Andres Koropecki
We prove that if a homeomorphism of a closed orientable surface S has no wandering points and leaves invariant a compact, connected set K which contains no periodic points, then either K=S and S is a torus, or $K$ is the intersection of a decreasing sequence of annuli. A version for non-orientable surfaces is given.
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Let $f$ be a chain mixing continuous onto mapping from the Cantor set onto itself.Let $g$ be an aperiodic homeomorphism on the Cantor set. We show that homeomorphisms that are topologically conjugate to g approximate $f$ in the topology of uniform convergence if a trivial necessary condition on periodic points is satisfied.In particular, let $f$ be a chain mixing continuous onto mapping from the Cantor set onto itself with a fixed point and g, an aperiodic homeomorphism on the Cantor set. Then, homeomorphisms that are topologically conjugate to $g$ approximate $f$.
For a Cantor set $X$, let $Homeo(X)$ denote the group of all homeomorphisms of $X$. The main result of this note is the following theorem. Let $T\in Homeo(X)$ be an aperiodic homeomorphism, let $μ_1,μ_2,...,μ_k$ be Borel probability measures on $X$, $\e> 0$, and $n \ge 2$. Then there exists a clopen set $E\subset X$ such that the sets $E,TE,..., T^{n-1}E$ are disjoint and $μ_i(E\cup TE\cup...\cup T^{n-1}E) > 1 - \e, i= 1,...,k$. Several corollaries of this result are given. In particular, it is proved that for any aperiodic $T\in Homeo(X)$ the set of all homeomorphisms conjugate to $T$ is dens…
On a Peano continuum, all local stable and unstable components of a continuum-wise expansive homeomorphism are non trivial. In particular, there is sensitive dependence on initial conditions. This generalizes results in \cite{h,l} about lack of Lyapunov stable points (weak sinks) and existence of non trivial stable and unstable components for expansive homeomorphisms on Peano continua. We also use this fact to generalize a result in \cite{ktt}: a Peano curve $X$ admitting a continuum-wise expansive homeomorphism is nowhere rim-countable. However, it is not resolved the question of whether such…
Let $f$ be an orientation preserving homeomorphism of $S^2$ which has a (nontrivial) continuum $X$ as a minimal set. Then there are exactly two connected components of $S^2\setminus X$ which are left invariant by $f$ and all the others are wandering. The Carathéodory rotation number of an invariant component is irrational.
In this article we show that there are homeomorphisms of plane continua whose conjugacy class is residual and have the shadowing property.
Dimensions associated with surjective local homeomorphisms and subshifts with low complexity
本文证明了具有有限左特殊元素的满射非周期子位移的Cuntz-Pimsner代数核维数有限，并推广到局部同胚，证明其相关维数及动态渐近维数均有限。
In the context of the Franks-Misiurewicz Conjecture, we study homeomorphisms of the two-torus semiconjugate to an irrational rotation of the circle. As a special case, this conjecture asserts uniqueness of the rotation vector in this class of systems. We first characterise these maps by the existence of an invariant foliation by essential annular continua (essential subcontinua of the torus whose complement is an open annulus) which are permuted with irrational combinatorics. This result places the considered class close to skew products over irrational rotations. Generalising a well-known res…
Let $f$ be a homeomorphism of the closed annulus $A$ isotopic to the identity, and let $X\subset {\rm Int}A$ be an $f$-invariant continuum which separates $A$ into two domains, the upper domain $U_+$ and the lower domain $U_-$. Fixing a lift of $f$ to the universal cover of $A$, one defines the rotation set $\tilde ρ(X)$ of $X$ by means of the invariant probabilities on $X$. For any rational number $p/q\in \tildeρ(X)$, $f$ is shown to admit a $p/q$ periodic point in $X$, provided that (1) $X$ consists of nonwandering points or (2) $X$ is an attractor and the frontiers of $U_-$ and $U_+$ coinci…
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 …
In this paper, we define a new notion of "freely tracing property by free chains" on $G$-like continua and we prove that a positive topological entropy homeomorphism on a $G$-like continuum admits a Cantor set $Z$ such that every tuple of finite points in $Z$ is an $IE$-tuple of $f$ and $Z$ has the freely tracing property by free chains. Also, by use of this notion, we prove the following theorem: If $G$ is any graph and a homeomorphism $f$ on a $G$-like continuum $X$ has positive topological entropy, then there is a Cantor set $Z$ which is related to both the chaotic behaviors of Kerr and Li …
A flexibility result for polynomial entropy of pointwise periodic homeomorphisms
构造一族连续统和逐点周期同胚，实现任意多项式熵值，证明正多项式熵的存在性，与流形和局部树突上的零熵形成对比。
We show that if $f$ is an annular homeomorphism admitting an attractor which is an irreducible annular continua with two different rotation numbers, then the entropy of $f$ is positive. Further, the entropy is shown to be associated to a $C^0$-robust rotational horseshoe. On the other hand, we construct examples of annular homeomorphisms with such attractors so that the rotation interval is uniformly large but the entropy approaches zero as much as desired. The developed techniques allow us to obtain similar results in the context of Birkhoff attractors.
It is proved that whenever two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, their associated complexity functions are asymptotically equivalent in a certain sense (which implies, if the complexity is polynomial, that the exponent of the leading term is preserved by homeomorphism). This theorem can be reworded in terms of $d$-dimensional infinite words: if two $\mathbb{Z}^d$-subshifts (with the same conditions as above) are flow equivalent, their complexity functions are equivalent. An analogue theorem is proved for the repetitivity function, which …
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.
Universal minimal flows of homeomorphism groups of continua
通过定义射影Fraïssé范畴的近似Ramsey性质，证明了该性质与群极端可安性等价，并应用于伪螺线管同胚群得到非可度量化泛最小流。
Transitive mappings on the Cantor fan
本文在拓扑结构相对简单的 Cantor 扇上构造了传递同胚，通过商空间和 Mahavier 乘积等四种不同技术实现，并展示了该方法可构造出逆极限为 Lelek 扇的传递函数。
This article includes an almost self-contained exposition on the discrete Conley index and its duality. We work with a local homeomorphism of $\mathds{R}^d$ and an invariant and isolated acyclic continuum, such as a cellular set or a fixed point. In this setting, we obtain a complete description of the first discrete homological Conley index, which is periodic, that enforces a combinatorial behavior of higher indices. As a consequence, we prove that isolated (as an invariant set) fixed points of orientation-reversing homeomorphisms of $\mathds{R}^3$ have fixed point index $\le 1$ and, as a cor…
Continuum-wise hyperbolic homeomorphisms on surfaces
本文研究曲面连续统双曲同胚的动力学，证明含有限多个脊的 $cw_F$-双曲同胚为 $cw_2$-双曲，并证明 $cw_3$-双曲性蕴含 $cw_2$-双曲性，同时揭示脊的孤立性。
This paper is a survey devoted to the study of probability and infinite ergodic invariant measures for aperiodic homeomorphisms of a Cantor set. We focus mostly on the cases when a homeomorphism has either a unique ergodic invariant measure or finitely many such measures (finitely ergodic homeomorphisms). Since every Cantor dynamical system $(X,T)$ can be realized as a Vershik map acting on the path space of a Bratteli diagram, we use combinatorial methods developed in symbolic dynamics and Bratteli diagrams during the last decade to study the simplex of invariant measures.
Maximal equicontinuous factor and minimal map on finitely suslinean continua
研究通过负区域邻近对证明局部连通连续统满射的极大等度连续因子单调，并推出有限 Suslinean 连续统上的极小映射必为拓扑圆上的无理旋转。
This paper studies homeomorphisms of the closed annulus that are isotopic to the identity from the viewpoint of rotation theory, using a newly developed forcing theory for surface homeomorphisms. Our first result is a solution to the so called strong form of Boyland's Conjecture on the closed annulus: Assume $f$ is a homeomorphism of $\overline{\mathbb{A}}:=(\mathbb{R}/\mathbb{Z})\times [0,1]$ which is isotopic to the identity and preserves a Borel probability measure $μ$ with full support. We prove that if the rotation set of $f$ is a non-trivial segment, then the rotation number of the measu…
Accessibility of countable sets in plane embeddings of arc-like continua
本文研究弧状连续统的平面嵌入问题，证明了在特定条件下逆向极限存在使给定可数集所有点均可接近的嵌入，并应用于 Knaster 连续统及构造了作为平面同胚吸引子的新嵌入。
Essential factors for surface homeomorphisms
针对亏格至少为1的闭可定向曲面上同痕于伪阿诺索夫自同胚的自同胚，通过坍缩万有覆盖中迭代直径一致有界的极大连续统构造单调因子，该因子具强混沌特征且可保面积，其极大熵测度为全支撑伯努利测度。
E.D. Tymchatyn constructed a hereditarily locally connected continuum which can be approximated by a sequence of mutually disjoint arcs. We show the example re-opens a conjecture of G.T. Seidler and H. Kato about continua which admit positive entropy homeomorphisms. We prove that every indecomposable semicontinuum can be approximated by a sequence of disjoint subcontinua, and no composant of an indecomposable continuum can be embedded into a Suslinian continuum. We also prove that if $Y$ is a hereditarily unicoherent Suslinian continuum, then there exists $\varepsilon>0$ such that every two $\…
An annular continuum is a compact connected set $K$ which separates a closed annulus $A$ into exactly two connected components, one containing each boundary component. The topology of such continua can be very intricate (for instance, non-locally connected). We adapt a result proved by Handel in the case where $K=A$, showing that if $K$ is an invariant annular continuum of a homeomorphism of $A$ isotopic to the identity, then the rotation set in $K$ is closed. Moreover, every element of the rotation set is realized by an ergodic measure supported in $K$ (and by a periodic orbit if the rotation…
Surfaces and other Peano Continua with no Generic Chains
本文证明了一大类拓扑空间（包括维数≥3的流形、多数紧致曲面及齐性Peano连续统）不存在generic链，并在强局部齐性空间中证明链无法被可数结构分类且同胚群具有不可度化泛极小流。
For an orientation-preserving homeomorphism of the sphere, we prove that if a translation line does not accumulate in a fixed point, then it necessarily spirals towards a topological attractor. This is in analogy with the description of flow lines given by Poincaré-Bendixson theorem. We then apply this result to the study of invariant continua without fixed points, in particular to circloids and boundaries of simply connected open sets. Among the applications, we show that if the prime ends rotation number of such an open set $U$ vanishes, then either there is a fixed point in the boundary, or…
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