We consider a diffeomorphism $f$ of a compact manifold $M$ which is almost Axiom A, i.e. $f$ is hyperbolic in a neighborhood of some compact $f$-invariant set, except in some singular set of neutral points. We prove that if there exists some $f$-invariant set of hyperbolic points with positive unstable Lebesgue measure such that for every point in this set the stable and unstable leaves are ‘long enough’, then $f$ admits an SRB (probability) measure.

We combine effective mixing and Duke’s theorem on closed geodesics on the modular surface to show that certain subcollections of the collection of geodesics with a given discriminant still equidistribute. These subcollections are only assumed to have sufficiently large total length without any further restrictions.

Model sets are always Meyer sets but the converse is generally not true. In this work we show that for a repetitive Meyer multiple set of $\mathbb{R}^{d}$ with associated dynamical system $(\mathbb{X},\mathbb{R}^{d})$ , the property of being a model multiple set is equivalent to $(\mathbb{X},\mathbb{R}^{d})$ being almost automorphic. We deduce this by showing that a repetitive Meyer multiple set can always be embedded into a repetitive model multiple set having a smaller group of topological eigenvalues.

We prove that if ${\rm\Gamma}$ is a subgroup of $\text{Diff}_{+}^{\,1+{\it\epsilon}}(I)$ and $N$ is a natural number such that every non-identity element of ${\rm\Gamma}$ has at most $N$ fixed points, then ${\rm\Gamma}$ is solvable. If in addition ${\rm\Gamma}$ is a subgroup of $\text{Diff}_{+}^{\,2}(I)$, then we can claim that ${\rm\Gamma}$ is meta-abelian.

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.

Let $Q\rightarrow X$ be a continuous principal bundle whose group $G$ is reductive. A flow ${\it\phi}$ of automorphisms of $Q$ endowed with an ergodic probability measure on the compact base space $X$ induces two decompositions of the flag bundles associated to $Q$: a continuous one given by the finest Morse decomposition and a measurable one furnished by the multiplicative ergodic theorem. The second is contained in the first. In this paper we find necessary and sufficient conditions so that they coincide. The equality between the two decompositions implies continuity of the Lyapunov spectra under perturbations leaving unchanged the flow on the base space.

In this note we give an example of an ergodic non-singular map whose unitary operator admits a Lebesgue component of multiplicity one in its spectrum.

Let $f$ and $g$ be two class $P$ -homeomorphisms of the circle $S^{1}$ with break point singularities, which are differentiable maps except at some singular points where the derivative has a jump. Assume that $f$ and $g$ have irrational rotation numbers and the derivatives $\text{Df}$ and $\text{Dg}$ are absolutely continuous on every continuity interval of $\text{Df}$ and $\text{Dg}$ , respectively. We prove that if the product of the $f$ -jumps along all break points of $f$ is distinct from that of $g$ then the homeomorphism $h$ conjugating $f$ and $g$ is a singular function, i.e. it is continuous on $S^{1}$ , but $\text{Dh}(x)=0$  almost everywhere with respect to the Lebesgue measure. This result generalizes previous results for one and two break points obtained by Dzhalilov, Akin and Temir, and Akhadkulov, Dzhalilov and Mayer. As a consequence, we get in particular Dzhalilov–Mayer–Safarov’s theorem: if the product of the $f$ -jumps along all break points of $f$ is distinct from $1$ , then the invariant measure $\unicode[STIX]{x1D707}_{f}$ is singular with respect to the Lebesgue measure.

We prove a rigidity theorem for dominated Hölder cocycles with values on diffeomorphism groups of a compact manifold over hyperbolic homeomorphisms. More precisely, we show that if two such cocycles have equal periodic data, then they are cohomologous.

A criterion of joint ergodicity of several sequences of transformations of a probability measure space $X$ of the form $T_{i}^{\unicode[STIX]{x1D711}_{i}(n)}$ is given for the case where $T_{i}$ are commuting measure-preserving transformations of $X$ and $\unicode[STIX]{x1D711}_{i}$ are integer-valued generalized linear functions, that is, the functions formed from conventional linear functions by an iterated use of addition, multiplication by constants, and the greatest integer function. We also establish a similar criterion for joint ergodicity of families of transformations depending on a continuous parameter, as well as a condition of joint ergodicity of sequences $T_{i}^{\unicode[STIX]{x1D711}_{i}(n)}$ along primes.

In this paper we give some positive and negative results about the contact property for the energy levels ${\rm\Sigma}_{c}$ of a symplectic magnetic field on $S^{2}$. In the first part we focus on the case of the area form on a surface of revolution. We state a sufficient condition for an energy level to be of contact type and give an example where the contact property fails. If the magnetic curvature is positive, the dynamics and the action of invariant measures can be numerically computed. The collected data hint at the conjecture that an energy level of a symplectic magnetic field with positive magnetic curvature should be of contact type. In the second part we show that, for a small energy $c$, there exist a convex hypersurface $N_{c}$ in $\mathbb{C}^{2}$ and a double cover $N_{c}\rightarrow {\rm\Sigma}_{c}$ such that the pull-back of the characteristic distribution on ${\rm\Sigma}_{c}$ is the standard characteristic distribution on $N_{c}$. As a corollary, we prove that there are either two or infinitely many periodic orbits on ${\rm\Sigma}_{c}$. The second alternative holds if there exists a contractible prime periodic orbit.

We generalize the polynomial Szemerédi theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new polynomial configurations in positive-density subsets of $\mathbb{Z}^{m}$ and strengthens and extends recent results of Bergelson, Leibman and Lesigne [Intersective polynomials and the polynomial Szemerédi theorem. Adv. Math.219(1) (2008), 369–388] on polynomials over the integers.

Let ${\it\beta}_{n}>1$ be a root of $x^{n}-x-1$ for $n=4,5,\ldots$ ; we will prove that ${\it\beta}_{n}$ is not a Parry number, i.e., the associated beta transformation does not correspond to a sofic symbolic system. A generalization is shown in the last section.

In this paper we extend Bourgain’s double recurrence result to the Wiener–Wintner averages. Let $(X,{\mathcal{F}},{\it\mu},T)$ be a standard ergodic system. We will show that for any $f_{1},f_{2}\in L^{\infty }(X)$, the double recurrence Wiener–Wintner average

$$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)e^{2{\it\pi}int}\end{eqnarray}$$

$X$

$t$

$N\rightarrow \infty$

$f_{1}$

$f_{2}$

$t$

$0$

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.

We prove a concentration inequality for sequential dynamical systems of the unit interval enjoying an exponential loss of memory in the BV norm and we investigate several of its consequences. In particular, this covers compositions of $\unicode[STIX]{x1D6FD}$-transformations, with all $\unicode[STIX]{x1D6FD}$ lying in a neighborhood of a fixed $\unicode[STIX]{x1D6FD}_{\star }>1$, and systems satisfying a covering-type assumption.

We show that a partially hyperbolic $C^{1}$-diffeomorphism $f:M\rightarrow M$ with a uniformly compact $f$-invariant center foliation ${\mathcal{F}}^{c}$ is dynamically coherent. Further, the induced homeomorphism $F:M/{\mathcal{F}}^{c}\rightarrow M/{\mathcal{F}}^{c}$ on the quotient space of the center foliation has the shadowing property, i.e. for every ${\it\epsilon}>0$ there exists ${\it\delta}>0$ such that every ${\it\delta}$-pseudo-orbit of center leaves is ${\it\epsilon}$-shadowed by an orbit of center leaves. Although the shadowing orbit is not necessarily unique, we prove the density of periodic center leaves inside the chain recurrent set of the quotient dynamics. Other interesting properties of the quotient dynamics are also discussed.

We prove that the spaces of ${\mathcal{C}}^{\infty }$ orientation preserving actions of $\mathbb{Z}^{n}$ on $[0,1]$ and non-free actions of $\mathbb{Z}^{2}$ on the circle are connected.

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 $\mathbb{Z}$.

We consider actions of locally compact groups $G$ on certain CAT(0) spaces $X$ by isometries. The CAT(0) spaces we consider have finite dimension at large scale. In case $B$ is a $G$-boundary, that is a measurable $G$-space with some amenability and ergodicity properties, we prove the existence of equivariant maps from $B$ to the visual boundary $\partial X$.