scholarly journals Fully essential dynamics for area-preserving surface homeomorphisms

2017 ◽  
Vol 38 (5) ◽  
pp. 1791-1836 ◽  
Author(s):  
ANDRES KOROPECKI ◽  
FABIO ARMANDO TAL
Keyword(s):  
Initial Conditions ◽  
Essential Dynamics ◽  
Empty Interior ◽  
Fixed Point Set ◽  
Higher Genus ◽  
Point Set ◽  
Surface Homeomorphisms ◽  
Rotation Set ◽  
Bounded Sets ◽  
Area Preserving

We study the interplay between the dynamics of area-preserving surface homeomorphisms homotopic to the identity and the topology of the surface. We define fully essential dynamics and generalize the results previously obtained on strictly toral dynamics to surfaces of higher genus. Non-fully essential dynamics are, in a way, reducible to surfaces of lower genus, while in the fully essential case the dynamics is decomposed into a disjoint union of periodic bounded disks and a complementary invariant externally transitive continuum $C$. When the Misiurewicz–Ziemian rotation set has non-empty interior the dynamics is fully essential, and the set $C$ is (externally) sensitive on initial conditions and realizes all of the rotational dynamics. As a fundamental tool we introduce the notion of homotopically bounded sets and we prove a general boundedness result for invariant open sets when the fixed point set is inessential.

scholarly journals Area-preserving diffeomorphisms of the torus whose rotation sets have non-empty interior

2013 ◽  
Vol 35 (1) ◽  
pp. 1-33 ◽  
Author(s):  
SALVADOR ADDAS-ZANATA
Keyword(s):  
Periodic Point ◽  
Interior Point ◽  
Connected Component ◽  
Empty Interior ◽  
Dehn Twists ◽  
Topologically Mixing ◽  
Rotation Set ◽  
Area Preserving ◽  
Uniformly Bounded

AbstractIn this paper we consider${C}^{1+ \epsilon } $area-preserving diffeomorphisms of the torus $f$, either homotopic to the identity or to Dehn twists. We suppose that$f$has a lift$\widetilde {f} $to the plane such that its rotation set has interior and prove, among other things, that if zero is an interior point of the rotation set, then there exists a hyperbolic$\widetilde {f} $-periodic point$\widetilde {Q} \in { \mathbb{R} }^{2} $such that${W}^{u} (\widetilde {Q} )$intersects${W}^{s} (\widetilde {Q} + (a, b))$for all integers$(a, b)$, which implies that$ \overline{{W}^{u} (\widetilde {Q} )} $is invariant under integer translations. Moreover,$ \overline{{W}^{u} (\widetilde {Q} )} = \overline{{W}^{s} (\widetilde {Q} )} $and$\widetilde {f} $restricted to$ \overline{{W}^{u} (\widetilde {Q} )} $is invariant and topologically mixing. Each connected component of the complement of$ \overline{{W}^{u} (\widetilde {Q} )} $is a disk with diameter uniformly bounded from above. If$f$is transitive, then$ \overline{{W}^{u} (\widetilde {Q} )} = { \mathbb{R} }^{2} $and$\widetilde {f} $is topologically mixing in the whole plane.

scholarly journals Existence of periodic points near an isolated fixed point with Lefschetz index one and zero rotation for area preserving surface homeomorphisms

2015 ◽  
Vol 36 (7) ◽  
pp. 2293-2333
Author(s):  
JINGZHI YAN
Keyword(s):  
Fixed Point ◽  
Periodic Orbits ◽  
Periodic Points ◽  
Dirac Measure ◽  
Lefschetz Index ◽  
Weak Star ◽  
Surface Homeomorphisms ◽  
Rotation Set ◽  
Area Preserving ◽  
Measure Sense

Let $f$ be an orientation and area preserving diffeomorphism of an oriented surface $M$ with an isolated degenerate fixed point $z_{0}$ with Lefschetz index one. Le Roux conjectured that $z_{0}$ is accumulated by periodic orbits. In this paper, we will approach Le Roux’s conjecture by proving that if $f$ is isotopic to the identity by an isotopy fixing $z_{0}$ and if the area of $M$ is finite, then $z_{0}$ is accumulated not only by periodic points, but also by periodic orbits in the measure sense. More precisely, the Dirac measure at $z_{0}$ is the limit in the weak-star topology of a sequence of invariant probability measures supported on periodic orbits. Our proof is purely topological. It works for homeomorphisms and is related to the notion of local rotation set.

A condition that implies full homotopical complexity of orbits for surface homeomorphisms

2019 ◽  
Vol 41 (1) ◽  
pp. 1-47
Author(s):  
SALVADOR ADDAS-ZANATA ◽  
BRUNO DE PAULA JACOIA
Keyword(s):  
Compact Convex ◽  
Universal Cover ◽  
Invariant Sets ◽  
Maximal Dimension ◽  
Rotation Vectors ◽  
Surface Homeomorphisms ◽  
Rotation Set ◽  
Poincaré Disk ◽  
Area Preserving ◽  
Uniformly Bounded

We consider closed orientable surfaces $S$ of genus $g>1$ and homeomorphisms $f:S\rightarrow S$ isotopic to the identity. A set of hypotheses is presented, called a fully essential system of curves $\mathscr{C}$ and it is shown that under these hypotheses, the natural lift of $f$ to the universal cover of $S$ (the Poincaré disk $\mathbb{D}$), denoted by $\widetilde{f},$ has complicated and rich dynamics. In this context, we generalize results that hold for homeomorphisms of the torus isotopic to the identity when their rotation sets contain zero in the interior. In particular, for $C^{1+\unicode[STIX]{x1D716}}$ diffeomorphisms, we show the existence of rotational horseshoes having non-trivial displacements in every homotopical direction. As a consequence, we found that the homological rotation set of such an $f$ is a compact convex subset of $\mathbb{R}^{2g}$ with maximal dimension and all points in its interior are realized by compact $f$-invariant sets and by periodic orbits in the rational case. Also, $f$ has uniformly bounded displacement with respect to rotation vectors in the boundary of the rotation set. This implies, in case where $f$ is area preserving, that the rotation vector of Lebesgue measure belongs to the interior of the rotation set.

Iterative algorithm for a split equilibrium problem and fixed problem for finite asymptotically nonexpansive mappings in Hlbert space

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1423-1434 ◽  
Author(s):  
Sheng Wang ◽  
Min Chen
Keyword(s):  
Equilibrium Problem ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Fixed Point Set ◽  
Split Equilibrium Problem ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Point Set ◽  
The Common

In this paper, we propose an iterative algorithm for finding the common element of solution set of a split equilibrium problem and common fixed point set of a finite family of asymptotically nonexpansive mappings in Hilbert space. The strong convergence of this algorithm is proved.

scholarly journals Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Zhao-Rong Kong ◽  
Lu-Chuan Ceng ◽  
Qamrul Hasan Ansari ◽  
Chin-Tzong Pang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Hybrid Extragradient Method ◽  
Point Set ◽  
Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.

Stein manifolds of nonnegative curvature

2018 ◽  
Vol 18 (3) ◽  
pp. 285-287
Author(s):  
Xiaoyang Chen
Keyword(s):  
Fixed Point ◽  
Sectional Curvature ◽  
Normal Bundle ◽  
Stein Manifold ◽  
Riemannian Metric ◽  
Nonnegative Curvature ◽  
Fixed Point Set ◽  
Nonnegative Sectional Curvature ◽  
Point Set ◽  
Nonempty Compact

AbstractLet X bea Stein manifold with an anti-holomorphic involution τ and nonempty compact fixed point set Xτ. We show that X is diffeomorphic to the normal bundle of Xτ provided that X admits a complete Riemannian metric g of nonnegative sectional curvature such that τ*g = g.

scholarly journals On soft quasi-pseudometric spaces

2021 ◽  
Vol 22 (1) ◽  
pp. 17
Author(s):  
Hope Sabao ◽  
Olivier Olela Otafudu
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Soft Set ◽  
Fixed Point Set ◽  
Point Set ◽  
The Absolute ◽  
Pseudometric Space

<p>In this article, we introduce the concept of a soft quasi-pseudometric space. We show that every soft quasi-pseudometric induces a compatible quasi-pseudometric on the collection of all soft points of the absolute soft set whenever the parameter set is finite. We then introduce the concept of soft Isbell convexity and show that a self non-expansive map of a soft quasi-metric space has a nonempty soft Isbell convex fixed point set.</p>

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