subspaces. We detail examples of cocycles of expanding maps with isolated Lyapunov spectrum and calculate the Oseledets subspaces, which lead to an identification of the underlying coherent structures. Our constructions generalize the notions of almost-invariant and almost-cyclic sets to non-autonomous dynamical systems and provide a new ensemble-based formalism for coherent structures in one-dimensional non-autonomous dynamics.

In this note, we obtain verifiable sufficient conditions for the extreme-value distribution for a certain class of skew-product extensions of non-uniformly hyperbolic base maps. We show that these conditions, formulated in terms of the decay of correlations on the product system and the measure of rapidly returning points on the base, lead to a distribution for the maximum of Φ(p)=−log(d(p,p0)) that is of the first type. In particular, we establish the type I distribution for S1 extensions of piecewise C2 uniformly expanding maps of the interval, non-uniformly expanding maps of the interval modeled by a Young tower, and a skew-product extension of a uniformly expanding map with a curve of neutral points.

An extended version of the ergodic closing lemma of Mañé is proved. As an application, we show that, C1 densely in the complement of the closure of Morse–Smale diffeomorphisms and those with a homoclinic tangency, there exists a weakly hyperbolic structure (dominated splittings with average hyperbolicity at almost every point on hyperbolic parts, and one-dimensional center direction when zero Lyapunov exponents are involved) over the supports of all non-atomic ergodic measures. As another application, we prove an approximation theorem, which claims that approximating the Lyapunov exponents of any non-atomic ergodic measure by those of an atomic ergodic measure by a C1 small perturbation is possible.

We investigate algebraic properties of the automorphism group of multidimensional shifts of finite type (SFTs). We show that positive entropy implies that the automorphism group contains every finite group and, together with transitivity, implies that the center of the automorphism group is trivial (i.e. consists only of the shift action). We also show that positive entropy and dense minimal points (in particular, dense periodic points) imply that the automorphism group of X contains a copy of the automorphism group of the one-dimensional full shift, and hence contains non-trivial elements of infinite order. On the other hand we construct a mixing, positive-entropy SFT whose automorphism group is, modulo the shift action, a union of finite groups.

A basic nilsequence is a sequence of the form ψ(n)=f(Tnx), where x is a point of a compact nilmanifold X, T is a translation on X, and f∈C(X); a nilsequence is a uniform limit of basic nilsequences. Let X=G/Γ be a compact nilmanifold, Y be a subnilmanifold of X, g(n) be a polynomial sequence in G, and f∈C(X); we show that the sequence ∫ g(n)Yf, n∈ℤ, is the sum of a basic nilsequence and a sequence that converges to zero in uniform density. This implies that, given an ergodic invertible measure-preserving system (W,ℬ,μ,T), with μ(W)<∞, polynomials p1,…,pk∈ℤ[n], and sets A1,…,Ak∈ℬ, the sequence μ(Tp1(n)A1∩⋯∩Tpk(n)Ak) is the sum of a nilsequence and a sequence that converges to zero in uniform density. We also obtain a version of this result for the case where pi are polynomials in several variables.

We consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets goes to 2. We prove that the Lebesgue measure of these Julia sets tend to zero. An important part of the proof consists in applying martingale theory to a stochastic process with non-integrable increments.

Let ℳ={f(z)=(zm/sinh m z) for z∈ℂ∣ either m or m/2 is an odd natural number}. For each f∈ℳ, the set of singularities of the inverse function of f is an unbounded subset of the real line ℝ. In this paper, the iteration of functions in one-parameter family 𝒮={fλ(z)=λf(z)∣λ∈ℝ∖{0}} is investigated for each f∈ℳ. It is shown that, for each f∈ℳ, there is a critical parameter λ*>0 depending on f such that a period-doubling bifurcation occurs in the dynamics of functions fλ in 𝒮 when the parameter |λ| passes through λ*. The non-existence of Baker domains and wandering domains in the Fatou set of fλ is proved. Further, it is shown that the Fatou set of fλ is infinitely connected for 0<∣λ∣≤λ* whereas for ∣λ∣≥λ*, the Fatou set of fλ consists of infinitely many components and each component is simply connected.

In this paper we study characteristic exponents of flows in relation with the dynamics of flows on flag bundles. The starting point is a flow on a principal bundle with semi-simple group G. Projection against the Iwasawa decomposition G=KAN defines an additive cocycle over the flow with values in 𝔞=log A. Its Lyapunov exponents (limits along trajectories) and Morse exponents (limits along chains) are studied. A symmetric property of these spectral sets is proved, namely invariance under the Weyl group. We also prove that these sets are located in certain Weyl chambers, defined from the dynamics on the associated flag bundles. As a special case linear flows on vector bundles are considered.

In this paper, we study some skew product diffeomorphisms with non-uniformly hyperbolic structure along fibers and show that there is an invariant measure with zero entropy which has atomic conditional measures along fibers. For such diffeomorphisms, our result gives an affirmative answer to the question posed by Herman as to whether a smooth diffeomorphism of positive topological entropy would fail to be uniquely ergodic. The proof is based on some techniques that are analogous to those developed by Pesin and Katok, together with an investigation of certain combinatorial properties of the projected return map on the base.

This paper deals with random perturbations of diffeomorphisms on n-dimensional Riemannian manifolds with distributions supported on k-dimensional disks, where k<n. First we demonstrate general but not very intuitive conditions which guarantee that all invariant measures for rank-k random perturbations of C2 diffeomorphisms are absolutely continuous with respect to the Riemannian measure on M. For two subclasses of Anosov diffeomorphisms, hyperbolic toral automorphisms and Anosov diffeomorphisms with codimension 1 stable manifolds, the above conditions are modified in order to relate k-dimensional disks that support the distributions to certain foliations that arise from Anosov diffeomorphisms. We conclude that generic rank-k random perturbations have absolutely continuous invariant measures.