Synchronisation in Invertible Random Dynamical Systems on the Circle

Minimal non-invertible maps on the pseudo-circle

In this article, we show that R.H. Bing's pseudo-circle admits a minimal non-invertible map. This resolves a problem raised by Bruin, Kolyada and Snoha in the negative. The main tool is the Denjoy-Rees technique, further developed by Béguin-Crovisier-Le Roux, combined with detailed study into the structure of the pseudo-circle.

随机圆同胚向混沌过渡的间歇双点动力学

Intermittent two-point dynamics at the transition to chaos for random circle endomorphisms

研究揭示了随机圆同胚在从同步到混沌过渡过程中间歇双点动力学与无限平稳测度的存在机制。

非可逆随机动力系统的原子物理测度

Atomic physical measures for non-invertible random dynamical systems

研究非可逆随机动力系统，构造圆上由局部可逆映射构成的系统并证明其有原子物理平稳测度$\nu$，指出平稳测度赫尔德正则性不能推广，还给出相关示例。

Minimal sets of fibre-preserving maps in graph bundles

Topological structure of minimal sets is studied for a dynamical system $(E,F)$ given by a fibre-preserving, in general non-invertible, continuous selfmap $F$ of a graph bundle $E$. These systems include, as a very particular case, quasiperiodically forced circle homeomorphisms. Let $M$ be a minimal set of $F$ with full projection onto the base space $B$ of the bundle. We show that $M$ is nowhere dense or has nonempty interior depending on whether the set of so called endpoints of $M$ is dense in $M$ or is empty. If $M$ is nowhere dense, we prove that either a typical fibre of $M$ is a Cantor …

Some Limit Theorems Regarding Products of Random Matrices I: Directional Derivative of the Lyapunov Exponent

Given an i.i.d. sequence $\{A_n(ω)\}_{n\ge 1}$ of invertible matrices and a random matrix $B(ω)$, we consider the random matrix sequences inductively defined by $S_n(ω) = A_n(ω)S_{n-1}(ω)$ and $T_n(ω) = B(σ^{n-1}ω)S_{n-1}(ω)+A_n(ω)T_{n-1}(ω)$, and study several limit theorems involving $T_n(ω)$ as well as the asymptotic behaviour of the action of $T_n(ω)$ on the projective space and on the unit circle.