We consider the dimensions of a family of self-affine sets related to the Bedford–McMullen carpets. In particular, we fix a Bedford–McMullen system and then randomize the translation vectors with the stipulation that the column structure is preserved. As such, we maintain one of the key features in the Bedford–McMullen set-up in that alignment causes the dimensions to drop from the affinity dimension. We compute the Hausdorff, packing and box dimensions outside of a small set of exceptional translations, and also for some explicit translations even in the presence of overlapping. Our results rely on, and can be seen as a partial extension of, Hochman’s recent work on the dimensions of self-similar sets and measures.

We consider the space of complex polynomials of degree $n\geq 3$ with $n-1$ distinct marked periodic orbits of given periods. We prove that this space is irreducible and the multipliers of the marked periodic orbits, considered as algebraic functions on that space, are algebraically independent over $\mathbb{C}$. Equivalently, this means that at its generic point the moduli space of degree-$n$ polynomial maps can be locally parameterized by the multipliers of $n-1$ arbitrary distinct periodic orbits. We also prove a similar result for a certain class of affine subspaces of the space of complex polynomials of degree $n$.

A simple proof of the fact that each rank-one infinite measure preserving (i.m.p.) transformation is subsequence weakly rationally ergodic is found. Some classes of funny rank-one i.m.p. actions of Abelian groups are shown to be subsequence weakly rationally ergodic. A constructive definition of finite funny rank for actions of arbitrary infinite countable groups is given. It is shown that the ergodic i.m.p. transformations of balanced finite funny rank are subsequence weakly rationally ergodic. It is shown that the ergodic probability preserving transformations of exact finite rank, the ergodic Bratteli–Vershik maps corresponding to the ‘consecutively ordered’ Bratteli diagrams of finite rank, some their generalizations and the ergodic interval exchange transformations are partially rigid.

In this article, Conley’s connection matrix theory and a spectral sequence analysis of a filtered Morse chain complex $(C,{\rm\Delta})$ are used to study global continuation results for flows on surfaces. The briefly described unfoldings of Lyapunov graphs have been proved to be a well-suited combinatorial tool to keep track of continuations. The novelty herein is a global dynamical cancellation theorem inferred from the differentials of the spectral sequence $(E^{r},d^{r})$ . The local version of this theorem relates differentials $d^{r}$ of the $r$ th page $E^{r}$ to Smale’s theorem on cancellation of critical points.

We investigate the rate of convergence of the iterates of an $n$-dimensional quasiregular mapping within the basin of attraction of a fixed point of high local index. A key tool is a refinement of a result that gives bounds on the distortion of the image of a small spherical shell. This result also has applications to the rate of growth of quasiregular mappings of polynomial type, and to the rate at which the iterates of such maps can escape to infinity.

We investigate functorial properties of Putnam’s homology theory for Smale spaces. Our analysis shows that the addition of a conjugacy condition is necessary to ensure functoriality. Several examples are discussed that elucidate the need for our additional hypotheses. Our second main result is a natural generalization of Putnam’s Pullback Lemma from shifts of finite type to non-wandering Smale spaces.

We show some level-2 large deviation principles for real and complex one-dimensional maps satisfying a weak form of hyperbolicity. More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages.

Let $G=(G_{n})_{n}$ be a strictly increasing sequence of positive integers with $G_{0}=1$. We study the system of numeration defined by this sequence by looking at the corresponding compactification ${\mathcal{K}}_{G}$ of $\mathbb{N}$ and the extension of the addition-by-one map ${\it\tau}$ on ${\mathcal{K}}_{G}$ (the ‘odometer’). We give sufficient conditions for the existence and uniqueness of ${\it\tau}$-invariant measures on ${\mathcal{K}}_{G}$ in terms of combinatorial properties of $G$.

It is proved that positive entropy implies mean Li–Yorke chaos for a $G$ -system, where $G$ is a countable, infinite, discrete, bi-orderable amenable group. Examples are given for the cases of integer lattice groups and groups of integer unipotent upper triangular matrices.

D’après un célèbre lemme de John Franks, toute perturbation de la différentielle d’un difféomorphisme $f$ le long d’une orbite périodique est réalisée par une $C^{1}$-perturbation $g$ du difféomorphisme sur un petit voisinage de ladite orbite. On n’a cependant aucune information sur le comportement des variétés invariantes de l’orbite périodique après perturbation. Nous montrons que si la perturbation de la dérivée est obtenue par une isotopie le long de laquelle existent les variétés stables/instables fortes de certaines dimensions, alors on peut faire la perturbation ci-dessus en préservant les variétés stables/instables semi-locales correspondantes. Ce résultat a de nombreuses applications en systèmes dynamiques de classes $C^{1}$. Nous en démontrons quelques unes.

We study the dimension spectrum of Lyapunov exponents for multimodal maps of the interval and their generalizations. We also present related results for rational maps on the Riemann sphere.

Let $f$ be a germ of holomorphic diffeomorphism with an irrationally indifferent fixed point at the origin in $\mathbb{C}$ (i.e. $f(0)=0,f^{\prime }(0)=e^{2{\it\pi}i{\it\alpha}},{\it\alpha}\in \mathbb{R}-\mathbb{Q}$ ). Perez-Marco showed the existence of a unique family of non-trivial invariant full continua containing the fixed point called Siegel compacta. When $f$ is non-linearizable (i.e. not holomorphically conjugate to the rigid rotation $R_{{\it\alpha}}(z)=e^{2{\it\pi}i{\it\alpha}}z$ ) the invariant compacts obtained are called hedgehogs. Perez-Marco developed techniques for the construction of examples of non-linearizable germs [Fixed points and circle maps. Acta Math. 179(2) (1997), 243–294]; these were used by the author to construct hedgehogs of Hausdorff dimension one, and adapted by Cheritat [Relatively compact Siegel disks with non-locally connected boundaries, Preprint, 2009] to construct Siegel disks with pseudo-circle boundaries. We use these techniques to construct hedgehogs of positive area and hedgehogs with inaccessible fixed points.

Let $T$ be an ergodic measure-preserving transformation on a non-atomic probability space $(X,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$ . We prove uniform extensions of the Wiener–Wintner theorem in two settings: for averages involving weights coming from Hardy field functions  $p$ ,

$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(p(n))T^{n}f(x)\bigg\}; & & \displaystyle \nonumber\end{eqnarray}$$

$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(n\unicode[STIX]{x1D703})T^{P(n)}f(x)\bigg\} & & \displaystyle \nonumber\end{eqnarray}$$

$\unicode[STIX]{x1D703}\in [0,1]$

$\unicode[STIX]{x1D707}$

$f\in L^{p}(X),p>1,$

$\unicode[STIX]{x1D703}\in [0,1]$

In this paper some aspects on chaotic behavior and minimality in planar piecewise smooth vector fields theory are treated. The occurrence of non-deterministic chaos is observed and the concept of orientable minimality is introduced. Some relations between minimality and orientable minimality are also investigated and the existence of new kinds of non-trivial minimal sets in chaotic systems is observed. The approach is geometrical and involves the ordinary techniques of non-smooth systems.

By a transfer operator approach to Maass cusp forms and the Selberg zeta function for cofinite Hecke triangle groups, Möller and the present author found a factorization of the Selberg zeta function into a product of Fredholm determinants of transfer-operator-like families:

$$\begin{eqnarray}Z(s)=\det (1-{\mathcal{L}}_{s}^{+})\det (1-{\mathcal{L}}_{s}^{-}).\end{eqnarray}$$

${\mathcal{L}}_{s}^{\pm }$

$s\in \mathbb{C}$

$\text{Re}s={\textstyle \frac{1}{2}}$

${\mathcal{L}}_{s}^{+}$

${\mathcal{L}}_{s}^{-}$

$s(1-s)$

In this article we will extend ‘the weak mixing theorem’ for certain locally compact Polish groups (Moore groups and minimally weakly mixing groups). In addition, we will show that the Gaussian action associated with the infinite-dimensional irreducible representation of the continuous Heisenberg group, $H_{3}(\mathbb{R})$, is weakly mixing but not mildly mixing.

We prove that a nilpotent subgroup of orientation-preserving $C^{1}$ diffeomorphisms of $\mathbb{S}^{2}$ has a finite orbit of cardinality at most two. We also prove that a finitely generated nilpotent subgroup of orientation-preserving $C^{1}$ diffeomorphisms of $\mathbb{R}^{2}$ preserving a compact set has a global fixed point. These results generalize theorems of Franks et al for the abelian case. We show that a nilpotent subgroup of orientation-preserving $C^{1}$ diffeomorphisms of $\mathbb{S}^{2}$ that has a finite orbit of odd cardinality also has a global fixed point. Moreover, we study the properties of the 2-points orbits of nilpotent fixed-point-free subgroups of orientation-preserving $C^{1}$ diffeomorphisms of $\mathbb{S}^{2}$.

Let $G,H$ be two Kleinian groups with homeomorphic quotients $\mathbb{H}^{3}/G$ and $\mathbb{H}^{3}/H$ . We assume that $G$ is of divergence type, and consider the Patterson–Sullivan measures of $G$ and $H$ . The measurable rigidity theorem by Sullivan and Tukia says that a measurable and essentially directly measurable equivariant boundary map $\widehat{k}$ from the limit set $\unicode[STIX]{x1D6EC}_{G}$ of $G$ to that of $H$ is either the restriction of a Möbius transformation or totally singular. In this paper, we shall show that such $\widehat{k}$ always exists. In fact, we shall construct $\widehat{k}$ concretely from the Cannon–Thurston maps of $G$ and $H$ .

In this paper we consider diffeomorphisms preserving hyperbolic Sinaĭ–Ruelle–Bowen (SRB) probability measures ${\it\mu}$ having intersections for almost every pair of the stable and unstable manifolds. In this context, when the dimension of the unstable manifold is constant almost everywhere, we show the ergodicity of ${\it\mu}$ . As an application we obtain another proof of the ergodicity of a hyperbolic SRB measure for transitive surface diffeomorphisms, which is shown by Rodriguez Hertz, Rodriguez Hertz, Tahzibi and Ures [Uniqueness of SRB measures for transitive diffeomorphisms on surfaces. Comm. Math. Phys.306(1) (2011), 35–49].

We prove that in any totally irrational cut-and-project setup with codimension (internal space dimension) one, it is possible to choose sections (windows) in non-trivial ways so that the resulting sets are bounded displacement equivalent to lattices. Our proof demonstrates that for any irrational ${\it\alpha}$, regardless of Diophantine type, there is a collection of intervals in $\mathbb{R}/\mathbb{Z}$ which is closed under translation, contains intervals of arbitrarily small length, and along which the discrepancy of the sequence $\{n{\it\alpha}\}$ is bounded above uniformly by a constant.