Ergodic properties of folding maps on spheres

Quantum Ergodicity and Averaging Operators on the Sphere

We prove quantum ergodicity for certain orthonormal bases of $L^2(\mathbb{S}^2)$, consisting of joint eigenfunctions of the Laplacian on $\mathbb{S}^2$ and the discrete averaging operator over a finite set of rotations, generating a free group. If in addition the rotations are algebraic we give a quantified version of this result. The methods used also give a new, simplified proof of quantum ergodicity for large regular graphs.

The exact convergence rate in the ergodic theorem of Lubotzky Phillips Sarnak

We compute exact convergence rates in von Neumann type ergodic theorems when the acting group of measure preserving transformations is free and the means are taken over spheres or over balls defined by a word metric. Relying on the upper bounds on the spectra of Koopman operators deduced by Lubozky, Phillips, and Sarnak from Deligne's work on the Weil conjecture, we compute the exact convergence rate for the free groups (of rank $(p+1)/2$ where $p\equiv 1\mod 4$ is prime) of isometries of the round sphere defined by Lipschitz quaternions. We also show that any finite rank free group of automor…

Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres

We establish an asymptotic formula for the number of lattice points in the sets \[ \mathbf S_{h_1, h_2, h_3}(λ): =\{x\in\mathbb Z_+^3:\lfloor h_1(x_1)\rfloor+\lfloor h_2(x_2)\rfloor+\lfloor h_3(x_3)\rfloor=λ\} \quad \text{with}\quad λ\in\mathbb Z_+; \] where functions $h_1, h_2, h_3$ are constant multiples of regularly varying functions of the form $h(x):=x^c\ell_h(x)$, where the exponent $c>1$ (but close to $1$) and a function $\ell_h(x)$ is taken from a certain wide class of slowly varying functions. Taking $h_1(x)=h_2(x)=h_3(x)=x^c$ we will also derive an asymptotic formula for the number o…

等谱联络、标架流的遍历性以及球面之间的多项式映射

Isospectral connections, ergodicity of frame flows, and polynomial maps between spheres

本文证明闭负曲率流形上Bochner Laplacian的谱在低秩条件下决定向量丛同构类与联络的规范类，并证明无和乐约化的低秩标架流是遍历的，核心方法是建立其与球面间多项式映射分类的联系。

Harmonic analysis, Ergodic theory and Counting for thin groups

For a geometrically finite group Gamma of G=SO(n,1), we survey recent developments on counting and equidistribution problems for orbits of Gamma in a homogeneous space H\G where H is trivial, symmetric or horospherical. Main applications are found in an affine sieve on orbits of thin groups as well as in sphere counting problems for sphere packings invariant under a geometrically finite group. In our sphere counting problems, spheres can be ordered with respect to a general conformal metric.

A note on the space of evolutionary operators in population genetics and folding dynamics

Discrete dynamical systems defined by the iteration of a polynomial map of the unit simplex to itself appear in the context of population genetic systems evolving under mutation, recombination and weak selection. Although exceptional progress has been made in finding particular solutions to these systems, our knowledge of the general properties of the space of all possible dynamical systems of this kind is still limited. We prove that the space of bounded-degree polynomial maps of the unit simplex to itself is a compact and convex subset of a Euclidean space. We provide an explicit characteriz…

$p$-adic dynamical systems of $(3,1)$-rational functions with unique fixed point

We describe the set of all $(3,1)$-rational functions given on the set of complex $p$-adic field $\mathbb C_p$ and having a unique fixed point. We study $p$-adic dynamical systems generated by such $(3,1)$-rational functions and show that the fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. We obtain Siegel disks of these dynamical systems. Moreover an upper bound for the set of limit points of each trajectory is given. For each $(3,1)$-rational function on $\mathbb C_p$ there is a point $\hat x=\hat x(f)\in \mathbb…

Ergodic properties of linked-twist maps

We study a class of homeomorphisms of surfaces collectively known as linked-twist maps. We introduce an abstract definition which enables us to give a precise characterisation of a property observed by other authors, namely that such maps fall into one of two classes termed co- and counter-twisting. We single out three specific linked-twist maps, one each on the two-torus, in the plane and on the two-sphere and for each prove a theorem concerning its ergodic properties with respect to the invariant Lebesgue measure. For the map on the torus we prove that there is an invariant, zero-measure Can…

Ergodic Transformations of the Space of $p$-adic Integers

Let $\mathcal L_1$ be the set of all mappings $f\colon\Z_p\Z_p$ of the space of all $p$-adic integers $\Z_p$ into itself that satisfy Lipschitz condition with a constant 1. We prove that the mapping $f\in\mathcal L_1$ is ergodic with respect to the normalized Haar measure on $\Z_p$ if and only if $f$ induces a single cycle permutation on each residue ring $\Z/p^k\Z$ modulo $p^k$, for all $k=1,2,3,...$. The multivariate case, as well as measure-preserving mappings, are considered also. Results of the paper in a combination with earlier results of the author give explicit description of ergodic …

正则树上自同构不变过程的混合与等分

Mixing and equipartition for automorphism invariant processes on regular trees

针对正则树上保奇偶性树自同构不变的有限状态过程，证明其遍历过程在水平球面内沿球面和球几乎处处等分，并在定量混合条件下得到偶数半径度量球面的香农-麦克米兰-布莱曼定理。

Origami, affine maps, and complex dynamics

We investigate the combinatorial and dynamical properties of so-called nearly Euclidean Thurston maps, or NET maps. These maps are perturbations of many-to-one folding maps of an affine two-sphere to itself. The close relationship between NET maps and affine maps makes computation of many invariants tractable. In addition to this, NET maps are quite diverse, exhibiting many different behaviors. We discuss data, findings, and new phenomena.

$p$-adic dynamical systems of $(2,2)$-rational functions with unique fixed point

We consider a family of $(2,2)$-rational functions given on the set of complex $p$-adic field $\mathbb{C}_p$. Each such function has a unique fixed point. We study $p$-adic dynamical systems generated by the $(2,2)$-rational functions. We show that the fixed point is indifferent and therefore the convergence of the trajectories is not the typical case for the dynamical systems. Siegel disks of these dynamical systems are found. We obtain an upper bound for the set of limit points of each trajectory, i.e., we determine a sufficiently small set containing the set of limit points. For each $(2,2)…

Reduced dynamical systems

We consider the dynamics of complex rational maps on the Riemann sphere. We prove that, after reducing their orbits to a fixed number of positive values representing the Fubini-Study distances between finitely many initial elements of the orbit and the origin, ergodic properties of the rational map are preserved.

无不动点伪 Anosov 映射，以及 Poincaré 球中的亏格二 L-space 纽结

Fixed-point-free pseudo-Anosovs, and genus-two L-space knots in the Poincare sphere

本文利用 train track 与 folding automata 及映射类不动点理论，分类 Poincaré 同调球中的亏格二 L-space 纽结，并推广刻画任意三维流形中相应次最高阶秩为 1 的双曲纤维纽结。

Equilibrium States for Expanding Thurston Maps

In this paper, we use the thermodynamical formalism to show that there exists a unique equilibrium state $μ_ϕ$ for each expanding Thurston map $f: S^2\rightarrow S^2$ together with a real-valued Hölder continuous potential $ϕ$. Here the sphere $S^2$ is equipped with a natural metric induced by $f$, called a visual metric. We also prove that identical equilibrium states correspond to potentials which are co-homologous upto a constant, and that the measure-preserving transformation $f$ of the probability space $(S^2,μ_ϕ)$ is exact, and in particular, mixing and ergodic. Moreover, we establish ve…

C*-algebras of minimal dynamical systems of the product of a Cantor set and an odd dimensional sphere

Let β: S^n \to S^n, for n = 2k + 1, k \geq 1, be one of the known examples of a non-uniquely ergodic minimal diffeomorphism of an odd dimensional sphere. For every such minimal dynamical system (S^n, β) there is a Cantor minimal system (X, α) such that the corresponding product system (X x S^n, αx β) is minimal and the resulting crossed product C*-algebra C(X x S^n) \rtimes_{αx β} \mathbb{Z} is tracially approximately an interval algebra (TAI). This entails classification for such C*-algebras. Moreover, the minimal Cantor system (X, α) is such that each tracial state on C(X x S^n) \rtimes_β \m…

平坦射影丛上的叶状 horocycle 流：拓扑注记与应用

Horocycle flow on flat projective bundles: topological remarks and applications

本文研究双曲曲面上平坦射影丛的叶状 horocycle 流，在 Conze-Guivarc'h 条件下用近端动力学刻画其拓扑与遍历性质，并证明凸余紧时限制流具有唯一保守遍历 Radon 测度。

无限体积局部对称秩一流形的刚性

Rigidity of locally symmetric rank one manifolds of infinite volume

本文通过引入幂零 Sierpiński 地毯及其拉伸，构造了非刚性离散 $\mathbb{F}$-双曲群，并揭示了其非刚性与离散群在极限集上作用的非遍历动力学之间的联系。