We give conditions under which non-uniformly expanding maps exhibit lower bounds of polynomial type for the decay of correlations and for a large class of observables. We show that if the Lasota–Yorke-type inequality for the transfer operator of a first return map is satisfied in a Banach space ${\mathcal{B}}$, and the absolutely continuous invariant measure obtained is weak mixing, in terms of aperiodicity, then, under some renewal condition, the maps have polynomial decay of correlations for observables in ${\mathcal{B}}.$ We also provide some general conditions that give aperiodicity for expanding maps in higher dimensional spaces. As applications, we obtain lower bounds for piecewise expanding maps with an indifferent fixed point and for which we also allow non-Markov structure and unbounded distortion. The observables are functions that have bounded variation or satisfy quasi-Hölder conditions and have their support bounded away from the neutral fixed points.

We study the homotopical rotation vectors and the homotopical rotation sets for the billiard flow on the unit flat torus with two disjoint and orthogonal toroidal (cylindrical) scatterers removed from it. The natural habitat for these objects is the infinite cone erected upon the Cantor set $\text{Ends}(G)$ of all ‘ends’ of the hyperbolic group $G=\unicode[STIX]{x1D70B}_{1}(\mathbf{Q})$. An element of $\text{Ends}(G)$ describes the direction in (the Cayley graph of) the group $G$ in which the considered trajectory escapes to infinity, whereas the height function $s$ ($s\geq 0$) of the cone gives us the average speed at which this escape takes place. The main results of this paper claim that the orbits can only escape to infinity at a speed not exceeding $\sqrt{3}$ and, in any direction $e\in \text{Ends}(\unicode[STIX]{x1D70B}_{1}({\mathcal{Q}}))$, the escape is feasible with any prescribed speed $s$, $0\leq s\leq 1/(\sqrt{6}+2\sqrt{3})$. This means that the radial upper and lower bounds for the rotation set $R$ are actually pretty close to each other. Furthermore, we prove the convexity of the set $\mathit{AR}$ of constructible rotation vectors, and that the set of rotation vectors of periodic orbits is dense in $\mathit{AR}$. We also provide effective lower and upper bounds for the topological entropy of the studied billiard flow.

We consider the actions of (semi)groups on a locally compact group by automorphisms. We show the equivalence of distality and pointwise distality for the actions of a certain class of groups. We obtain a decomposition for contraction groups of an automorphism under certain conditions. We give a necessary and sufficient condition for distality of an automorphism in terms of its contraction group. We compare classes of (pointwise) distal groups and groups whose closed subgroups are unimodular. In particular, we study relations between distality, unimodularity and contraction subgroups.

In this article pattern statistics of typical cubical cut and project sets are studied. We give estimates for the rate of convergence of appearances of patches to their asymptotic frequencies. We also give bounds for repetitivity and repulsivity functions. The proofs use ideas and tools developed in discrepancy theory.

We give an example of a $C^{\infty }$ vector field $X$, defined in a neighbourhood $U$ of $0\in \mathbb{R}^{8}$, such that $U-\{0\}$ is foliated by closed integral curves of $X$, the differential $DX(0)$ at $0$ defines a one-parameter group of non-degenerate rotations and $X$ is not orbitally equivalent to its linearization. Such a vector field $X$ has the first integral $I(x)=\Vert x\Vert ^{2}$, and its main feature is that its period function is locally unbounded near the stationary point. This proves in the $C^{\infty }$ category that the classical Poincaré centre theorem, true for planar non-degenerate centres, is not generalizable to multicentres. Such an example is obtained through a careful study and a suitable modification of a celebrated example by Sullivan [A counterexample to the periodic orbit conjecture. Publ. Math. Inst. Hautes Études Sci. 46 (1976), 5–14], by blowing up the stationary point at the origin and through the construction of a smooth one-parameter family of foliations by circles of $S^{7}$ whose orbits have unbounded lengths (equivalently, unbounded periods) for each value of the parameter and which smoothly converges to the Hopf fibration $S^{1}{\hookrightarrow}S^{7}\rightarrow \mathbb{CP}^{3}$.

We investigate the polynomial lower and upper bounds for decay of correlations of a class of two-dimensional almost Anosov diffeomorphisms with respect to their Sinai–Ruelle–Bowen (SRB) measures. The degrees of the bounds are determined by the expansion and contraction rates as the orbits approach the indifferent fixed point, and are expressed by using coefficients of the third-order terms in the Taylor expansions of the diffeomorphisms at the indifferent fixed point.

In this paper, we provide a new criterion for the stable transitivity of volume-preserving finite generated groups on any compact Riemannian manifold. As one of our applications, we generalize a result of Dolgopyat and Krikorian [On simultaneous linearization of diffeomorphisms of the sphere. Duke Math. J. 136 (2007), 475–505] and obtain stable transitivity for random rotations on the sphere in any dimension. As another application, we show that for $\infty \geq r\geq 2$, for any $C^{r}$ volume-preserving partially hyperbolic diffeomorphism $g$ on any compact Riemannian manifold $M$ having sufficiently Hölder stable or unstable distribution, for any sufficiently large integer $K$ and for any $(f_{i})_{i=1}^{K}$ in a $C^{1}$ open $C^{r}$ dense subset of $\text{Diff}^{r}(M,m)^{K}$, the group generated by $g,f_{1},\ldots ,f_{K}$ acts transitively.

We simplify a criterion (due to Ibarlucía and the author) which characterizes dynamical simplices, that is, sets $K$ of probability measures on a Cantor space $X$ for which there exists a minimal homeomorphism of $X$ whose set of invariant measures coincides with $K$. We then point out that this criterion is related to Fraïssé theory, and use that connection to provide a new proof of Downarowicz’ theorem stating that any non-empty metrizable Choquet simplex is affinely homeomorphic to a dynamical simplex. The construction enables us to prove that there exist minimal homeomorphisms of a Cantor space which are speedup equivalent but not orbit equivalent, answering a question of Ash.

We prove several results for the arithmetic dynamics of monomial maps, including Silverman’s conjectures on height growth, dynamical Mordell–Lang conjecture, and dynamical Manin–Mumford conjecture. These results were originally known for monomial maps on algebraic tori. We extend them to arbitrary toric varieties.

We consider the deviation of Birkhoff sums along fixed orbits of substitution dynamical systems. We show distributional convergence for the Birkhoff sums of eigenfunctions of the substitution matrix. For non-coboundary eigenfunctions with eigenvalue of modulus $1$, we obtain a central limit theorem. For other eigenfunctions, we show convergence to distributions supported on Cantor sets. We also give a new criterion for such an eigenfunction to be a coboundary, as well as a new characterization of substitution dynamical systems with bounded discrepancy.

Let ${\mathcal{R}}$ be a strongly compact $C^{2}$ map defined in an open subset of an infinite-dimensional Banach space such that the image of its derivative $D_{F}{\mathcal{R}}$ is dense for every $F$. Let $\unicode[STIX]{x1D6FA}$ be a compact, forward invariant and partially hyperbolic set of ${\mathcal{R}}$ such that${\mathcal{R}}:\unicode[STIX]{x1D6FA}\rightarrow \unicode[STIX]{x1D6FA}$ is onto. The $\unicode[STIX]{x1D6FF}$-shadow $W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$ of $\unicode[STIX]{x1D6FA}$ is the union of the sets

$$\begin{eqnarray}W_{\unicode[STIX]{x1D6FF}}^{s}(G)=\{F:\operatorname{dist}({\mathcal{R}}^{i}F,{\mathcal{R}}^{i}G)\leq \unicode[STIX]{x1D6FF}\text{for every }i\geq 0\},\end{eqnarray}$$

$G\in \unicode[STIX]{x1D6FA}$

$W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$

$C^{1+\text{Lip}}$

$n$

$M$

$\unicode[STIX]{x1D6FA}$

$W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$

$M\cap W_{\unicode[STIX]{x1D6FF}}^{s}(\unicode[STIX]{x1D6FA})$

$M$

$n$

$\unicode[STIX]{x1D6FF}^{\prime }$

$$\begin{eqnarray}\mathop{\bigcup }_{i\geq 0}{\mathcal{R}}^{-i}W_{\unicode[STIX]{x1D6FF}^{\prime }}^{s}(\unicode[STIX]{x1D6FA})\end{eqnarray}$$

$C^{k}$

$k\geq 0$

We prove that if $n\geq 2$, then there is no $C^{1}$-diffeomorphism $f$ of the $n$-torus such that $f$ is semi-conjugate to a minimal translation and its wandering domains are geometric balls. This improves a recent result of Navas, who proved it assuming $C^{n+1}$ regularity of $f$.

We show that linear analytic cocycles where all Lyapunov exponents are negative infinite are nilpotent. For such one-frequency cocycles we show that they can be analytically conjugated to an upper triangular cocycle or a Jordan normal form. As a consequence, an arbitrarily small analytic perturbation leads to distinct Lyapunov exponents. Moreover, in the one-frequency case where the $k$th Lyapunov exponent is finite and the $(k+1)$st negative infinite, we obtain a simple criterion for domination in which case there is a splitting into a nilpotent part and an invertible part.