We study the structure of two-fold self-joinings for a typical transformation of the standard non-atomic Borel space $(X,\mu)$. We prove that the typical automorphism is not simple and semisimple. This is an answer to questions posed by del Junco, Rudolph, Lemanczyk and Mentzen. In order to answer these questions we proved that for a typical transformation there exists an ergodic self-joining concentrated on two fibers, i.e. there exists the simplest non-trivial ergodic self-joining.

Let $(X,\mathcal{B},\mu, T)$ be an ergodic dynamical system on the finite measure space $(X,\mathcal{B},\mu )$. Let $\mathcal{K}$ denote the Kronecker factor of T, i.e. the closed linear span in L2 of the eigenfunctions for T. We say that $(X,\mathcal{B},\mu ,T)$ is a Wiener–Wintner (WW) dynamical system of power type $\alpha$ in L1 if there exists in $\mathcal{K}^{\bot}$ a dense set of functions f for which the following holds: there exists a finite positive constant Cf such that \[\bigg\| \sup_{\varepsilon} \bigg|\frac{1}{N} \sum_{n=1}^Nf\circ T^n e^{2\pi in\varepsilon}\bigg| \bigg\|_1\leq \frac{C_f}{N^{\alpha}}\] for all positive integers N. Examples of ergodic dynamical systems with this WW property include K automorphisms as well as some skew products over irrational rotations. For WW dynamical systems a simpler proof of the almost everywhere double recurrence property, random weights with a break of duality can be obtained. They also provide naturally almost everywhere continuous random Fourier series related to the spectral measure of the transformation.

In an explicit family of partially hyperbolic diffeomorphisms of the torus T3, Shub and Wilkinson recently succeeded in perturbing the Lyapunov exponents of the center direction. We present here a local version of their argument, allowing one to perturb the center Lyapunov exponents of any partially hyperbolic system, in any dimension and with arbitrary dimension of the center bundle.

This paper shows that any symplectic diffeomorphism $\mathbf{F} :(\Sigma^{2n},\eta) \to (\Sigma^{2n},\eta)$ can be embedded as a subsystem of a Liouville-integrable Hamiltonian flow on some symplectic manifold. If F is real-analytic, then the flow can be chosen to be real-analytic, but it is Liouville integrable with smooth first integrals. Examples are constructed of integrable, volume-preserving Hamiltonian flows on Poisson manifolds whose metric entropy with respect to the volume form is positive. Completely integrable Hamiltonian flows on a symplectic manifold are constructed which have positive metric entropy with respect to an invariant probability measure that is absolutely continuous with respect to the canonical volume form.

The purpose of this article is two fold. First, we show how quasi-periodic solutions for the N-body problem can be constructed by variational methods. We illustrate this by constructing uncountably many quasi-periodic solutions for the four- and six-body problems with equal masses. Second, we show by examples that a system of N masses can possess infinitely many simple or multiple choreographic solutions. In particular, it is shown that the four-body problem with equal masses has infinitely many double choreographic solutions and the six-body problem with equal masses has infinitely many simple and double choreographic solutions. Our approach is based on the technique of binary decomposition and some variational properties of Keplerian orbits.

We give a partial answer to a question posed by C. Pugh and M. Shub in [PS2]. A partially hyperbolic diffeomorphism $f\in \mathit{Diff}^{2}(M)$ is accessible if every two points of M can be connected by a path which is piecewise in a stable or an unstable leaf. We prove that accessibility is open in $\mathit{Diff}^{2}(M)$ among the partially hyperbolic diffeomorphisms with one-dimensional center bundle.

We give a new definition for the crossed-product of a C*-algebra A by a *-endomorphism $\alpha$, which depends not only on the pair $(A,\alpha)$ but also on the choice of a transfer operator. With this we generalize some of the earlier constructions in the situations in which they behave best (e.g. for monomorphisms with hereditary range), but we get a different and perhaps more natural outcome in other situations. For example, we show that the Cuntz–Krieger algebra $\mathcal{O}_{\mathcal A}$ arises as the result of our construction when applied to the corresponding Markov subshift and a very natural transfer operator.

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.

We study the relation between topological entropy and the dispersion of preimages. Symbolic dynamics plays a crucial role in our investigation. For forward expansive maps, we show that the two pointwise preimage entropy invariants defined by Hurley agree with each other and with topological entropy, and are reflected in the growth rate of the number of preimages of a single point, called a preimage growth point for the map. We extend this notion to that of an entropy point for a system, in which the dispersion of preimages of an $\varepsilon$-stable set measures topological entropy. We show that for maps satisfying a weak form of the specification property, every point is an entropy point and that every asymptotically h-expansive homeomorphism (in particular, every smooth diffeomorphism of a compact manifold) has entropy points. Examples are given of maps in which Hurley's invariants differ and of homeomorphisms with no entropy points.

For a sequence (cn) of complex numbers, the quadratic polynomials $f_{c_n}:=z^2+c_n$ and the sequence (Fn) of iterates $F_n:=f_{c_n}\circ\dotsb \circ f_{c_1}$ are considered. The Fatou set $\mathcal{F}(c_n)$ is defined as the set of all $z\in \hat{\mathbb{C}}: =\mathbb{C}\cup \{\infty\}$ such that (Fn) is normal in some neighbourhood of z, while the complement $\mathcal{J}(c_n)$ of $\mathcal{F}(c_n)$ (in $\hat{\mathbb{C}}$) is called the Julia set. In this paper we discuss the conditions for $\mathcal{J}(c_n)$ to be totally disconnected. A problem posed by Brück is solved.

Let $\Gamma$ be a lattice in $G={\rm SL}(n,\mathbb{R})$ and X = G/S be a homogeneous space of G, where S is a closed subgroup of G which contains a real algebraic subgroup H such that G/H is compact. We establish the uniform distribution of orbits of $\Gamma$ in X analogous to the classical equidistribution on a torus. To obtain this result, we first prove an ergodic theorem along balls in the connected component of a Borel subgroup of G.

A locally free Lie group action on a closed manifold is called parameter rigid, if given another action with the same orbit foliation, there is a $C^\infty$ orbit-preserving diffeomorphism which conjugates one action to the other, up to an automorphism of the Lie group. We show some relationships between parameter rigidity and the first leafwise cohomology of the orbit foliation, and give a new example of a parameter rigid solvable group action. We also compute the first leafwise cohomology of the stable foliation of geodesic flows of closed orientable surfaces of constant negative curvature.

We prove measurable Livšic theorems for Lie group valued cocycles over certain classes of uniformly hyperbolic maps with discontinuities, in particular the $\beta$-transformation, C2 Markov maps and a simple class of toral linked twist maps. As an application, we establish stable ergodicity results for compact group extensions of certain classes of uniformly hyperbolic systems with discontinuities.

We construct equilibrium states, including measures of maximal entropy, for a large (open) class of non-uniformly expanding maps on compact manifolds. Moreover, we study uniqueness of these equilibrium states, as well as some of their ergodic properties.

In this paper we consider random dynamical systems generated by compositions of one-sided independent and identically distributed random diffeomorphisms of class C2 on a compact manifold. We prove an entropy formula for such random dynamical systems without assuming the SRB condition. This result is the random version of the main result obtained by Ledrappier and Young (The metric entropy of diffeomorphisms. Part II: Relations between entropy, exponents and dimension. Ann. Math. 122 (1985), 540–574).