Entropy Theory for Cross Sections

Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

Let c be a real parameter in the Mandelbrot set, and f_c(z):= z^2 + c. We prove a formula relating the topological entropy of f_c to the Hausdorff dimension of the set of rays landing on the real Julia set, and to the Hausdorff dimension of the set of rays landing on the real section of the Mandelbrot set, to the right of the given parameter c. We then generalize the result by looking at the entropy of Hubbard trees: namely, we relate the Hausdorff dimension of the set of external angles which land on a certain slice of a principal vein in the Mandelbrot set to the topological entropy of the q…

三维 Reeb 流的一般性质：Birkhoff 截面与熵

Generic properties of $3$-dimensional Reeb flows: Birkhoff sections and entropy

本文利用破碎书分解研究闭三维流形上的 Reeb 流，证明在 Liouville 测度可被周期轨道逼近的条件下存在 Birkhoff 截面，并由此得到具有 Birkhoff 截面及正拓扑熵的接触形式集合在 $C^\infty$-拓扑中开且稠密。

Cross sections of Borel flows with restrictions on the distance set

Given a set of positive reals, we provide a necessary and sufficient condition for a free Borel flow to admit a cross section with all distances between adjacent points coming from this set.

Regular cross sections of Borel flows

Any free Borel flow is shown to admit a cross section with only two possible distances between adjacent points. Non smooth flows are proved to be Lebesgue orbit equivalent if and only if they admit the same number of invariant ergodic probability measures.