In this paper, we prove a cocycle version of marked length spectrum rigidity. There are two consequences: the first one is marked length pattern rigidity for arithmetic hyperbolic locally symmetric manifolds, and the second one is a strengthened marked length spectrum rigidity for surfaces and closed locally symmetric manifolds.

We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, we calculate the Hausdorff dimension of the uniform Diophantine set

$$ \begin{align*} {\mathcal{U}(\hat{\nu})}= &\ \{x\in[0,1)\colon \text{for all }N\gg1,\text{ there exists }n\in[1,N],\\&\ \ \text{ such that }|T^{n}(x)-y| < |I_{N}(y)|^{\hat{\nu}}\} \end{align*} $$

$y\in [0,1)$

We study the dynamics of dissipative billiard maps within planar convex domains. Such maps have a global attractor. We are interested in the topological and dynamical complexity of the attractor, in terms both of the geometry of the billiard table and of the strength of the dissipation. We focus on the study of an invariant subset of the attractor, the so-called Birkhoff attractor. On the one hand, we show that for a generic convex table with ‘pinched’ curvature, the Birkhoff attractor is a normally contracted manifold when the dissipation is strong. On the other hand, for a mild dissipation, we prove that, generically, the Birkhoff attractor is complicated, both from the topological and the dynamical points of view.

For an arbitrary countable discrete infinite group G, non-singular rank-one actions are introduced. It is shown that the class of non-singular rank-one actions coincides with the class of non-singular $(C,F)$-actions. Given a decreasing sequence of cofinite subgroups in G with $\bigcap _{n=1}^\infty \bigcap _{g\in G}g\Gamma _ng^{-1}=\{1_G\}$, the projective limit of the homogeneous G-spaces $G/\Gamma _n$ as $n\to \infty $ is a G-space. Endowing this G-space with an ergodic non-singular non-atomic measure, we obtain a dynamical system which is called a non-singular odometer. Necessary and sufficient conditions are found for a rank-one non-singular G-action to have a finite factor and a non-singular odometer factor in terms of the underlying $(C,F)$-parameters. Similar conditions are also found for a rank-one non-singular G-action to be isomorphic to an odometer. Minimal Radon uniquely ergodic locally compact Cantor models are constructed for the non-singular rank-one extensions of odometers. Several concrete examples are constructed and several facts are proved that illustrate a sharp difference of the non-singular non-commutative case from the classical finite measure preserving one: odometer actions which are not of rank-one and factors of rank-one systems which are not of rank one; however, each probability preserving odometer is a factor of an infinite measure preserving rank-one system, etc.

We study actions of higher rank lattices $\Gamma <G$ on hyperbolic spaces and we show that all such actions satisfying mild properties come from the rank-one factors of G. In particular, all non-elementary isometric actions on an unbounded hyperbolic space are of this type.

Let A be a rational function of one complex variable of degree at least two, and $z_0$ its repelling fixed point with the multiplier $\unicode{x3bb} .$ A Poincaré function associated with $z_0$ is a function meromorphic on ${\mathbb C}$ such that , and In this paper, we study the following problem: given Poincaré functions and , find out if there is an algebraic relation between them and, if such a relation exists, describe the corresponding algebraic curve $f(x,y)=0.$ We provide a solution, which can be viewed as a refinement of the classical theorem of Ritt about commuting rational functions. We also reprove and extend previous results concerning algebraic dependencies between Böttcher functions.

We study the vectorial length compactification of the space of conjugacy classes of maximal representations of the fundamental group $\Gamma$ of a closed hyperbolic surface $\Sigma$ in $\textrm{PSL}(2,{\mathbb{R}})^n$. We identify the boundary with the sphere ${\mathbb{P}}(({\mathcal{ML}})^n)$, where $\mathcal{ML}$ is the space of measured geodesic laminations on $\Sigma$. In the case $n=2$, we give a geometric interpretation of the boundary as the space of homothety classes of ${\mathbb{R}}^2$-mixed structures on $\Sigma$. We associate to such a structure a dual tree-graded space endowed with an ${\mathbb{R}}_+^2$-valued metric, which we show to be universal with respect to actions on products of two $\mathbb{R}$-trees with the given length spectrum.

Consider the quadratic family $T_a(x) = a x (1 - x)$ for $x \in [0, 1]$ and mixing Collet–Eckmann (CE) parameters $a \in (2,4)$. For bounded $\varphi $, set $\tilde \varphi _{a} := \varphi - \int \varphi \, d\mu _a$, with $\mu _a$ the unique acim of $T_a$, and put $(\sigma _a (\varphi ))^2 := \int \tilde \varphi _{a}^2 \, d\mu _a + 2 \sum _{i>0} \int \tilde \varphi _{a} (\tilde \varphi _{a} \circ T^i_{a}) \, d\mu _a$. For any mixing Misiurewicz parameter $a_{*}$, we find a positive measure set $\Omega _{*}$ of mixing CE parameters, containing $a_{*}$ as a Lebesgue density point, such that for any Hölder $\varphi $ with $\sigma _{a_{*}}(\varphi )\ne 0$, there exists $\epsilon _\varphi>0$ such that, for normalized Lebesgue measure on $\Omega _{*}\cap [a_{*}-\epsilon _\varphi , a_{*}+\epsilon _\varphi ]$, the functions $\xi _i(a)=\tilde \varphi _a(T_a^{i+1}(1/2))/\sigma _a (\varphi )$ satisfy an almost sure invariance principle (ASIP) for any error exponent $\gamma>2/5$. (In particular, the Birkhoff sums satisfy this ASIP.) Our argument goes along the lines of Schnellmann’s proof for piecewise expanding maps. We need to introduce a variant of Benedicks–Carleson parameter exclusion and to exploit fractional response and uniform exponential decay of correlations from Baladi et al [Whitney–Hölder continuity of the SRB measure for transversal families of smooth unimodal maps. Invent. Math. 201 (2015), 773–844].

We devise schemes for producing, in the least possible time, p identical objects with n agents that work at differing speeds. This involves halting the process to transfer production across agent types. For the case of two types of agent, we construct schemes based on the Euclidean algorithm that seeks to minimize the number of pauses in production.

Let G be a torsion-free, finitely generated, nilpotent and metabelian group. In this work, we show that G embeds into the group of orientation-preserving $C^{1+\alpha }$-diffeomorphisms of the compact interval for all $\alpha < 1/k$, where k is the torsion-free rank of $G/A$ and A is a maximal abelian subgroup. We show that, in many situations, the corresponding $1/k$ is critical in the sense that there is no embedding of G with higher regularity. A particularly nice family where this happens is the family of $(2n+1)$-dimensional Heisenberg groups, for which we can show that the critical regularity is equal to $1+1/n$.

For a class of robustly transitive diffeomorphisms on ${\mathbb T}^4$ introduced by Shub [Topologically transitive diffeomorphisms of $T^4$. Proceedings of the Symposium on Differential Equations and Dynamical Systems (Lecture notes in Mathematics, 206). Ed. D. Chillingworth. Springer, Berlin, 1971, pp. 39–40], satisfying an additional bunching condition, we show that there exists a $C^2$ open and $C^r$ dense subset ${\mathcal U}^r$, $2\leq r\leq \infty $, such that any two hyperbolic points of $g\in {\mathcal U}^r$ with stable index $2$ are homoclinically related. As a consequence, every $g\in {\mathcal U}^r$ admits a unique homoclinic class associated to the hyperbolic periodic points with index $2$, and this homoclinic class coincides with the whole ambient manifold. Moreover, every $g\in {\mathcal U}^r$ admits at most one measure of maximal entropy, and every $g\in {\mathcal U}^{\infty }$ admits a unique measure of maximal entropy.

We introduce and study two conditions on groups of homeomorphisms of Cantor space, namely the conditions of being vigorous and of being flawless. These concepts are dynamical in nature, and allow us to study a certain interplay between the dynamics of an action and the algebraic properties of the acting group. A group $G\leq \operatorname {Homeo}(\mathfrak {C})$ is vigorous if for any clopen set A and proper clopen subsets B and C of A, there is $\gamma \in G$ in the pointwise stabiliser of $\mathfrak {C}\backslash A$ with $B\gamma \subseteq C$. A nontrivial group $G\leq \operatorname {Homeo}(\mathfrak {C})$ is flawless if for all k and w a nontrivial freely reduced product expression on k variables (including inverse symbols), a particular subgroup $w(G)_\circ $ of the verbal subgroup $w(G)$ is the whole group. We show: 1) simple vigorous groups are either two-generated by torsion elements, or not finitely generated, 2) flawless groups are both perfect and lawless, 3) vigorous groups are simple if and only if they are flawless, and, 4) the class of vigorous simple subgroups of $\operatorname {Homeo}(\mathfrak {C})$ is fairly broad (the class is closed under various natural constructions and contains many well known groups, such as the commutator subgroups of the Higman–Thompson groups $G_{n,r}$, the Brin-Thompson groups $nV$, Röver’s group $V(\Gamma )$, and others of Nekrashevych’s ‘simple groups of dynamical origin’).

We show that an infinite group G definable in a $1$-h-minimal field admits a strictly K-differentiable structure with respect to which G is a (weak) Lie group, and we show that definable local subgroups sharing the same Lie algebra have the same germ at the identity. We conclude that infinite fields definable in K are definably isomorphic to finite extensions of K and that $1$-dimensional groups definable in K are finite-by-abelian-by-finite. Along the way, we develop the basic theory of definable weak K-manifolds and definable morphisms between them.

We obtain a new interpretation of the cohomological Hall algebra $\mathcal {H}_Q$ of a symmetric quiver Q in the context of the theory of vertex algebras. Namely, we show that the graded dual of $\mathcal {H}_Q$ is naturally identified with the underlying vector space of the principal free vertex algebra associated to the Euler form of Q. Properties of that vertex algebra are shown to account for the key results about $\mathcal {H}_Q$. In particular, it has a natural structure of a vertex bialgebra, leading to a new interpretation of the product of $\mathcal {H}_Q$. Moreover, it is isomorphic to the universal enveloping vertex algebra of a certain vertex Lie algebra, which leads to a new interpretation of Donaldson–Thomas invariants of Q (and, in particular, re-proves their positivity). Finally, it is possible to use that vertex algebra to give a new interpretation of CoHA modules made of cohomologies of non-commutative Hilbert schemes.

We characterize the fractional Dehn twist coefficient (FDTC) on the n-stranded braid group as the unique homogeneous quasimorphism to $\mathbb {R}$ of defect at most 1 that equals 1 on the positive full twist and vanishes on the $(n-1)$-stranded braid subgroup. In a different direction, we establish that the slice-Bennequin inequality holds with the FDTC in place of the writhe. In other words, we establish an affine linear lower bound for the smooth slice genus of the closure of a braid in terms of the braid’s FDTC. We also discuss connections between these two seemingly unrelated results. In the appendix, we provide a unifying framework for the slice-Bennequin inequality and its counterpart for the FDTC.

We prove a general formula that relates the parity of the Langlands parameter of a conjugate self-dual discrete series representation of $\operatorname { {GL}}_n$ to the parity of its Jacquet-Langlands image. It gives a generalization of a partial result by Mieda concerning the case of invariant $1/n$ and supercuspidal representations. It also gives a variation of the result on the self-dual case by Prasad and Ramakrishnan.

For microscale heterogeneous partial differential equations (PDEs), this article further develops novel theory and methodology for their macroscale mathematical/asymptotic homogenization. This article specifically encompasses the case of quasi-periodic heterogeneity with finite scale separation: no scale separation limit is required. A key innovation herein is to analyse the ensemble of all phase-shifts of the heterogeneity. Dynamical systems theory then frames the homogenization as a slow manifold of the ensemble. Depending upon any perceived scale separation within the quasi-periodic heterogeneity, the homogenization may be done in either one step or two sequential steps: the results are equivalent. The theory not only assures us of the existence and emergence of an exact homogenization at finite scale separation, it also provides a practical systematic method to construct the homogenization to any specified order. For a class of heterogeneities, we show that the macroscale homogenization is potentially valid down to lengths which are just twice that of the microscale heterogeneity! This methodology complements existing well-established results by providing a new rigorous and flexible approach to homogenization that potentially also provides correct macroscale initial and boundary conditions, treatment of forcing and control, and analysis of uncertainty.