The trigonometric moment problem arises from the study of one-parameter families of centers in polynomial vector fields. It seeks to classify the trigonometric polynomials $Q$ which are orthogonal to all powers of a trigonometric polynomial $P$. We show that this problem has a simple and natural solution under certain conditions on the monodromy group of the Laurent polynomial associated to $P$. In the case of real trigonometric polynomials, which is the primary motivation of the problem, our conditions are shown to hold for all trigonometric polynomials of degree 15 or less. In the complex case, we show that there are a small number of exceptional monodromy groups up to degree 30 where the conditions fail to hold and show how counterexamples can be constructed in several of these cases.

We prove distributional limit theorems for random walk adic transformations obtaining ergodic distributional limits of exponential $\chi ^2$form.

We examine the class of increasing sequences of natural numbers which are IP-rigidity sequences for some weakly mixing probability-preserving transformation. This property is closely related to the uncountability of the eigenvalue group of a corresponding non-singular transformation. We give examples, including a super-lacunary sequence which is not IP-rigid.

We study the Lyapunov spectrum of the Kontsevich–Zorich cocycle on SL(2,ℝ)-invariant subbundles of the Hodge bundle over the support of SL(2,ℝ)-invariant probability measures on the moduli space of Abelian differentials. In particular, we prove formulas for partial sums of Lyapunov exponents in terms of the second fundamental form (the Kodaira–Spencer map) of the Hodge bundle with respect to the Gauss–Manin connection and investigate the relations between the central Oseledets subbundle and the kernel of the second fundamental form. We illustrate our conclusions in two special cases.

We study the motion of a particle on a Riemannian 2-torus under the influence of a magnetic field and a Gaussian thermostat. We prove a Hopf type rigidity for this dynamical system without conjugate points.

Most results in multifractal analysis are obtained using either a thermodynamic approach based on the existence and uniqueness of equilibrium states or an orbit-gluing approach based on some version of the specification property. A general framework incorporating the most important multifractal spectra was introduced by Barreira and Saussol, who used the thermodynamic approach to establish the multifractal formalism in the uniformly hyperbolic setting, unifying many existing results. We extend this framework to apply to a broad class of non-uniformly hyperbolic systems, including examples with phase transitions, and obtain new results for a number of examples that have already been studied using the orbit-gluing approach. We compare the thermodynamic and orbit-gluing approaches and give a survey of many of the multifractal results in the literature.

We present the general notion of Borel fields of metric spaces and show some properties of such fields. Then we make the study specific to the Borel fields of proper CAT(0) spaces and we show that the standard tools we need behave in a Borel way. We also introduce the notion of the action of an equivalence relation on Borel fields of metric spaces and we obtain a rigidity result for the action of an amenable equivalence relation on a Borel field of proper finite dimensional CAT(0) spaces. This main theorem is inspired by the result obtained by Adams and Ballmann regarding the action of an amenable group on a proper CAT(0) space.

In this paper, we consider torus homeomorphisms $f$ homotopic to Dehn twists. We prove that if the vertical rotation set of $f$ is reduced to zero, then there exists a compact connected essential ‘horizontal’ set $K$, invariant under $f$. In other words, if we consider the lift $\hat {f}$ of $f$ to the cylinder, which has zero vertical rotation number, then all points have uniformly bounded motion under iterates of $\hat {f}$. Also, we give a simple explicit condition which, when satisfied, implies that the vertical rotation set contains an interval and thus also implies positive topological entropy. As a corollary of the above results, we prove a version of Boyland’s conjecture to this setting: if $f$ is area preserving and has a lift $\hat {f}$ to the cylinder with zero Lebesgue measure vertical rotation number, then either the orbits of all points are uniformly bounded under $\hat {f}$, or there are points in the cylinder with positive vertical velocity and others with negative vertical velocity.

The squiral inflation rule is equivalent to a bijective block substitution rule and leads to an interesting lattice dynamical system under the action of ${ \mathbb{Z} }^{2} $. In particular, its balanced version has purely singular continuous diffraction. The dynamical spectrum is of mixed type, with pure point and singular continuous components. We present a constructive proof that admits a generalization to bijective block substitutions of trivial height on ${ \mathbb{Z} }^{d} $.

Let $F: \mathbb{R} \rightarrow \mathbb{R} $ be a real analytic increasing diffeomorphism with $F- \mathrm{Id} $ being 1-periodic. Consider the translated family of maps $\mathop{({F}_{t} : \mathbb{R} \rightarrow \mathbb{R} )}\nolimits_{t\in \mathbb{R} } $ defined as ${F}_{t} (x)= F(x)+ t$. Let $\mathrm{Trans} ({F}_{t} )$ be the translation number of ${F}_{t} $ defined by

$$\mathrm{Trans} ({F}_{t} ): = \lim _{n\rightarrow + \infty }\frac{{ F}_{t}^{\circ n} - \mathrm{Id} }{n} .$$

$2\tau $

$F$

${p}_{n} / {q}_{n} $

$n$

$\mathrm{Trans} (F)= \alpha \in \mathbb{R} \setminus \mathbb{Q} $

${\ell }_{\theta } $

$\{ t\in \mathbb{R} ~\mid ~\mathrm{Trans} ({F}_{t} )= \theta \} $

$({\ell }_{{p}_{n} / {q}_{n} } )$

${q}_{n} $

$$\mathop {\mathrm{lim\hphantom{,}sup} }\limits _{n\rightarrow + \infty } \frac{1}{{q}_{n} } \log {\ell }_{{p}_{n} / {q}_{n} } \leq - 2\pi \tau .$$

${\ell }_{{p}_{n} / {q}_{n} } $

Let fi∈C2+α(S1∖{ai,bi}), α>0,i=1,2, be circle homeomorphisms with two break points ai,bi, that is, discontinuities in the derivative Dfi, with identical irrational rotation number ρ and μ1([a1,b1])=μ2([a2,b2]), where μi are the invariant measures of fi,i=1,2. Suppose that the products of the jump ratios of Df1 and Df2do not coincide, that is, Df1 (a1 −0)/Df1 (a1 +0)⋅Df1 (b1 −0)/Df1 (b1 +0)≠Df2(a2−0)/Df2(a2+0)⋅Df2(b2−0)/Df2(b2+0) . Then the map ψ conjugating f1 and f2 is a singular function, that is, it is continuous on S1, but Dψ(x)=0 almost everywhere with respect to Lebesgue measure.

For any $1\leq r\lt \infty $, we build on the disk, and therefore on any manifold, a ${C}^{r} $-diffeomorphism with no measure of maximal entropy.

In this paper we consider the sequence of Kakutani’s $\alpha $-refinements corresponding to the inverse of the golden ratio (which we call the Kakutani–Fibonacci sequence of partitions) and associate to it an ergodic interval exchange (which we call the Kakutani–Fibonacci transformation) using the ‘cutting–stacking’ technique. We prove that the orbit of the origin under this map coincides with a low discrepancy sequence (which we call the Kakutani–Fibonacci sequence of points), which has also been considered by other authors.

Let f be a class P-homeomorphism of the circle with break point singularities, that is, differentiable except at some singular points where the derivative has a jump. Let f have irrational rotation number and Df be absolutely continuous on every continuity interval of Df. We prove that if the product of the f-jumps along any subset of break points is distinct from 1 then the invariant measure μf is singular with respect to the Haar measure. This result generalizes previous results obtained by Dzhalilov and Khanin, Dzhalilov, Akhadkulov, Dzhalilov–Liousse and Mayer. Moreover, we prove that if the rotation number ρ(f) is irrational of bounded type then (a) if the product of the f-jumps on some orbit is distinct from 1 then the invariant measure μf is singular with respect to the Haar measure m, and (b) if the product of the f-jumps on each orbit is equal to 1 and D2f∈Lp (S1) for some p>1 then μfis equivalent to the Haar measure.

We study two properties of a finite measure-preserving dynamical system and a given sequence $({n}_{m} )$ of positive integers, namely rigidity and non-recurrence. Our goal is to find conditions on the sequence which ensure that it is, or is not, a rigid sequence or a non-recurrent sequence for some weakly mixing system or more generally for some ergodic system. The main focus is on weakly mixing systems. For example, we show that for any integer $a\geq 2$ the sequence ${n}_{m} = {a}^{m} $ is a sequence of rigidity for some weakly mixing system. We show the same for the sequence of denominators of the convergents in the continued fraction expansion of any irrational $\alpha $. We also consider the stronger property of IP-rigidity. We show that if $({n}_{m} )$ grows fast enough then there is a weakly mixing system which is IP-rigid along $({n}_{m} )$ and non-recurrent along $({n}_{m} + 1)$.

We prove that if ${\Sigma }_{\mathbf{A} } ( \mathbb{N} )$ is an irreducible Markov shift space over $ \mathbb{N} $ and $f: {\Sigma }_{\mathbf{A} } ( \mathbb{N} )\rightarrow \mathbb{R} $ is coercive with bounded variation then there exists a maximizing probability measure for $f$, whose support lies on a Markov subshift over a finite alphabet. Furthermore, the support of any maximizing measure is contained in this same compact subshift. To the best of our knowledge, this is the first proof beyond the finitely primitive case in the general irreducible non-compact setting. It is also noteworthy that our technique works for the full shift over positive real sequences.

Using the works of Mañé [On the topological entropy of the geodesic flows. J. Differential Geom.45 (1989), 74–93] and Paternain [Topological pressure for geodesic flows. Ann. Sci. Éc. Norm. Supér. (4) 33 (2000), 121–138] we study the distribution of geodesic arcs with respect to equilibrium states of the geodesic flow on a closed manifold, equipped with a $\mathcal {C}^{\infty }$Riemannian metric. We prove large-deviation lower and upper bounds and a contraction principle for the geodesic flow in the space of probability measures of the unit tangent bundle. We deduce a way of approximating equilibrium states for continuous potentials.

Every sufficiently regular non-periodic space of tilings of $\mathbb {R}^d$ has at least one pair of distinct tilings that are asymptotic under translation in all the directions of some open $(d-1)$-dimensional hemisphere. If the tiling space comes from a substitution, there is a way of defining a location on such tilings at which asymptoticity ‘starts’. This leads to the definition of the branch locus of the tiling space: this is a subspace of the tiling space, of dimension at most $d-1$, that summarizes the ‘asymptotic in at least a half-space’ behavior in the tiling space. We prove that if a $d$-dimensional self-similar substitution tiling space has a pair of distinct tilings that are asymptotic in a set of directions that contains a closed $(d-1)$-hemisphere in its interior, then the branch locus is a topological invariant of the tiling space. If the tiling space is a two-dimensional self-similar Pisot substitution tiling space, the branch locus has a description as an inverse limit of an expanding Markov map on a zero- or one-dimensional simplicial complex.

It is shown that if an infinite synchronized system has a flip, then it has infinitely many non-conjugate flips, and that the result cannot be extended to the class of coded systems.

We prove that a ${C}^{1} $ generic symplectic diffeomorphism is either Anosov or its topological entropy is bounded from below by the supremum over the smallest positive Lyapunov exponent of its periodic points. We also prove that ${C}^{1} $ generic symplectic diffeomorphisms outside the Anosov ones do not admit symbolic extension and, finally, we give examples of volume preserving surface diffeomorphisms which are not points of upper semicontinuity of the entropy function in the ${C}^{1} $ topology.