Let α be an automorphism of a finite von Neumann algebra and let H(α) be its Connes-Størmer's entropy. We show that H(α) = 0 if α is the induced automorphism on the crossed product of a Lebesgue space by a pure point spectrum transformation. We also show that H is not continuous in α and we compute H(α) for some α.

The pentagram map, introduced by Schwartz [The pentagram map. Exp. Math. 1(1) (1992), 71–81], is a dynamical system on the moduli space of polygons in the projective plane. Its real and complex dynamics have been explored in detail. We study the pentagram map over an arbitrary algebraically closed field of characteristic not equal to 2. We prove that the pentagram map on twisted polygons is a discrete integrable system, in the sense of algebraic complete integrability: the pentagram map is birational to a self-map of a family of abelian varieties. This generalizes Soloviev’s proof of complex integrability [F. Soloviev. Integrability of the pentagram map. Duke Math. J. 162(15) (2013), 2815–2853]. In the course of the proof, we construct the moduli space of twisted n-gons, derive formulas for the pentagram map, and calculate the Lax representation by characteristic-independent methods.

We survey results about, and results using, central sets since their introduction in 1981.

We study actions of higher rank lattices $\Gamma <G$ on hyperbolic spaces and we show that all such actions satisfying mild properties come from the rank-one factors of G. In particular, all non-elementary isometric actions on an unbounded hyperbolic space are of this type.

Given a topological process (X, µ, T) where T is a homeomorphism of the compact metric space X which preserves the probability measure µ and is ergodic, we show that there exists an uncountable family {(Xi, µi, Ti)}i∈I of topological processes such that for every i, (Xi, µi, Ti) is measure-theoretically isomorphic to (X, µ, T) but for every i ≠ j, (Xi, µi, Ti) and (Xj, µj, Tj) are not almost topologically conjugate.

We introduce a notion of sensitivity with respect to a continuous real-valued bounded map which provides a sufficient condition for a continuous transformation, acting on a Baire metric space, to exhibit a Baire generic subset of points with historic behavior (also known as irregular points). The applications of this criterion recover, and extend, several known theorems on the genericity of the irregular set, in addition to yielding a number of new results, including information on the irregular set of geodesic flows, in both negative and non-positive curvature, and semigroup actions.

Given a measure-preserving transformation T on a Lebesgue σ-algebra, a complete T-invariant sub-σ-algebra is said to split if there is another complete T-invariant sub-σ-algebra on which T is Bernoulli which is completely independent of the given sub-σ-algebra and such that the two sub-σ-algebras together generate the entire σ-algebra. It is easily shown that two splitting sub-σ-algebras with nothing in common imply T to be K. Here it is shown that T does not have to be Bernoulli by exhibiting two such non-intersecting σ-algebras for the T,T−1 transformation, negatively answering a question posed by Thouvenot in 1975.

An iterative square root of a self-map f is a self-map g such that $g(g(\cdot ))=f(\cdot )$ . We obtain new characterizations for detecting the non-existence of such square roots for self-maps on arbitrary sets. They are used to prove that continuous self-maps with no square roots are dense in the space of all continuous self-maps for various topological spaces. The spaces studied include those that are homeomorphic to the unit cube in ${\mathbb R}^{m}$ and to the whole of $\mathbb {R}^{m}$ for every positive integer $m.$ However, we also prove that every continuous self-map on a space homeomorphic to the unit cube in $\mathbb {R}^{m}$ with a fixed point on the boundary can be approximated by iterative squares of continuous self-maps.

In this paper, we give an elementary proof of the absence of invariant line fields on the conical Julia set of an analytic function of one variable. This proof applies not only to rational and transcendental meromorphic functions (where it was previously known), but even to the extremely general setting of Ahlfors islands maps as defined by Epstein. In fact, we prove a more general result on the absence of invariant differentials, measurable with respect to a conformal measure that is supported on the (unbranched) conical Julia set. This includes the study of cohomological equations for log ∣f′∣, which are relevant to a number of well-known rigidity questions. In particular, we prove the absence of continuous line fields on the Julia set of any transcendental entire function.

Let $(M,g,J)$ be a closed Kähler manifold with negative sectional curvature and complex dimension $m := \dim _{\mathbb {C}} M \geq 2$. In this article, we study the unitary frame flow, that is, the restriction of the frame flow to the principal $\mathrm {U}(m)$-bundle $F_{\mathbb {C}}M$ of unitary frames. We show that if $m \geq 6$ is even and $m \neq 28$, there exists $\unicode{x3bb} (m) \in (0, 1)$ such that if $(M, g)$ has negative $\unicode{x3bb} (m)$-pinched holomorphic sectional curvature, then the unitary frame flow is ergodic and mixing. The constants $\unicode{x3bb} (m)$ satisfy $\unicode{x3bb} (6) = 0.9330...$, $\lim _{m \to +\infty } \unicode{x3bb} (m) = {11}/{12} = 0.9166...$, and $m \mapsto \unicode{x3bb} (m)$ is decreasing. This extends to the even-dimensional case the results of Brin and Gromov [On the ergodicity of frame flows. Invent. Math. 60(1) (1980), 1–7] who proved ergodicity of the unitary frame flow on negatively curved compact Kähler manifolds of odd complex dimension.

Although detailed descriptions of the possible types of behaviour inside periodic Fatou components have been known for over 100 years, a classification of wandering domains has only recently been given. Recently, simply connected wandering domains were classified into nine possible types and examples of escaping wandering domains of each of these types were constructed. Here we consider the case of oscillating wandering domains, for which only six of these types are possible. We use a new technique based on approximation theory to construct examples of all six types of oscillating simply connected wandering domains. This requires delicate arguments since oscillating wandering domains return infinitely often to a bounded part of the plane. Our technique is inspired by that used by Eremenko and Lyubich to construct the first example of an oscillating wandering domain, but with considerable refinements which enable us to show that the wandering domains are bounded, to specify the degree of the mappings between wandering domains and to give precise descriptions of the dynamical behaviour of these mappings.

Let α1,…,αn be irrational numbers which are linearly independent over the rationals, and f1,…,fn commuting germs of holomorphic diffeomorphisms in ℂ such that fk(0)=0,f′k(0)=e2πiαk,k=1,…,n. Moser showed that f 1,…,fn are simultaneously linearizable (i.e. conjugate by a germ of holomorphic diffeomorphism h to the rigid rotations Rαk(z)=e2πiαkz) if the vector of rotation numbers (α 1,…,αn) satisfies a Diophantine condition. Adapting Yoccoz’s renormalization to the setting of commuting germs, we show that simultaneous linearization holds in the presence of a weaker Brjuno-type condition ℬ(α1,…,αn)<+∞, where ℬ(α1,…,αn) is a multivariable analogue of the Brjuno function. If there are no periodic orbits for the action of the germs f1,…,fn in a neighbourhood of the origin, then a weaker arithmetic condition ℬ′ (α 1,…,αn)<+∞ analogous to Perez-Marco’s condition for linearization in the absence of periodic orbits is shown to suffice for linearizability. Normalizing the germs to be univalent on the unit disc, in both cases the Siegel discs are shown to contain discs of radii for some universal constant C.

We obtain quenched almost sure invariance principles (with convergence rates) for random Young towers if the average measure of the tail of return times to the base of random towers decays sufficiently fast. We apply our results to some independent and identically distributed perturbations of some non-uniformly expanding maps. These imply that the random systems under study tend to a Brownian motion under various scalings.

We show that among Cr extensions (r>0) of a uniformly hyperbolic dynamical system with fiber the standard real Heisenberg group ℋn of dimension 2n+1, those that avoid an obvious obstruction to topological transitivity are generically topologically transitive. Moreover, if one considers extensions with fiber a connected nilpotent Lie group with a compact commutator subgroup (for example ℋn/ℤ), among those that avoid the obvious obstruction, topological transitivity is open and dense.

We prove that for each ergodic automorphism T:(X, ℬ, μ)→(X, ℬ, μ) for which we can find an element S∈C(T) such that the corresponding Z2-action (S, T) on (X, ℬ, μ) is free, there exists a circle valued cocycle φ such that the group extension Tφ is ergodic but is not coalescent. In particular, the existence of such a cocycle is proved for all ergodic rigid automorphisms. As a corollary, in the class of ergodic transformations of [0,1) × [0,1) given by

for each irrational α we find φ such that Tφ is not coalescent. In some special cases the group law of the centralizer is given.

We construct linked twist maps of the two-dimensional torus which are Bernoulli and possess infinite entropy. Inparticular, we construct a Bernoulli toral linked twist map B* of infinite entropy which has smooth, absolutely continuous local (un)stable manifolds and positive Lyapunov exponents defined almost everywhere. This map is continuous at each point save those on two line segments.

B* is shown to be stochastically stable under the following random perturbation: apply the map to a point p and then jump (all points move the same distance and in the same direction) according to a B-process (not necessarily an independent process) such that the expected distance moved is equal to r. Stochastic stability means that given α > 0 if r > 0 is sufficiently small then the perturbed and unperturbed systems are α-congruent. We prove a similar stability result for B* under a perturbation in which the random jump described above is distributed according to a general stochastic process. These stability results are also shown to hold (in a slightly modified form) for a general class of finite-entropy toral linked twist maps under the same perturbations.

Let f be a continuous map of the circle into itself of degree one. We introduce the notion of rotation algorithms. One of these algorithms associates each z ∈ S1 with an interval, the so-called speed interval S(z, f), which is contained in the rotation interval ρ(f) of f. In contrast with the rotation set ρ(z, f), the interval S(z, f) sometimes allows us to ascertain that ρ(f) is non-degenerate, by using only finitely many elements of {fn (z) | n ≥ 0}. We further show that all choices for ρ(z, f) and S(z, f) occur, for certain z ∈ S1 provided that ρ(z, f) ⊂ S(z, f) ⊂ ρ(f).

Relationships between the permutations associated to periodic orbits of maps of the interval and the topology of the space of these maps are given. These extend the author's previous work on periodic orbits to orbit types.