We extend the recent progress on the cocycle rigidity of partially hyperbolic homogeneous abelian actions to the setting with rank-one factors in the universal cover. The method of proof relies on the periodic cycle functional and analysis of the cycle structure, but uses a new argument to give vanishing.

We consider the random $\unicode[STIX]{x1D6FD}$-transformation $K_{\unicode[STIX]{x1D6FD}}$ introduced by Dajani and Kraaikamp [Random $\unicode[STIX]{x1D6FD}$-expansions. Ergod. Th. & Dynam. Sys.23 (2003), 461–479], which is defined on $\{0,1\}^{\mathbb{N}}\times [0,[\unicode[STIX]{x1D6FD}]/(\unicode[STIX]{x1D6FD}-1)]$. We give an explicit formula for the density function of a unique $K_{\unicode[STIX]{x1D6FD}}$-invariant probability measure absolutely continuous with respect to the product measure $m_{p}\otimes \unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FD}}$, where $m_{p}$ is the $(1-p,p)$-Bernoulli measure on $\{0,1\}^{\mathbb{N}}$ and $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FD}}$ is the normalized Lebesgue measure on $[0,[\unicode[STIX]{x1D6FD}]/(\unicode[STIX]{x1D6FD}-1)]$. We apply the explicit formula for the density function to evaluate its upper and lower bounds and to investigate its continuity as a function of the two parameters $p$ and $\unicode[STIX]{x1D6FD}$.

We study the effect of the mapping class group of a reducible 3-manifold $M$ on each incompressible surface that is invariant under a self-homeomorphism of $M$. As an application of this study we answer a question of F. Rodriguez Hertz, M. Rodriguez Hertz, and R. Ures: a reducible 3-manifold admits an Anosov torus if and only if one of its prime summands is either the 3-torus, the mapping torus of $-\text{id}$, or the mapping torus of a hyperbolic automorphism.

In this paper, we consider the product space of several non-compact finite-volume hyperbolic spaces, $V_{1},V_{2},\ldots ,V_{k}$ of dimension $n$. Let $\text{T}^{1}(V_{i})$ denote the unit tangent bundle of $V_{i}$ and $g_{t}$ denote the geodesic flow on $\text{T}^{1}(V_{i})$ for each $i=1,\ldots ,k$. We define

$$\begin{eqnarray}{\mathcal{D}}_{k}:=\{(v_{1},\ldots ,v_{k})\,\in \,\text{T}^{1}(V_{1})\times \cdots \times \text{T}^{1}(V_{k})\,:\,(g_{t}(v_{1}),\ldots ,g_{t}(v_{k}))\text{ diverges as }t\rightarrow \infty \}.\end{eqnarray}$$

${\mathcal{D}}_{k}$

$k(2n-1)-((n-1)/2)$

We consider suspension flows over uniquely ergodic skew-translations on a $d$-dimensional torus $\mathbb{T}^{d}$ for $d\geq 2$. We prove that there exists a set $\mathscr{R}$ of smooth functions, which is dense in the space $\mathscr{C}(\mathbb{T}^{d})$ of continuous functions, such that every roof function in $\mathscr{R}$ which is not cohomologous to a constant induces a mixing suspension flow. We also construct a dense set of mixing examples which is explicitly described in terms of their Fourier coefficients. In the language of nilflows on nilmanifolds, our result implies that, for every uniquely ergodic nilflow on a quasi-abelian filiform nilmanifold, there exists a dense subspace of smooth time-changes in which mixing occurs if and only if the time-change is not cohomologous to a constant. This generalizes a theorem by Avila, Forni and Ulcigrai [Mixing for time-changes of Heisenberg nilflows. J. Differential Geom.89(3) (2011), 369–410] for the classical Heisenberg group.

Recently, there has been an increasing interest in non-autonomous composition of perturbed hyperbolic systems: composing perturbations of a given hyperbolic map $F$ results in statistical behaviour close to that of $F$. We show this fact in the case of piecewise regular expanding maps. In particular, we impose conditions on perturbations of this class of maps that include situations slightly more general than what has been considered so far, and prove that these are stochastically stable in the usual sense. We then prove that the evolution of a given distribution of mass under composition of time-dependent perturbations (arbitrarily—rather than randomly—chosen at each step) close to a given map $F$ remains close to the invariant mass distribution of $F$. Moreover, for almost every point, Birkhoff averages along trajectories do not fluctuate wildly. This result complements recent results on memory loss for non-autonomous dynamical systems.

We first consider the dynamics of a class of meromorphic skew products having superattracting fixed points or fixed indeterminacy points at the origin. Our theorem asserts that, if a map has a suitable weight, then it is conjugate to the associated monomial map on an invariant open set whose closure contains the origin. We next extend this result to a wider class of meromorphic maps such that the eigenvalues of the associated matrices are real and greater than $1$.

Motivated by the classical results by Halmos and Rokhlin on the genericity of weakly but not strongly mixing transformations and the Furstenberg tower construction, we show that weakly but not strongly mixing extensions on a fixed product space with both measures non-atomic are generic. In particular, a generic extension does not have an intermediate nilfactor.