Arnol'd flows are a class of area-preserving flows on surfaces. In this paper, we prove that typical Arnol'd flows have the minimal self-joinings property. Consequently, we can classify centralizers and factors of typical Arnol'd flows.
This paper is about topological rigidity of diagonal group actions on the homogeneous _4(((t^-1)))/_4([t]) where is a finite field of characteristic 3. We show that there is a non-closed relatively compact orbit of the diagonal group.
We introduce an algebraic structure which encodes a collection of countable graphs through a set of states, generators and relations. These structures, which we call blueprints, can capture standard algebraic objects such as groups, monoids or small categories, as well as geometric tiling spaces with finite local complexity. We provide a general framework for symbolic dynamics on blueprints under a partial monoid action, and for transferring invariants of their symbolic dynamics through quasi-isometries. In particular, we show that the undecidability of the domino problem, the existence of strongly aperiodic subshifts of finite type, and the existence of subshifts of finite type without computable points are all quasi-isometry invariants for finitely presented blueprints. As an application of this model, we show that two variants of the domino problem for geometric tilings of ℝ^d are undecidable for d ≥ 2 on any underlying tiling space with finite local complexity.
. Let A- and A+ be properly immersed, closed, locally convex subsets of a Riemannian manifold M with pinched negative sectional curvature. When the Bowen-Margulis measure on T 1M is finite and mixing for the geodesic flow, we prove that the Lebesgue measures along the common perpendiculars of length at most t from A- to A+, counted with multiplicities and lifted to T 1M, equidistribute to the Bowen-Margulis measure as t -> +infinity. When M is locally symmetric with finite volume and the geodesic flow is exponentially mixing, we give an error term for the asymptotic. When T 1M is endowed with a bounded H & ouml;lder-continuous potential, and when the associated equilibrium state is finite and mixing for the geodesic flow, we prove the equidistribution of these Lebesgue measures weighted by the amplitudes of the potential to the equilibrium state.
Let S be a compact surface of genus >= 2 equipped with a metric that is flat everywhere except at finitely many cone points with angles greater than 2 pi. We examine the geodesic flow on S and prove local product structure for a wide class of equilibrium states. Using this, we establish the Bernoulli property for these systems. We also establish local product structure for a similar class of equilibrium states for geodesic flows on rank 1, nonpositively curved manifolds.
We prove that for a generic sub-Riemannian and reversible sub-Finsler metrics defined on a fixed co-rank 1 distribution, all strictly normal periodic orbits are non-degenerate.
We investigate the limiting behavior of multiple ergodic averages along sparse sequences evaluated at prime numbers. Our sequences arise from smooth and well-behaved functions that have polynomial growth. Central to this topic is a comparison result between standard Cesáro averages along positive integers and averages weighted by the (modified) von Mangoldt function. The main ingredients are a recent result of Matomäki, Shao, Tao and Teräväinen on the Gowers uniformity of the latter function in short intervals, a lifting argument that allows one to pass from actions of integers to flows, a simultaneous (variable) polynomial approximation in appropriate short intervals, and some quantitative equidistribution results for the former polynomials. We derive numerous applications in multiple recurrence, additive combinatorics, and equidistribution in nilmanifolds along primes. In particular, we deduce that any set of positive density contains arithmetic progressions with step $\lfloor p^c \rfloor$, where $c$ is a positive non-integer and $p$ denotes a prime, establishing a conjecture of Frantzikinakis.
We establish arithmeticity in the sense of A. Katok and F. Rodriguez Hertz of smooth actions $\alpha$ of $\mathbb{R}^k$ on an anonymous manifold $M$ of dimension $2k+1$ provided that there is an ergodic invariant Borel probability measure on $M$ w/r/t which each nontrivial time-$t$ map $\alpha_t$ of the action has positive entropy. Arithmeticity in this context means that the action $\alpha$ is measure theoretically isomorphic to a constant time change of the suspension of an affine Cartan action of $\mathbb{Z}^k$. This in particular solves, up to measure theoretical isomorphism, Problem 4 from a prequel paper of Katok and Rodriguez Hertz, joint with B. Kalinin.
This paper studies polynomials with core entropy zero. We give several characterizations of polynomials with core entropy zero. In particular, we show that a degree d post-critically finite polynomial f has core entropy zero if and only if f is in the degree d main molecule. The characterizations define several quantities which measure the complexities of polynomials with core entropy zero. We show that these measures are all comparable.
Let $T$ be the Koopman operator of a measure preserving transformation $\theta$ of a probability space $(X,\Sigma,\mu)$. We study the convergence properties of the averages $M_nf:=\frac1n\sum_{k=0}^{n-1}T^kf$ when $f \in L^r(\mu)$, $0
In honor of Zhiren Wang on the occasion of being awarded the Brin Prize, we report on his exciting and deep work on rigidity of higher rank abelian groups and lattices in higher rank semisimple groups.
Consider the sequence of continued fraction convergents pn/ qn to a random irrational number. We study the distribution of the sequences pn (mod m) and qn (mod m) with a fixed modulus m, and more generally, the distribution of the 2 x 2 matrix with entries pn-1, pn, qn-1, qn (mod m). Improving the strong law of large numbers due to Sz & uuml;sz, Moeckel, Jager and Liardet, we establish the central limit theorem and the law of the iterated logarithm, as well as the weak and the almost sure invariance principles. As an application, we find the limit distribution of the maximum and the minimum of the Birkhoff sum for the irrational rotation with the indicator of an interval as test function. We also compute the normalizing constant in a classical limit law for the same Birkhoff sum due to Kesten, and dispel a misconception about its dependence on the test interval.
. Let S be an oriented closed surface of genus g >= 1, furnished with an area form omega. We show that for 1 <= r <= infinity, there exists an open and dense set Or of the space of Hamiltonian diffeomorphisms of class Cr, endowed with the Cr -topology, such that every f E Or possesses infinitely many periodic orbits with nonzero rotation vector. Similar results hold if one replaces the space of Hamiltonian diffeomorphisms with the space of symplectic diffeomorphisms that are isotopic to the identity. Moreover, we give some details about the possible homological directions. In the Hamiltonian case, we obtain a positive answer to a question asked by Viktor Ginzburg and Ba,sak G & uuml;rel concerning existence of non-contractible periodic orbits. The proof is a consequence of recent previous works of the authors [15].
Consider the sequence of continued fraction convergents $p_n/q_n$ to a random irrational number. We study the distribution of the sequences $p_n \pmod{m}$ and $q_n \pmod{m}$ with a fixed modulus $m$, and more generally, the distribution of the $2 \times 2$ matrix with entries $p_{n-1}, p_n, q_{n-1}, q_n \pmod{m}$. Improving the strong law of large numbers due to Sz\"usz, Moeckel, Jager and Liardet, we establish the central limit theorem and the law of the iterated logarithm, as well as the weak and the almost sure invariance principles. As an application, we find the limit distribution of the maximum and the minimum of the Birkhoff sum for the irrational rotation with the indicator of an interval as test function. We also compute the normalizing constant in a classical limit law for the same Birkhoff sum due to Kesten, and dispel a misconception about its dependence on the test interval.
In this paper we study connection points on the double regular n-gon translation surface, for n >= 7 odd and its staircase model. For n not equal 9, we provide a large family of points with coordinates in the trace field that are not connection points. This family includes the central points, and for n = 7 we conjecture that all the remaining points are connection points. Further, in the case where n >= 7 is a prime number, we provide a constructive proof by exhibiting an explicit separatrix passing through a central point that does not extend to a saddle connection.
