scholarly journals Abundance of Fast Growth of the Number of Periodic Points in 2-Dimensional Area-Preserving Dynamics

2017 ◽  
Vol 356 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Masayuki Asaoka
Keyword(s):  
Periodic Points ◽  
Fast Growth ◽  
Area Preserving

scholarly journals On reversible maps and symmetric periodic points

2016 ◽  
Vol 38 (4) ◽  
pp. 1479-1498
Author(s):  
JUNGSOO KANG
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Periodic Points ◽  
Analogous Statement ◽  
Twist Map ◽  
Planar Domains ◽  
Area Preserving Maps ◽  
Symmetric Periodic Orbits ◽  
Area Preserving ◽  
Double Symmetries

In reversible dynamical systems, it is of great importance to understand symmetric features. The aim of this paper is to explore symmetric periodic points of reversible maps on planar domains invariant under a reflection. We extend Franks’ theorem on a dichotomy of the number of periodic points of area-preserving maps on the annulus to symmetric periodic points of area-preserving reversible maps. Interestingly, even a non-symmetric periodic point guarantees infinitely many symmetric periodic points. We prove an analogous statement for symmetric odd-periodic points of area-preserving reversible maps isotopic to the identity, which can be applied to dynamical systems with double symmetries. Our approach is simple, elementary, and far from Franks’ proof. We also show that a reversible map has a symmetric fixed point if and only if it is a twist map which generalizes a boundary twist condition on the closed annulus in the sense of Poincaré–Birkhoff. Applications to symmetric periodic orbits in reversible dynamical systems with two degrees of freedom are briefly discussed.

scholarly journals A symplectic proof of a theorem of Franks

2012 ◽  
Vol 148 (6) ◽  
pp. 1969-1984 ◽  
Author(s):  
Brian Collier ◽  
Ely Kerman ◽  
Benjamin M. Reiniger ◽  
Bolor Turmunkh ◽  
Andrew Zimmer
Keyword(s):  
Crucial Role ◽  
Additional Assumption ◽  
Periodic Points ◽  
Two Dimensional ◽  
Symplectic Topology ◽  
Hamiltonian Diffeomorphisms ◽  
Area Preserving

AbstractA celebrated theorem in two-dimensional dynamics due to John Franks asserts that every area-preserving homeomorphism of the sphere has either two or infinitely many periodic points. In this work we re-prove Franks’ theorem under the additional assumption that the map is smooth. Our proof uses only tools from symplectic topology and thus differs significantly from previous proofs. A crucial role is played by the results of Ginzburg and Kerman concerning resonance relations for Hamiltonian diffeomorphisms.

scholarly journals Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics

2016 ◽  
Vol 368 (3-4) ◽  
pp. 1277-1309 ◽  
Author(s):  
Masayuki Asaoka ◽  
Katsutoshi Shinohara ◽  
Dmitry Turaev
Keyword(s):  
Periodic Points ◽  
Fast Growth ◽  
Semigroup Actions

scholarly journals A Cr unimodal map with an arbitrary fast growth of the number of periodic points

2011 ◽  
Vol 32 (1) ◽  
pp. 159-165 ◽  
Author(s):  
V. KALOSHIN ◽  
O. S. KOZLOVSKI
Keyword(s):  
Critical Points ◽  
Acta Math ◽  
Periodic Points ◽  
Fast Growth ◽  
Unimodal Map ◽  
One Dimensional

AbstractIn this paper we present a surprising example of a Cr unimodal map of an interval f:I→I whose number of periodic points Pn(f)=∣{x∈I:fnx=x}∣ grows faster than any ahead given sequence along a subsequence nk=3k. This example also shows that ‘non-flatness’ of critical points is necessary for the Martens–de Melo–van Strien theorem [M. Martens, W. de Melo and S. van Strien. Julia–Fatou–Sullivan theory for real one-dimensional dynamics. Acta Math.168(3–4) (1992), 273–318] to hold.

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.

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