Rational weak mixing is a measure theoretic version of Krickeberg’s strong ratio mixing property for infinite measure preserving transformations. It requires ‘density’ ratio convergence for every pair of measurable sets in a dense hereditary ring. Rational weak mixing implies weak rational ergodicity and (spectral) weak mixing. It is enjoyed for example by Markov shifts with Orey’s strong ratio limit property. The power, subsequence version of the property is generic.

Extending a result of Mañé, Hayashi proved that every diffeomorphism $f$ which has a $C^1$-neighborhood $\mathcal {U}$ where all periodic points of any $g\in \mathcal {U}$are hyperbolic is an Axiom A diffeomorphism. Here we prove the analogous result in the volume-preserving scenario.

Let $G$ be a connected nilpotent Lie group. Given probability-preserving$G$-actions $(X_i,\Sigma _i,\mu _i,u_i)$, $i=0,1,\ldots ,k$, and also polynomial maps $\phi _i:\mathbb {R}\to G$, $i=1,\ldots ,k$, we consider the trajectory of a joining $\lambda $ of the systems $(X_i,\Sigma _i,\mu _i,u_i)$ under the ‘off-diagonal’ flow

\[ (t,(x_0,x_1,x_2,\ldots ,x_k))\mapsto (x_0,u_1^{\phi _1(t)}x_1,u_2^{\phi _2(t)}x_2,\ldots ,u_k^{\phi _k(t)}x_k). \]

$\lambda $

$\lambda '$

$\lambda '$

$G^{k+1}$

We prove the following dichotomy for vector fields in a $C^1$-residual subset of volume-preserving flows: for Lebesgue-almost every point, either all of its Lyapunov exponents are equal to zero or its orbit has a dominated splitting. Moreover, we prove that a volume-preserving and $C^1$-stably ergodic flow can be $C^1$-approximated by another volume-preserving flow which is non-uniformly hyperbolic.

We study orbit spaces of generalized gradient vector fields for Morse functions. Typically, these orbit spaces are non-Hausdorff. Nevertheless, they are quite structured topologically and are amenable to study. We show that these orbit spaces are locally contractible. We also show that the quotient map associated to each such orbit space is a weak homotopy equivalence and has the path lifting property.

We demonstrate essential coexistence of hyperbolic and non-hyperbolic behavior in the continuous-time case by constructing a smooth volume preserving flow on a five-dimensional compact smooth manifold that has non-zero Lyapunov exponents almost everywhere on an open and dense subset of positive but not full volume and is ergodic on this subset while having zero Lyapunov exponents on its complement. The latter is a union of three-dimensional invariant submanifolds, and on each of these submanifolds the flow is linear with Diophantine frequency vector.

We study dynamics of continuous maps on compact metrizable spaces containing a free interval (i.e. an open subset homeomorphic to an open interval). Special attention is paid to relationships between topological transitivity, weak and strong topological mixing, dense periodicity and topological entropy as well as to the topological structure of minimal sets. In particular, a trichotomy for minimal sets and a dichotomy for transitive maps are proved.

Leaves of laminations can be symbolically represented by deforming them into paths of labeled embedded carrier graphs, including train tracks. Here, we describe and characterize the languages of two-way infinite words coming from this kind of coding, called lamination languages, first, by using carrier graph sequences, and second, by using word combinatorics. These characterizations generalize those existing for interval exchange transformations. We also show that lamination languages have ultimately affine factor complexity, and we present effective techniques to build these languages.

We show that for a locally free $C^{\infty }$-action of a connected and simply connected nilpotent Lie group on a compact manifold, if every real-valued cocycle is cohomologous to a constant cocycle, then the action is parameter rigid. The converse is true if the action has a dense orbit. Using this, we construct parameter rigid actions of simply connected nilpotent Lie groups whose Lie algebras admit rational structures with graduations. This generalizes the results of dos Santos [Parameter rigid actions of the Heisenberg groups. Ergod. Th. & Dynam. Sys.27(2007), 1719–1735] concerning the Heisenberg groups.

A caustic of a billiard table is a curve such that any billiard trajectory, once tangent to the curve, stays tangent after every reflection at the boundary. When the billiard table is an ellipse, any non-singular billiard trajectory has a caustic, which can be either a confocal ellipse or a confocal hyperbola. Resonant caustics—those whose tangent trajectories are closed polygons—are destroyed under generic perturbations of the billiard table. We prove that none of the resonant elliptical caustics persists under a large class of explicit perturbations of the original ellipse. This result follows from a standard Melnikov argument and the analysis of the complex singularities of certain elliptic functions.

We consider subshifts of the full shift of bi-infinite sequences with alphabet $\{ 0, 1, \ldots , n- 1\} $ defined by not allowing the sum of finite words to exceed a value depending on its length. These shifts we call bounded density shifts. We study these shifts in detail and make a comparison on the similarities to but also differences from the well-known $\beta $-shifts.