On the non-robustness of intermingled basins

A robustly transitive diffeomorphism of Kan's type

We construct a family of partially hyperbolic skew-product diffeomorphisms on $\mathbb{T}^3$ that are robustly transitive and admitting two physical measures with intermingled basins. In particularly, all these diffeomorphisms are not topologically mixing. Moreover, for every such example, it exhibits a dichotomy under perturbation: every perturbation of such example either has a unique physical measure and is robustly topologically mixing, or has two physical measures with intermingled basins.

Kan型部分双曲微分同胚的拓扑传递性

Topological transitivity of Kan-type partially hyperbolic diffeomorphisms

本文证明了加厚环面上包括Kan 1994年部分双曲例子在内的一类微分同胚具有拓扑传递性，该例子以混合盆现象著称。

Measures maximizing the entropy for Kan endomorphisms

In 1994, Ittai Kan provided the first examples of maps with intermingled basins. The Kan example corresponds to a partially hyperbolic endomorphism defined on a surface with the boundary exhibiting two intermingled hyperbolic physical measures. Both measures are supported on the boundary, and they are also measures maximizing the topological entropy. In this work, we prove the existence of a third hyperbolic measure supported in the interior of the cylinder that maximizes the entropy for a larger class of maps including the Kan example. We also prove this statement for a larger class of invari…

The occurrence of riddled basins and blowout bifurcations in a parametric nonlinear system

In this paper, a two parameters family $F_{β_1,β_2}$ of maps of the plane living two different subspaces invariant is studied. We observe that, our model exhibits two chaotic attractors $A_i$, $i=0,1$, lying in these invariant subspaces and identify the parameters at which $A_i$ has a locally riddled basin of attraction or becomes a chaotic saddle. Then, the occurrence of riddled basin in the global sense is investigated in an open region of $β_1β_2$-plane. We semi-conjugate our system to a random walk model and define a fractal boundary which separates the basins of attraction of the two chao…

On the number of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms with compact center leaves

In this paper, we study the number of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms defined on $3-$torus with compact center leaves. Assuming the existence of a periodic leaf with Morse-Smale dynamics we prove a sharp upper bound for the number of maximal measures in terms of the number of sources and sinks of Morse-Smale dynamics. A well-known class of examples for which our results apply are the so-called Kan-type diffeomorphisms admitting physical measures with intermingled basins.

Geometric and measure-theoretical structures of maps with mostly contracting center

We show that every diffeomorphism with mostly contracting center direction exhibits a geometric-combinatorial structure, which we call \emph{skeleton}, that determines the number, basins and supports of the physical measures. Furthermore, the skeleton allows us to describe how the physical measure bifurcate as the diffeomorphism changes. In particular, we use this to construct examples with any given number of physical measures, with basins densely intermingled, and to analyse how these measures collapse into each other - through explosions of their basins - as the dynamics varies. This theory…

具有区间中心叶的部分双曲微分同胚的鲁棒传递性

Robust Transitivity of Partially Hyperbolic Diffeomorphisms with Interval Central Leaves

研究具有区间中心叶的边界保持部分双曲微分同胚，通过边界互连刻画其$C^k$鲁棒传递性，并将其应用于边界SRB测度的鲁棒混合盆现象刻画。

Elliptic Curves as Attractors in ${\mathbb P}^2$ Part 1: Dynamics

A study of rational maps of the real or complex projective plane of degree two or more, concentrating on those which map an elliptic curve onto itself, necessarily by an expanding map. We describe relatively simple examples with a rich variety of exotic dynamical behaviors which are perhaps familiar to the applied dynamics community but not to specialists in several complex variables. For example, we describe smooth attractors with riddled or intermingled attracting basins, and we observe ``blowout'' bifurcations when the transverse Lyapunov exponent for the invariant curve changes sign. In th…