Fabio Armando Tal
We show that, if $f$ is a homeomorphism of the 2--torus isotopic to the identity, and its lift $\widetilde f$ is transitive, or even if it is transitive outside of the lift of the elliptic islands, then $(0,0)$ is in the interior of the rotation set of $\widetilde f.$ This proves a particular case of Boyland's conjecture.
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Let $f$ be a transitive homeomorphism of the two-dimensional torus in the homotopy class of the identity. We show that a lift of $f$ to the universal covering is transitive if and only if the rotation set of the lift contains the origin in its interior.
Stable/unstable holonomies, density of periodic points, and transitivity for continuum-wise hyperbolic homeomorphisms
本文研究了连续wise超几何同胚的稳定/不稳定同宿线的正则性，证明了在特定条件下周期点的密度以及同胚的transitivity。
On sufficient conditions for the transitivity of homeomorphisms
本文推导了具有跟踪性质的同胚拓扑传递的充要条件（含稠密不变子集的重心性质），对比Anosov微分同胚的相关条件，描述$C^1$内部、讨论例子并给出应用，为动力学传递性研究提供新依据。
We study the relationship between transitivity and topological chaos for homeomorphisms of the two torus. We show that if a transitive homeomorphism of $\mathbb{T}^2$ is homotopic to the identity and has both a fixed point and a periodic point which is not fixed, then it has a topological horseshoe. We also show that if a transitive homeomorphims of $\mathbb{T}^2$ is homotopic to a Dehn twist, then either it is aperiodic or it has a topological horseshoe.
It is shown that if a non-invertible area preserving local homeomorphism on $\mathbb{T}^2$ is homotopic to a linear expanding or hyperbolic endomorphism, then it must be topologically transitive. This gives a complete characterization, in any smoothness category, of those homotopy classes of conservative endomorphisms that consist entirely of transitive maps.
On jointly transitive commuting minimal homeomorphisms
类比遍历理论，引入紧致度量空间上可交换极小同胚的联合拓扑传递性和联合极小性概念，证明了关于第一个概念的Berend - Bergelson定理的类似结果并研究其对第二个概念的适用程度。
Fully chaotic conservative models for some torus homeomorphisms
研究同伦于恒等映射且旋转集有非空内部的环面同胚，证明其单调半共轭于保勒贝格测度且旋转集相同的同胚，且商映射具有拓扑混合、周期点稠密和连续统扩张等混沌性质。
A homeomorphism f of a manifold M is called H_1-transitive if there is a transitive lift of an iterate of f to the universal Abelian cover \tM. Roughly speaking, this means that f has orbits which repeatedly and densely explore all elements of H_1(M). For a rel pseudo-Anosov map ϕof a compact surface M we show that the following are equivalent: (a) ϕis H_1-transitive, (b) the action of ϕon H_1(M) has spectral radius one, and (c) the lifts of the invariant foliations of ϕto \tM have dense leaves. The proof relies on a characterization of transitivity for twisted \Z^d-extensions of a transitive …
Let $f \colon X \to X$ be a continuous map on a compact metric space $X$ and let $α_f$, $ω_f$ and $ICT_f$ denote the set of $α$-limit sets, $ω$-limit sets and nonempty closed internally chain transitive sets respectively. In this paper we characterise, by introducing novel variants of shadowing, maps for which every element of $ICT_f$ is equal to (resp. may be approximated by) the $α$-limit set and the $ω$-limit set of the same full trajectory. We construct examples highlighting the difference between these properties.
For every group $G$, we show that either $G$ has a topologically transitive action on the line $\mathbb R$ by orientation-preserving homeomorphisms, or every orientation-preserving action of $G$ on $\mathbb R$ has a wandering interval. According to this result, all groups are divided into two types: transitive type and wandering type, and the types of several groups are determined. We also show that every finitely generated orderable group of wandering type is indicable. As a corollary, we show that if a higher rank lattice $Γ$ is orderable, then $Γ$ is of transitive type.
On Non-Wandering Sets with Non-empty Interior for Endomorphisms
本文研究自同态的传递性，证明在双曲非游荡集具有非空内部或可达部分双曲条件下，系统是传递的，推进了相关猜想。
We study the relationship between free curves and periodic points for torus homeomorphisms in the homotopy class of the identity. By free curve we mean a homotopically nontrivial simple closed curve that is disjoint from its image. We prove that every rational point in the rotation set is realized by a periodic point provided that there is no free curve and the rotation set has empty interior. This gives a topological version of a theorem of Franks. Using this result, and inspired by a theorem of Guillou, we prove a version of the Poincaré-Birkhoff Theorem for torus homeomorphisms: in the abse…
Let $X$ be a continuum and let $φ:X\rightarrow X$ be a homeomorphism. To construct a dynamical system $(X,φ)$ with interesting dynamical properties, the continuum $X$ often needs to have some complicated topological structure. In this paper, we are interested in one such dynamical property: transitivity. By now, various examples of continua $X$ have been constructed in such a way that the dynamical system $(X,φ)$ is transitive. Mostly, they are examples of continua that are not path-connected, such as the pseudo-arc or the pseudo-circle, or they are examples of locally connected continua (and …
Let $f \colon X \to X$ be a continuous map on a compact metric space $X$ and let $α_f$, $ω_f$ and $ICT_f$ denote the set of $α$-limit sets, $ω$-limit sets and nonempty closed internally chain transitive sets respectively. We show that if the map $f$ has shadowing then every element of $ICT_f$ can be approximated (to any prescribed accuracy) by both the $α$-limit set and the $ω$-limit set of a full-trajectory. Furthermore, if $f$ is additionally c-expansive then every element of $ICT_f$ is equal to both the $α$-limit set and the $ω$-limit set of a full-trajectory. In particular this means that …
Let $f$ be a homeomorphism of the closed annulus $A$ that preserves orientation, boundary components and that has a lift $\tilde f$ to the infinite strip $\tilde A$ which is transitive. We show that, if the rotation number of both boundary components of $A$ is strictly positive, then there exists a closed nonempty connected set $Γ\subset\tilde A$ such that $Γ\subset]-\infty,0]\times[0,1]$, $Γ$ is unlimited, the projection of $Γ$ to $A$ is dense, $Γ-(1,0)\subsetΓ$ and $\tilde{f}(Γ)\subset Γ.$ Also, if $p_1$ is the projection in the first coordinate in $\tilde A$, then there exists $d>0$ such th…
In this work we develop a new criterion for the existence of topological horseshoes for surface homeomorphisms in the isotopy class of the identity. Based on our previous work on forcing theory, this new criterion is purely topological and can be expressed in terms of equivariant Brouwer foliations and transverse trajectories. We then apply this new tool in the study of the dynamics of homeomorphisms of surfaces with zero genus and null topological entropy and we obtain several applications. For homeomorphisms of the open annulus $\mathbb{A}$ with zero topological entropy, we show that rotatio…
We determine all the normal subgroups of the group of C^r diffeomorphisms of R^n, r = 1,2,...,infinity, except when r=n+1 or n=4, and also of the group of homeomorphisms of R^n (r=0). We also study the group A_0 of diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a compact manifold with nonempty boundary, then the quotient of A_0 by the normal subgroup of diffeomorphisms that coincide with the identity near to a given end e of M is simple.
Minimal sets for torus homeomorphisms with an irrational circle factor
本文研究具有无理圆周因子的环面同胚，在纤维为Jordan曲线等条件下证明极小集唯一性，并构造具有多个极小Cantor集的例子。
Control sets on maximal compact subgroups
本文研究有限中心非紧连通半单Lie群中具非空内部的半群在极大紧连通子群上的作用，刻画控制集传递集并确定控制集数量。
Hyperbolic isometries of the fine curve graph of higher genus surfaces
该研究针对闭双曲曲面上同痕于恒等映射的同胚，证明了其在精细曲线图上的双曲作用、相对于有限不变集同痕于pseudo-Anosov映射、遍历同调旋转集非空内部这三者的等价性。
Let $(X,T)$ be a topological dynamical system and $\mathcal{F}$ be a Furstenberg family (a collection of subsets of $\mathbb{Z}_+$ with hereditary upward property). A point $x\in X$ is called an $\mathcal{F}$-transitive one if $\{n\in\mathbb{Z}_+:\, T^n x\in U\}\in\F$ for every nonempty open subset $U$ of $X$; the system $(X,T)$ is called $\F$-point transitive if there exists some $\mathcal{F}$-transitive point. In this paper, we aim to classify transitive systems by $\mathcal{F}$-point transitivity. Among other things, it is shown that $(X,T)$ is a weakly mixing E-system (resp.\@ weakly mixin…
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