Existence of torsion-low maximal isotopies for area preserving surface homeomorphisms

AbstractWe prove that if f is a homeomorphism of the annulus which is homotopic to the identity and has a compact invariant chain transitive set L, then either f has a fixed point or every point of L moves uniformly in one direction: clockwise or counterclockwise. If f is area-preserving, then the annulus itself is a chain transitive set, so, in the presence of a boundary twist condition, one obtains a fixed point. The same techniques apply to homeomorphisms of the torus T2. In this setting we show that if f is homotopic to the identity, preserves Lebesgue measure and has mean translation 0, then it has at least one fixed point.

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

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2015.18 ◽

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.

Fully essential dynamics for area-preserving surface homeomorphisms

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2016.110 ◽

2017 ◽

Vol 38 (5) ◽

pp. 1791-1836 ◽

Cited By ~ 2

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.

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

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2019.62 ◽

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.

A Primer on Mapping Class Groups (PMS-49)

10.23943/princeton/9780691147949.001.0001 ◽

2017 ◽

Cited By ~ 1

Author(s):

Benson Farb ◽

Dan Margalit

Keyword(s):

Graduate Students ◽

Moduli Space ◽

Riemann Surfaces ◽

Class Group ◽

Mapping Class ◽

Torelli Group ◽

Surface Bundles ◽

Surface Homeomorphisms ◽

Low Dimensional

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

On an area-preserving inverse curvature flow of convex closed plane curves

Journal of Functional Analysis ◽

10.1016/j.jfa.2021.108931 ◽

2021 ◽

Vol 280 (8) ◽

pp. 108931

Author(s):

Laiyuan Gao ◽

Shengliang Pan ◽

Dong-Ho Tsai

Keyword(s):

Curvature Flow ◽

Plane Curves ◽

Area Preserving ◽

Closed Plane

Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.113 ◽

2020 ◽

pp. 1-17

Author(s):

THOMAS BARTHELMÉ ◽

SERGIO R. FENLEY ◽

STEVEN FRANKEL ◽

RAFAEL POTRIE

Keyword(s):

Large Class ◽

Universal Cover ◽

Isotopy Class ◽

Seifert Manifold ◽

Hyperbolic Diffeomorphisms ◽

Surface Homeomorphisms ◽

Partially Hyperbolic ◽

Partially Hyperbolic Diffeomorphisms ◽

Partially Hyperbolic Diffeomorphism

Abstract We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

Thermodynamics of smooth models of pseudo–Anosov homeomorphisms

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2021.43 ◽

2021 ◽

pp. 1-43

Author(s):

DOMINIC VECONI

Keyword(s):

Thermodynamic Formalism ◽

Equilibrium States ◽

Maximal Entropy ◽

Srb Measure ◽

The Central Limit Theorem ◽

Exponential Decay Of Correlations ◽

Surface Homeomorphisms ◽

Young Towers ◽

Uniqueness Of Equilibrium ◽

Measure Of Maximal Entropy

Abstract We develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

Area-Preserving Surface Flattening Using Lie Advection

Lecture Notes in Computer Science - Medical Image Computing and Computer-Assisted Intervention – MICCAI 2011 ◽

10.1007/978-3-642-23629-7_41 ◽

2011 ◽

pp. 335-342 ◽

Cited By ~ 2

Author(s):

Guangyu Zou ◽

Jiaxi Hu ◽

Xianfeng Gu ◽

Jing Hua

Keyword(s):

Surface Flattening ◽

Area Preserving

Airfoil Optimization Using Area-Preserving Free-Form Deformation

Volume 1: Advances in Aerospace Technology ◽

10.1115/imece2015-50904 ◽

2015 ◽

Author(s):

Stavros N. Leloudas ◽

Giorgos A. Strofylas ◽

Ioannis K. Nikolos

Keyword(s):

Cross Sectional Area ◽

Equality Constraint ◽

Optimization Process ◽

Free Form ◽

Sectional Area ◽

Cross Sectional ◽

Airfoil Optimization ◽

Free Form Deformation ◽

Correction Problem ◽

Area Preserving

Given the importance of structural integrity of aerodynamic shapes, the necessity of including a cross-sectional area equality constraint among other geometrical and aerodynamic ones arises during the optimization process of an airfoil. In this work an airfoil optimization scheme is presented, based on Area-Preserving Free-Form Deformation (AP FFD), which serves as an alternative technique for the fulfillment of a cross-sectional area equality constraint. The AP FFD is based on the idea of solving an area correction problem, where a minimum possible offset is applied on all free-to-move control points of the FFD lattice, subject to the area preservation constraint. Due to the linearity of the area constraint in each axis, the extraction of an inexpensive closed-form solution to the area preservation problem is possible by using Lagrange Multipliers. A parallel Differential Evolution (DE) algorithm serves as the optimizer, assisted by two Artificial Neural Networks as surrogates. The use of multiple surrogate models, in conjunction with the inexpensive solution to the area correction problem, render the optimization process time efficient. The application of the proposed methodology for wind turbine airfoil optimization demonstrates its applicability and effectiveness.