Topological Dichotomy and Strict Ergodicity for Translation Surfaces

高秩子空间流的遍历二分性

Ergodic dichotomy for subspace flows in higher rank

本文研究高秩子空间流的遍历性，提出类似 Hopf-Tsuji-Sullivan 二分性的判据，并推广到任意维数连通对角子群，为一般 Anosov 子群建立余维数二分性，推广了 Burger-Landesberg-Lee-Oh 的工作。

A family of distal functions and multipliers for strict ergodicity

We give two proofs to an old result of E. Salehi, showing that the Weyl subalgebra $\mathcal{W}$ of $\ell^\infty(\mathbb{Z})$ is a proper subalgebra of $\mathcal{D}$, the algebra of distal functions. We also show that the family $\mathcal{S}^d$ of strictly ergodic functions in $\mathcal{D}$ does not form an algebra and hence in particular does not coincide with $\mathcal{W}$. We then use similar constructions to show that a function which is a multiplier for strict ergodicity, either within $\mathcal{D}$ or in general, is necessarily a constant. An example of a metric, strictly ergodic, distal…

多维Toeplitz子移位的熵与严格遍历性

Entropy and strict ergodicity for multidimensional Toeplitz subshifts

本研究通过自洽构造，证明了具有任意规定熵的严格遍历Toeplitz Z^d-子移位存在，且其极大等连续因子均为Z^d上的通用里程计。

马尔可夫链的严格不可约性与斜积的遍历性

Strict irreducibility of Markov chains and ergodicity of skew products

引入马尔可夫核的严格不可约性概念，刻画了使族遍历性蕴含斜积遍历性的马尔可夫链类，将Bufetov关于有限状态马尔可夫链的充分条件推广到一般状态空间并证明其必要性，应用于随机变换的遍历定理极限描述。

Unique ergodicity of the horocycle flow on Riemannnian foliations

A classic result due to Furstenberg is the strict ergodicity of the horocycle flow for a compact hyperbolic surface. Strict ergodicity is unique ergodicity with respect to a measure of full support, and therefore implies minimality. The horocycle flow has been previously studied on minimal foliations by hyperbolic surfaces on closed manifolds, where it is known not to be minimal in general. In this paper, we prove that for the special case of Riemannian foliations, strict ergodicity of the horocycle flow still holds. This in particular proves that this flow is minimal, which establishes a conj…

Ergodic components and topological entropy in geodesic flows of surfaces

We consider the geodesic flow of reversible Finsler metrics on the 2-sphere and the 2-torus, whose geodesic flow has vanishing topological entropy. Following a construction of A. Katok, we discuss examples of Finsler metrics on both surfaces, which have large ergodic components for the geodesic flow in the unit tangent bundle. On the other hand, using results of J. Franks and M. Handel, we prove that ergodicity and dense orbits cannot occur in the full unit tangent bundle of the 2-sphere, if the Finsler metric has positive flag curvatures and at least two closed geodesics. In the case of the 2…