Let $b\geq 2$ be an integer and $\hat{v}$ a real number. Among other results, we compute the Hausdorff dimension of the set of real numbers ${\it\xi}$ with the property that, for every sufficiently large integer $N$, there exists an integer $n$ such that $1\leq n\leq N$ and the distance between $b^{n}{\it\xi}$ and its nearest integer is at most equal to $b^{-\hat{v}N}$. We further solve the same question when replacing $b^{n}{\it\xi}$ by $T_{{\it\beta}}^{n}{\it\xi}$, where $T_{{\it\beta}}$ denotes the classical ${\it\beta}$-transformation.

We prove several new versions of the Hadamard–Perron theorem, which relates infinitesimal dynamics to local dynamics for a sequence of local diffeomorphisms, and in particular establishes the existence of local stable and unstable manifolds. Our results imply the classical Hadamard–Perron theorem in both its uniform and non-uniform versions, but also apply much more generally. We introduce a notion of ‘effective hyperbolicity’ and show that if the rate of effective hyperbolicity is asymptotically positive, then the local manifolds are well behaved with positive asymptotic frequency. By applying effective hyperbolicity to finite-orbit segments, we prove a closing lemma whose conditions can be verified with a finite amount of information.

In this article, we study the automorphism group $\text{Aut}(X,{\it\sigma})$ of subshifts $(X,{\it\sigma})$ of low word complexity. In particular, we prove that $\text{Aut}(X,{\it\sigma})$ is virtually $\mathbb{Z}$ for aperiodic minimal subshifts and certain transitive subshifts with non-superlinear complexity. More precisely, the quotient of this group relative to the one generated by the shift map is a finite group. In addition, we show that any finite group can be obtained in this way. The class considered includes minimal subshifts induced by substitutions, linearly recurrent subshifts and even some subshifts which simultaneously exhibit non-superlinear and superpolynomial complexity along different subsequences. The main technique in this article relies on the study of classical relations among points used in topological dynamics, in particular, asymptotic pairs. Various examples that illustrate the technique developed in this article are provided. In particular, we prove that the group of automorphisms of a $d$-step nilsystem is nilpotent of order $d$ and from there we produce minimal subshifts of arbitrarily large polynomial complexity whose automorphism groups are also virtually $\mathbb{Z}$.

Within a subclass of monoids (with zero) a structural characterization is given of those that are associated to topologically transitive subshifts with Property (A).

We investigate the thermodynamic formalism for recurrent potentials on group extensions of countable Markov shifts. Our main result characterizes recurrent potentials depending only on the base space, in terms of the existence of a conservative product measure and a homomorphism from the group into the multiplicative group of real numbers. We deduce that, for a recurrent potential depending only on the base space, the group is necessarily amenable. Moreover, we give equivalent conditions for the base pressure and the skew product pressure to coincide. Finally, we apply our results to analyse the Poincaré series of Kleinian groups and the cogrowth of group presentations.

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.

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:

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

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

$s\in \mathbb{C}$

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

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

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

$s(1-s)$

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 $\mathbb{S}^{2}$.

For a continuous semicascade on a metrizable compact set ${\rm\Omega}$, we consider the weak$^{\ast }$ convergence of generalized operator ergodic means in $\text{End}\,C^{\ast }({\rm\Omega})$. We discuss conditions under which: every ergodic net contains a convergent sequence; all ergodic nets converge; all ergodic sequences converge. We study the relationships between the convergence of ergodic means and the properties of transitivity of the proximality relation on ${\rm\Omega}$, minimality of supports of ergodic measures, and uniqueness of minimal sets in the closure of trajectories of a semicascade. These problems are solved in terms of three associated algebraic-topological objects: the Ellis semigroup $E$, the Köhler operator semigroup ${\rm\Gamma}\subset \text{End}\,C^{\ast }({\rm\Omega})$, and the semigroup $G=\overline{\text{co}}\,{\rm\Gamma}$. The main results are stated for semicascades with metrizable $E$ and for tame semicascades.

For strongly dissipative Hénon maps at the first bifurcation parameter where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we establish a thermodynamic formalism, i.e. we prove the existence and uniqueness of an invariant probability measure that minimizes the free energy associated with a non-continuous geometric potential $-t\log J^{u}$, where $t\in \mathbb{R}$ is in a certain large interval and $J^{u}$ denotes the Jacobian in the unstable direction. We obtain geometric and statistical properties of these measures.

In this paper we address the issues of absolute continuity for the center foliation, as well as the disintegration on the non-absolute continuous case and rigidity of volume-preserving partially hyperbolic diffeomorphisms isotopic to a linear Anosov automorphism on $\mathbb{T}^{3}$. It is shown that the disintegration of volume on center leaves for these diffeomorphisms may be neither atomic nor Lebesgue, in contrast to the dichotomy (Lebesgue or atomic) obtained by Avila, Viana and Wilkinson [Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows. Preprint, 2012, arXiv:1110.2365v2] for perturbations of time-one of geodesic flow. In the case of atomic disintegration of volume on the center leaves of an Anosov diffeomorphism on $\mathbb{T}^{3}$, we show that it has to be one atom per leaf. Moreover, we show that not even a $C^{1}$ center foliation implies a rigidity result. However, for a volume-preserving partially hyperbolic diffeomorphism isotopic to a linear Anosov automorphism, assuming the center foliation is $C^{1}$ and transversely absolutely continuous with bounded Jacobians, we obtain smooth conjugacy to its linearization.

Let $(X,{\it\sigma}_{X}),(Y,{\it\sigma}_{Y})$ be one-sided subshifts and ${\it\pi}:X\rightarrow Y$ a factor map. Suppose that $X$ has the specification property. Let ${\it\mu}$ be a unique invariant Gibbs measure for a sequence of continuous functions ${\mathcal{F}}=\{\log f_{n}\}_{n=1}^{\infty }$ on $X$, which is an almost additive potential with bounded variation. We show that ${\it\pi}{\it\mu}$ is a unique invariant Gibbs measure for a sequence of continuous functions ${\mathcal{G}}=\{\log g_{n}\}_{n=1}^{\infty }$ on $Y$. When $(X,{\it\sigma}_{X})$ is a full shift, we characterize ${\mathcal{G}}$ and ${\it\mu}$ by using relative pressure. This ${\mathcal{G}}$ is a generalization of a continuous function found by Pollicott and Kempton in their work on factors of Gibbs measures for continuous functions. We also consider the following question: given a unique invariant Gibbs measure ${\it\nu}$ for a sequence of continuous functions ${\mathcal{F}}_{2}$ on $Y$, can we find an invariant Gibbs measure ${\it\mu}$ for a sequence of continuous functions ${\mathcal{F}}_{1}$ on $X$ such that ${\it\pi}{\it\mu}={\it\nu}$? We show that such a measure exists under a certain condition. In particular, if $(X,{\it\sigma}_{X})$ is a full shift and ${\it\nu}$ is a unique invariant Gibbs measure for a function in the Bowen class, then there exists a preimage ${\it\mu}$ of ${\it\nu}$ which is a unique invariant Gibbs measure for a function in the Bowen class.

Concerning hyperbolic diffeomorphisms, one expects a better smoothness of linearization, but it may be confined by resonance among eigenvalues. Hartman gave a three-dimensional analytic mapping with resonance which cannot be linearized by a Lipschitz conjugacy. Since then, efforts have been made to give the ${\it\alpha}$-Hölder continuity of the conjugacy and hope the exponent ${\it\alpha}<1$ can be as large as possible. Recently, it was proved for some weakly resonant hyperbolic diffeomorphisms that ${\it\alpha}$ can be as large as we expect. In this paper we prove that this result holds for all $C^{\infty }$ weakly resonant hyperbolic diffeomorphisms.