We present a general setting to investigate 𝒰fin-cocycle superrigidity for Gaussian actions in terms of closable derivations on von Neumann algebras. In this setting we give new proofs to some 𝒰fin-cocycle superrigidity results of S. Popa and we produce new examples of this phenomenon. We also use a result of K. R. Parthasarathy and K. Schmidt to give a necessary cohomological condition on a group representation in order for the resulting Gaussian action to be 𝒰fin-cocycle superrigid.

Let G be a Lie group acting in Hamiltonian fashion on a symplectic manifold M with moment map Φ:M → g*. A function of the form ƒ∘Φ where ƒ is a function on g* is called ‘collective’. We obtain necessary conditions on the G action for there to exist enough Poisson commuting functions on g* so that the corresponding collective functions on M form a completely integrable system. For the case G = O(n) or U(n) these conditions are sufficient. This explains Thimm's proof [17] of the complete integrability of the geodesic flow on the real and complex grassmanians. We also discuss related questions in the geometry of the moment map.

We study Markov chains via invariants constructed from periodic orbits. Canonical extensions, based on these invariants, are used to establish a constraint on the degree of finite-to-one block homomorphisms from one Markov chain to another. We construct a polytope from the normalized weights of periodic orbits. Using this polytope, we find canonically-defined induced Markov chains inside the original Markov chain. Each of the invariants associated with these Markov chains gives rise to a scaffold of invariants for the original Markov chain. This is used to obtain counterexamples to the finite equivalence conjecture and to a conjecture regarding finitary isomorphism with finite expected coding time. Also included are results related to the problem of minimality (with respect to block homomorphism) of Bernoulli shifts in the class of Markov chains with beta function equal to the beta function of the Bernoulli shift.

We show the stability of an elliptic fixed point of an analytic area-preserving mapping under the Bruno condition, using Moser's Twist-mapping Theorem. We require that there are non-vanishing Birkhoff constants and prove the convergence of the transformation to the Birkhoff normal form if all Birkhoff constants are zero. The convergence theorem is formulated and proved for arbitrary dimensions.

Hill's lunar problem appears in celestial mechanics as a limit of the restricted three-body problem. It is parameter-free and thus globally far from any simple well-known problem, and has shed strong numerical evidence of its lack of integrability in the past. An algebraic proof of meromorphic non-integrability is presented here. Beyond the result itself, the paper can also be considered as an example of the application of differential Galois and Morales–Ramis theories to a significant problem.

For a given expanding d-fold covering transformation of the one-dimensional torus, the notion of weak Gibbs measure is defined by a natural generalization of the classical Gibbs property. For these measures, we prove that the singularity spectrum and the $L^q$-spectrum form a Legendre transform pair. The main difficulty comes from the possible existence of first-order phase transition points, that is, points where the $L^q$-spectrum is not differentiable. We give examples of weak Gibbs measure with phase transition, including the so-called Erdös measure.

In this paper we consider an analog of the regions of instability for twist maps in the context of area preserving diffeomorphisms which are not twist maps. Several properties analogous to those of classical regions of instability are proved.

We estimate from below the measure theoretic entropy of the system of spheres interacting by elastic collisions. We find the asymptotics of the entropy in the case of two disks on atoms as the radius increases. We establish that high density leads to large entropy. We introduce a general estimate for Lyapunov exponents in the case of special symplectic matrices.

Generalizing the one-parameter case, we prove that the orbit of a point on a compact nilmanifold X under a polynomial action of $\mathbb{Z}^{d}$ by translations on X is uniformly distributed on the union of several sub-nilmanifolds of X. As a corollary we obtain the pointwise ergodic theorem for polynomial actions of $\mathbb{Z}^{d}$ by translations on a nilmanifold.

We construct Bratteli–Vershik models for Toeplitz flows and characterize a class of properly ordered Bratteli diagrams corresponding to these flows. We use this result to extend by a novel approach—using basic theory of dimension groups—an interesting and non-trivial result about Toeplitz flows, first shown by Downarowicz. (Williams had previously obtained preliminary results in this direction.) The result states that to any Choquet simplex $K$, there exists a $0$–$1$ Toeplitz flow $(Y,\psi)$, so that the set of invariant probability measures of $(Y,\psi)$ is affinely homeomorphic to $K$. Not only do we give a conceptually new proof of this result, we also show that we may choose $(Y,\psi)$ to have zero entropy and to have full rational spectrum.

Furthermore, our Bratteli–Vershik model for a given Toeplitz flow explicitly exhibits the factor map onto the maximal equicontinuous (odometer) factor. We utilize this to give a simple proof of the existence of a uniquely ergodic 0–1 Toeplitz flow of zero entropy having a given odometer as its maximal equicontinuous factor and being strongly orbit equivalent to this factor. By the same token, we show the existence of 0–1 Toeplitz flows having the 2-odometer as their maximal equicontinuous factor, being strong orbit equivalent to the same, and assuming any entropy value in $[0,\ln 2)$.

Finally, we show by an explicit example, using Bratteli diagrams, that Toeplitz flows are not preserved under Kakutani equivalence (in fact, under inducing)—contrasting what is the case for substitution minimal systems. In fact, the example we exhibit is an induced system of a 0–1 Toeplitz flow which is conjugate to the Chacon substitution system, thus it is prime, i.e. it has no non-trivial factors.

The thrust of our paper is to demonstrate the relevance and usefulness of Bratteli–Vershik models and dimension group theory for the study of minimal symbolic systems. This is also exemplified in recent papers by Forrest and by Durand, Host and Skau, treating substitution minimal systems, and by papers by Boyle, Handelman and by Ormes.

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.

Given a measure-preserving equivalence relation R with countable classes, we study relations between the properties of R and metric invariants. We give applications to pseudogroups of measure-preserving homeomorphisms.

In this paper we consider the family of rational maps of the complex plane given by \[z^2+\frac{\lambda}{z^2}\] where $\lambda$ is a complex parameter. We regard this family as a singular perturbation of the simple function $z^2$. We show that, in any neighborhood of the origin in the parameter plane, there are infinitely many open sets of parameters for which the Julia sets of the corresponding maps are Sierpinski curves. Hence all of these Julia sets are homeomorphic. However, we also show that parameters corresponding to different open sets have dynamics that are not conjugate.

It is shown that systems with hyperbolic structure have the Bernoulli property. Some new results on smooth cross-sections of hyperbolic Bernoulli flows are also derived. The proofs involve an abstract version of our original methods for showing that the geodesic flow on surfaces of negative curvature are Bernoulli.

A partially hyperbolic diffeomorphism is dynamically coherent if its center, center-stable and center-unstable invariant distributions are integrable, i.e. tangent to foliations. Dynamical coherence is a key assumption in the theory of stable ergodicity. The main result: a partially hyperbolic diffeomorphism f\colon M\to M is dynamically coherent if the strong stable and unstable foliations are quasi-isometric in the universal cover \widetilde{M}, i.e. for any two points in the same leaf, the distance between them in \widetilde{M} is bounded from below by a linear function of the distance along the leaf.

We study aperiodic substitution dynamical systems arising from non-primitive substitutions. We prove that the Vershik homeomorphism φ of a stationary ordered Bratteli diagram is topologically conjugate to an aperiodic substitution system if and only if no restriction of φ to a minimal component is conjugate to an odometer. We also show that every aperiodic substitution system generated by a substitution with nesting property is conjugate to the Vershik map of a stationary ordered Bratteli diagram. It is proved that every aperiodic substitution system is recognizable. The classes of m-primitive substitutions and derivative substitutions associated with them are studied. We discuss also the notion of expansiveness for Cantor dynamical systems of finite rank.

Let f be an orientation preserving ℋ-diffeomorphism of the circle. If the rotation number α = ρ(f) is irrational and log Df is of bounded variation then, by a wellknown theorem of Denjoy, f is conjugate to the rigid rotation Rα. The conjugation means that there exists an essentially unique homeomorphism h of the circle such that f = h−lRαh. The general problem of relating the smoothness of h to that of f under suitable diophantine conditions on α has been studied extensively (cf. [H1], [KO], [Y] and the references given there). At the bottom of the scale of smoothness for f there is a theorem of M. Herman [H2] which states that if Df is absolutely continuous and D log Df ∈ Lp, p > 1, α = ρ (f) is of ‘constant type’ which means ‘the coefficients in the continued fraction expansion of α are bounded’, and if f is a perturbation of Rα, then h is absolutely continuous. Our purpose in this paper is to give a different proof and an improved version of Herman's theorem. The main difference in the result is that we do not need to assume that f is close to Rα; the proof is very different from Herman's and is very much in the spirit of [KO].

We define and study new invariants called pre-image entropies which are similar to the standard notions of topological and measure-theoretic entropies. These new invariants are only non-zero for non-invertible maps, and they give a quantitative measurement of how far a given map is from being invertible. We obtain analogs of many known results for topological and measure-theoretic entropies. In particular, we obtain product rules, power rules, analogs of the Shannon–Breiman–McMillan theorem, and a variational principle.

In this paper we introduce and explore conformal parabolic iterated function systems. We define and study topological pressure, Perron–Frobenius-type operators, semiconformal and conformal measures and the Hausdorff dimension of the limit set. With every parabolic system we associate an infinite hyperbolic conformal iterated function system and we employ it to study geometric and dynamical features (properly defined invariant measures for example) of the limit set.

A distortion theory is developed for S-unimodal maps. It will be used to get some geometric understanding of invariant Cantor sets. In particular attracting Cantor sets turn out to have Lebesgue measure zero. Furthermore the ergodic behavior of S-unimodal maps is classified according to a distortion property, called the Markov-property.