We define staircase $\mathbb{Z}^d$ actions. We first prove that staircase $\mathbb{Z}^2$ actions satisfying a general condition are mixing. Then we describe how to extend the results to the staircase $\mathbb{Z}^d$ actions. Thus we have constructed explicitly rank one mixing $\mathbb{Z}^d$ actions which include natural analogues to the well-known staircase transformation.

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a homeomorphism of the plane. We define open sets $P \subset \mathbb{R}^{2}$, called pruning fronts after the work of Cvitanović et al. in 1988, for which it is possible to construct an isotopy $H: \mathbb{R}^{2} \times [0,1] \rightarrow \mathbb{R}^{2}$ with open support contained in $\bigcup _{n \in {\mathbb{Z}} } f^{n} (P)$ such that $H(\cdot, 0 ) = f(\cdot)$ and $H(\cdot, 1) = f_{P} (\cdot)$, where $f_P$ is a homeomorphism under which every point of $P$ is wandering. Applying this construction when $f$ is Smale's horseshoe, it is possible to obtain an uncountable family of homeomorphisms, depending on infinitely many parameters, going from trivial to chaotic dynamic behavior. This family is a two-dimensional analog of a one-dimensional universal family.

In a recent article, Manouchehri proved a ‘Sternberg theorem’ for Liouville vector fields and noticed that it provided normal forms for implicit differential equations and first-order partial differential equations. We establish local and global versions of Moser's celebrated result on volume and symplectic forms when they admit a non-trivial one parameter (pseudo-) group of homotheties—by definition, a Liouville field is the generator of such a flow. The local version implies that two germs of Liouville fields of a symplectic or volume form are conjugate by a diffeomorphism germ which preserves the form if and only if they are conjugate. This contains Manouchehri's theorem.

We construct the Green bundles for an energy level without conjugate points of a convex Hamiltonian. In this case we give a formula for the metric entropy of the Liouville measure and prove that the exponential map is a local diffeomorphism. We prove that the Hamiltonian flow is Anosov if and only if the Green bundles are transversal. Using the Clebsch transformation of the index form we prove that if the unique minimizing measure of a generic Lagrangian is supported on a periodic orbit, then it is a hyperbolic periodic orbit.

We also show some examples of differences with the behaviour of a geodesic flow without conjugate points, namely: (non-contact) flows and periodic orbits without invariant transversal bundles, segments without conjugate points but with crossing solutions and non-surjective exponential maps.

We explore substitution minimal systems and their relation to stationary Bratteli diagrams and stationary dimension groups. The constructions involved are algorithmic and explicit and render an effective method to compute an invariant of (ordered) $K$-theoretic nature for these systems. This new invariant is independent of spectral invariants which have previously been extensively studied. Before we state the main results we give some background.

Let $f$ be a smooth homeomorphism of the circle having one cubic-exponent critical point and irrational rotation number of bounded combinatorial type. Using certain pull-back and quasiconformal surgical techniques, we prove that the scaling ratios of $f$ about the critical point are asymptotically independent of $f$. This settles in particular the golden mean universality conjecture. We introduce the notion of a holomorphic commuting pair, a complex dynamical system that, in the analytic case, represents an extension of $f$ to the complex plane and behaves somewhat as a quadratic-like mapping. We define a suitable renormalization operator that acts on such objects. Through careful analysis of the family of entire mappings given by $z\mapsto z+\theta -(1/2\pi)\sin{2\pi z}$, $\theta$ real, we construct examples of holomorphic commuting pairs, from which certain necessary limit set pre-rigidity results are extracted. The rigidity problem for $f$ is thereby reduced to one of renormalization convergence. We handle this last problem by means of Teichmüller extremal methods made available through the recent work of Sullivan on Riemann surface laminations and renormalization of unimodal mappings.

Let $\mu$ be a probability measure on a locally compact group $G$, and suppose $G$ acts measurably on a probability measure space $(X,m)$, preserving the measure $m$. We study ergodic theoretic properties of the action along $\mu$-i.i.d. random walks on $G$. It is shown that under a (necessary) spectral assumption on the $\mu$-averaging operator on $L^2(X,m)$, almost surely the mean and the pointwise (Kakutani's) random ergodic theorems have roughly $n^{-1/2}$ rate of convergence. We also prove a central limit theorem for the pointwise convergence. Under a similar spectral condition on the diagonal $G$-action on $(X\times X,m\times m)$, an almost surely exponential rate of mixing along random walks is obtained.

The imposed spectral condition is shown to be connected to a strengthening of the ergodicity property, namely, the uniqueness of $m$-integration as a $G$-invariant mean on $L^\infty(X,m)$. These related conditions, as well as the presented sharp ergodic theorems, never occur for amenable $G$. Nevertheless, we provide many natural examples, among them automorphism actions on tori and actions on Lie groups' homogeneous spaces, for which our results can be applied.

In this paper we prove that any ergodic measurably distal system can be realized as a minimal topologically distal system with an invariant Borel measure of full support. The proof depends upon a theorem stating that every measurable function from a measurable system with its base space being a compact metric space to a connected compact group is cohomologous to a continuous function.

The modified Jacobi–Perron algorithm was shown by Ito etal in 1993 to be strongly convergent (in the sense of Brentjes 1981) almost everywhere with exponential rate. Their proof relies on very complicated computations. We will show that the original article by Podsypanin in 1977 on the modified Jacobi–Perron algorithm almost contains a proof of this convergence with exponential rate. The only ingredients missing in that article are some ergodic-theoretical facts about the transformation generating the approximations. This leads to a very simple proof of the before mentioned exponential convergence in the modified Jacobi–Perron algorithm.

We answer a question proposed by Barge and Franks in 1993 by showing that a bounded connected set in the plane which is closed or open and meets an infinite number of its iterates under an orientation-preserving fixed point free homeomorphism must meet all of them.

We introduce a set, $Q({\bf T})$, of Borel probability measures on the circle such that each $\mu\in Q({\bf T})$ obeys the conclusion of the Kerckhoff–Masur–Smillie theorem [3]: if $q$ is a meromorphic quadratic differential with at worst simple poles on a closed Riemann surface, then for each $\mu\in Q({\bf T})$ and $\mu$-a.e. $\zeta\in{\bf T}$, $\zeta q$ has uniquely ergodic vertical foliation. As an example, the normalized Cantor–Lebesgue measure belongs to $Q({\bf T})$. The analysis also yields an analogue, for the Teichmüller horocycle flow, of a theorem of Dani: every locally finite ergodic invariant measure for the Teichmüller horocycle flow is finite.