There are no proper topological hyperbolic homoclinic classes for area-preserving maps

AbstractWe begin by defining a homoclinic class for homeomorphisms. Then we prove that if a topological homoclinic class Λ associated with an area-preserving homeomorphism f on a surface M is topologically hyperbolic (i.e. has the shadowing and expansiveness properties), then Λ = M and f is an Anosov homeomorphism.

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Elliptic isles in families of area-preserving maps

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385707000983 ◽

2008 ◽

Vol 28 (06) ◽

pp. 1781 ◽

Cited By ~ 19

Author(s):

P. DUARTE

Keyword(s):

Area Preserving Maps ◽

Area Preserving

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Area-Preserving Maps

The Transition to Chaos - Institute for Nonlinear Science ◽

10.1007/978-1-4757-4350-0_3 ◽

2004 ◽

pp. 58-133 ◽

Cited By ~ 1

Author(s):

Linda E. Reichl

Keyword(s):

Area Preserving Maps ◽

Area Preserving

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Fixed point index of area-preserving maps

Lecture Notes in Mathematics - Dynamical Systems—Warwick 1974 ◽

10.1007/bfb0082601 ◽

1975 ◽

pp. 59-60

Author(s):

Carl P. Simon

Keyword(s):

Fixed Point ◽

Fixed Point Index ◽

Point Index ◽

Area Preserving Maps ◽

Area Preserving

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Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants

Electronic Research Announcements ◽

10.3934/era.2013.20.97 ◽

2013 ◽

Vol 20 (0) ◽

pp. 97-102

Author(s):

Daniel N. Dore ◽

◽

Andrew D. Hanlon ◽

Keyword(s):

Lower Bound ◽

Spectral Invariants ◽

Area Preserving Maps ◽

Area Preserving

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COMMENSURATE-INCOMMENSURATE TRANSITIONS AND AREA-PRESERVING MAPS : THE FRENKEL-KONTOROVA MODEL

Nonlinear Phenomena in Fluids, Solids and Other Complex Systems ◽

10.1016/b978-0-444-88791-7.50021-1 ◽

1991 ◽

pp. 243-265

Author(s):

Robert B. GRIFFITHS

Keyword(s):

Area Preserving Maps ◽

Area Preserving ◽

Kontorova Model

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Global stability of a class of dincontinuous area-preserving maps

Physica D Nonlinear Phenomena ◽

10.1016/0167-2789(85)90202-7 ◽

1985 ◽

Vol 14 (2) ◽

pp. 285

Keyword(s):

Global Stability ◽

Area Preserving Maps ◽

Area Preserving

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Renormalization group analysis of bifurcations in area-preserving maps

Physica D Nonlinear Phenomena ◽

10.1016/0167-2789(82)90023-9 ◽

1982 ◽

Vol 5 (2-3) ◽

pp. 287-292 ◽

Cited By ~ 21

Author(s):

Michael Widom ◽

Leo P. Kadanoff

Keyword(s):

Renormalization Group ◽

Group Analysis ◽

Renormalization Group Analysis ◽

Area Preserving Maps ◽

Area Preserving

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A computer-assisted proof of universality for area-preserving maps

Memoirs of the American Mathematical Society ◽

10.1090/memo/0289 ◽

1984 ◽

Vol 47 (289) ◽

pp. 0-0 ◽

Cited By ~ 15

Author(s):

J.-P. Eckmann ◽

H. Koch ◽

P. Wittwer

Keyword(s):

Computer Assisted ◽

Computer Assisted Proof ◽

Area Preserving Maps ◽

Area Preserving

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Lyapunov stable homoclinic classes for smooth vector fields

Open Mathematics ◽

10.1515/math-2019-0068 ◽

2019 ◽

Vol 17 (1) ◽

pp. 990-997

Author(s):

Manseob Lee

Keyword(s):

Vector Fields ◽

Smooth Vector ◽

Homoclinic Class ◽

Shadowing Property ◽

Homoclinic Classes

Abstract In this paper, we show that for generic C1, if a flow Xt has the shadowing property on a bi-Lyapunov stable homoclinic class, then it does not contain any singularity and it is hyperbolic.

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Graphs in the plane invariant under an area preserving linear map and general continuous solutions of certain quadratic functional equations

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100062812 ◽

1985 ◽

Vol 97 (2) ◽

pp. 261-278 ◽

Cited By ~ 2

Author(s):

P. J. McCarthy ◽

M. Crampin ◽

W. Stephenson

Keyword(s):

Functional Equation ◽

Arbitrary Function ◽

Functional Equations ◽

Quadratic Functional ◽

Continuous Solutions ◽

Linear Map ◽

Area Preserving Maps ◽

Image Position ◽

Area Preserving

AbstractThe requirement that the graph of a function be invariant under a linear map is equivalent to a functional equation of f. For area preserving maps M(det (M) = 1), the functional equation is equivalent to an (easily solved) linear one, or to a quadratic one of the formfor all Here 2C = Trace (M). It is shown that (Q) admits continuous solutions ⇔ M has real eigenvalues ⇔ (Q) has linear solutions f(x) = λx ⇔ |C| ≥ 1. For |c| = 1 or C < – 1, (Q) only admits a few simple solutions. For C > 1, (Q) admits a rich supply of continuous solutions. These are parametrised by an arbitrary function, and described in the sense that a construction is given for the graphs of the functions which solve (Q).

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