Cubic polynomials: a measurable view on parameter space

On embeddability of automorphisms into measurable flows from the point of view of self-joining properties

We compare self-joining- and embeddability properties. In particular, we prove that a measure preserving flow $(T_t)_{t\in\mathbb{R}}$ with $T_1$ ergodic is 2-fold quasi-simple (2-fold distally simple) if and only if $T_1$ is 2-fold quasi-simple (2-fold distally simple). We also show that the Furstenberg-Zimmer decomposition for a flow $(T_t)_{t\in\mathbb{R}}$ with $T_1$ ergodic with respect to any flow factor is the same for $(T_t)_{t\in\mathbb{R}}$ and for $T_1$. We give an example of a 2-fold quasi-simple flow disjoint from simple flows and whose time-one map is simple. We describe two clas…

从可测观点看拓扑稳定性

Topological stability from a measurable viewpoint

本文从可测观点引入$\mu$-拓扑稳定性，刻画其与拓扑稳定点、跟踪性质的关系，并证明其在共轭、限制下的不变性及测度集合的凸性，最后建立与弱跟踪性质的联系。

Models for spaces of dendritic polynomials

Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called \emph{dendritic}. By results of Kiwi, any dendritic polynomial is semi-conjugate to a topological polynomial whose topological Julia set is a dendrite. We construct a continuous map of the space of all cubic dendritic polynomials onto a laminational model that is a quotient space of a subset of the closed bidisk. This construction generalizes the "pinched disk" model of the Mandelbrot set due to Douady and Thurston. It can be viewed as a step towards constructing a model of the cubic connecte…

A note on configurations in sets of positive density which occur at all large scales

Furstenberg, Katznelson and Weiss proved in the early 1980s that every measurable subset of the plane with positive density at infinity has the property that all sufficiently large real numbers are realised as the Euclidean distance between points in that set. Their proof used ergodic theory to study translations on a space of Lipschitz functions corresponding to closed subsets of the plane, combined with a measure-theoretical argument. We consider an alternative dynamical approach in which the phase space is given by the set of measurable functions from $\mathbb{R}^d$ to $[0,1]$, which we vie…

流视图与可识别替换的无限区间交换变换

Flow views and infinite interval exchange transformations for recognizable substitutions

本文提出流视图概念，将替换子移位与无限区间交换变换通过可测共轭联系，证明对偶替换可保证自相似性，并讨论谱类型意义，附常长度替换的谱结果。