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.

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.

The Abel differential equation $y^{\prime }=p(x)y^{3}+q(x)y^{2}$ with polynomial coefficients $p,q$ is said to have a center on $[a,b]$ if all its solutions, with the initial value $y(a)$ small enough, satisfy the condition $y(a)=y(b)$. The problem of giving conditions on $(p,q,a,b)$ implying a center for the Abel equation is analogous to the classical Poincaré center-focus problem for plane vector fields. Center conditions are provided by an infinite system of ‘center equations’. During the last two decades, important new information on these equations has been obtained via a detailed analysis of two related structures: composition algebra and moment equations (first-order approximation of the center ones). Recently, one of the basic open questions in this direction—the ‘polynomial moments problem’—has been completely settled in Pakovich and Muzychuk [Solution of the polynomial moment problem. Proc. Lond. Math. Soc. (3)99(3) (2009), 633–657] and Pakovich [Generalized ‘second Ritt theorem’ and explicit solution of the polynomial moment problem. Compositio Math.149 (2013), 705–728]. In this paper, we present a progress in the following two main directions: first, we translate the results of Pakovich and Muzychuk [Solution of the polynomial moment problem. Proc. Lond. Math. Soc. (3)99(3) (2009), 633–657] and Pakovich [Generalized ‘second Ritt theorem’ and explicit solution of the polynomial moment problem. Compositio Math.149 (2013), 705–728] into the language of algebraic geometry of the center equations. Applying these new tools, we show that the center conditions can be described in terms of composition algebra, up to a ‘small’ correction. In particular, we significantly extend the results of Briskin, Roytvarf and Yomdin [Center conditions at infinity for Abel differential equations. Ann. of Math. (2)172(1) (2010), 437–483]. Second, applying these tools in combination with explicit computations, we start in this paper the study of the ‘second Melnikov coefficients’ (second-order approximation of the center equations), showing that in many cases vanishing of the moments and of these coefficients is sufficient in order to completely characterize centers.

Let $T$ be an $\mathbb{R}$-tree with dense orbits in the boundary of outer space. When the free group $\mathbb{F}_{N}$ acts freely on $T$, we prove that the number of projective classes of ergodic currents dual to $T$ is bounded above by $3N-5$. We combine Rips induction and splitting induction to define unfolding induction for such an $\mathbb{R}$-tree $T$. Given a current ${\it\mu}$ dual to $T$, the unfolding induction produces a sequence of approximations converging towards ${\it\mu}$. We also give a unique ergodicity criterion.

We produce infinitely many examples of Anosov flows in closed $3$-manifolds where the set of periodic orbits is partitioned into two infinite subsets. In one subset every closed orbit is freely homotopic to infinitely other closed orbits of the flow. In the other subset every closed orbit is freely homotopic to only one other closed orbit. The examples are obtained by Dehn surgery on geodesic flows. The manifolds are toroidal and have Seifert pieces and atoroidal pieces in their torus decompositions.

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.

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 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.

We consider the general non-vanishing, divergence-free vector fields defined on a domain in $3$-space and tangent to its boundary. Based on the theory of finite-type invariants, we define a family of invariants for such fields, in the style of Arnold’s asymptotic linking number. Our approach is based on the configuration space integrals due to Bott and Taubes.

We study double averages along orbits for measure-preserving actions of $\mathbb{A}^{{\it\omega}}$, the direct sum of countably many copies of a finite abelian group $\mathbb{A}$. We show an $\text{L}^{p}$ norm-variation estimate for these averages, which in particular re-proves their convergence in $\text{L}^{p}$ for any finite $p$ and for any choice of two $\text{L}^{\infty }$ functions. The result is motivated by recent questions on quantifying convergence of multiple ergodic averages.

We study countably piecewise continuous, piecewise monotone interval maps. We establish a necessary and sufficient criterion for the existence of a non-decreasing semiconjugacy to a map of constant slope in terms of the existence of an eigenvector of an operator acting on a space of measures. Then we give sufficient conditions under which this criterion is not satisfied. Finally, we give examples of maps not semiconjugate to a map of constant slope via a non-decreasing map. Our examples are continuous and transitive.

We formulate two natural but different extensions of the weak separation condition to infinite iterated function systems of conformal contractions with overlaps, and study the associated topological pressure functions. We obtain a formula for the Hausdorff dimension of the limit sets under these weak separation conditions.

In this paper, we consider a $\mathbb{Z}^{d}$ extension of the well known fact that subshifts with only finitely many follower sets are sofic. As in Kass and Madden [A sufficient condition for non-soficness of higher-dimensional subshifts. Proc. Amer. Math. Soc.141 (2013), 3803–3816], we adopt a natural $\mathbb{Z}^{d}$ analogue of a follower set called an extender set. The extender set of a finite word $w$ in a $\mathbb{Z}^{d}$ subshift $X$ is the set of all configurations of symbols on the rest of $\mathbb{Z}^{d}$ which form a point of $X$ when concatenated with $w$. As our main result, we show that for any $d\geq 1$ and any $\mathbb{Z}^{d}$ subshift $X$, if there exists $n$ so that the number of extender sets of words on a $d$-dimensional hypercube of side length $n$ is less than or equal to $n$, then $X$ is sofic. We also give an example of a non-sofic system for which this number of extender sets is $n+1$ for every $n$. We prove this theorem in two parts. First we show that if the number of extender sets of words on a $d$-dimensional hypercube of side length $n$ is less than or equal to $n$ for some $n$, then there is a uniform bound on the number of extender sets for words on any sufficiently large rectangular prism; to our knowledge, this result is new even for $d=1$. We then show that such a uniform bound implies soficity. Our main result is reminiscent of the classical Morse–Hedlund theorem, which says that if $X$ is a $\mathbb{Z}$ subshift and there exists an $n$ such that the number of words of length $n$ is less than or equal to $n$, then $X$ consists entirely of periodic points. However, most proofs of that result use the fact that the number of words of length $n$ in a $\mathbb{Z}$ subshift is non-decreasing in $n$, and we present an example (due to Martin Delacourt) which shows that this monotonicity does not hold for numbers of extender sets (or follower sets) of words of length $n$.

In the present article, we introduce beta-expansions in the ring $\mathbb{Z}_{p}$ of $p$-adic integers. We characterise the sets of numbers with eventually periodic and finite expansions.

We develop a theory that allows us to code dynamical systems induced by primitive substitutions continuously as shifts of finite type in many different ways. The well-known prefix–suffix coding turns out to correspond to one special case. We precisely analyse the basic properties of these codings (injectivity, coding of the periodic points, properties of the presentation graph, interaction with the shift map). A lot of examples illustrate the theory and show that, depending on the particular coding, several amazing effects may occur. The results give new insights into the theory of substitution dynamical systems and might serve as a powerful tool for further researches.

We prove that for typical rotation numbers $0<{\it\theta}<1$, the boundary of the Siegel disk of $f_{{\it\theta}}(z)=e^{2{\it\pi}i{\it\theta}}\sin (z)$ centered at the origin is a Jordan curve which passes through exactly two critical points ${\it\pi}/2$ and $-{\it\pi}/2$.