Discretized rotation has infinitely many periodic orbits (original) (raw)

Cascades of periodic orbits in two dimensions

For orientation preserving-homeomorphisms of the 2-disk, we consider the forcing relation among braid types of periodic orbits. We show that if a braid type forces a braid type , then either is an extension of or forces infinitely many extensions of . This generalizes a well known result in one-dimensional dynamics

Asymptotic linking of periodic orbits for diffeomorphisms of the two-disk

Let f be a C'lf orientation preserving diffeomorphism of the two-disk with positive topological entropy. We define for f an interval of topological invariants. Each point in this interval describes the way the elements of an infinite sequence of periodic orbits with arbitrarily large periods, are asymptotically linked one around the other, AMS classification scheme numbers: 3405.54EO. 58F15

Long periodic orbits of the triangle map

Proceedings of the American Mathematical Society, 1986

Let τ : [ 0 , 1 ] → [ 0 , 1 ] \tau :[0,1] \to [0,1] be defined by τ ( x ) = 2 x \tau (x) = 2x on [ 0 , 1 / 2 ] [0,1/2] and τ ( x ) = 2 ( 1 − x ) \tau (x) = 2(1 - x) on [ 1 / 2 , 1 ] [1/2,1] . We consider τ \tau restricted to the domain D N = { 2 a / p N , N ⩾ 1 , 0 ⩽ 2 a ⩽ p N , ( a , p ) = 1 } {D_N} = \{ 2a/{p^N},N \geqslant 1,0 \leqslant 2a \leqslant {p^N},(a,p) = 1\} where p p is any odd prime. Let k ⩾ 1 k \geqslant 1 be the minimum integer such that p N | 2 k ± 1 {p^N}|{2^k} \pm 1 . Then there are ( ( p − 1 ) ⋅ p N − 1 ) / 2 k (({\text {p}} - 1) \cdot {p^{N - 1}})/2k periodic orbits of τ | D N \tau {|_{{D_N}}} , having equal length, and there are k k points in each orbit. Furthermore, the proportion of points in any of these periodic orbits which lie in an interval ( c , d ) (c,d) approaches d − c d - c as p N − 1 → ∞ {p^{N - 1}} \to \infty . An application to the irreducibility of certain nonnegative matrices is given.

Periodic orbits for a class of reversible quadratic vector field on

Journal of Mathematical Analysis and Applications, 2007

For a class of reversible quadratic vector fields on R 3 we study the periodic orbits that bifurcate from a heteroclinic loop having two singular points at infinity connected by an invariant straight line in the finite part and another straight line at infinity in the local chart U 2. More specifically, we prove that for all n ∈ N, there exists ε n > 0 such that the reversible quadratic polynomial differential systeṁ x = a 0 + a 1 y + a 3 y 2 + a 4 y 2 + ε a 2 x 2 + a 3 xz , y = b 1 z + b 3 yz + εb 2 xy, z = c 1 y + c 4 z 2 + εc 2 xz in R 3 , with a 0 < 0, b 1 c 1 < 0, a 2 < 0, b 2 < a 2 , a 4 > 0, c 2 < a 2 and b 3 / ∈ {c 4 , 4c 4 }, for ε ∈ (0, ε n) has at least n periodic orbits near the heteroclinic loop.

Dynamical Systems on the Circle

2019

In this paper we introduce dynamical systems on the circle. Beginning with elementary notions of dynamical systems, we develop several tools to study these systems, leading to the rotation number, a key invariant of circle homeomorphisms. We then show that any non-periodic circle homeomorphism is semiconjugate to the rotation by its rotation number. Finally, we present a proof of Denjoy’s Theorem, which states that any non-periodic circle diffeomorphism with derivative of bounded variation is conjugate to the rotation by its rotation number.

Periodic Orbits of Maps of Y

Transactions of the American Mathematical Society, 1989

We introduce some notions that are useful for studying the behavior of periodic orbits of maps of one-dimensional spaces. We use them to characterize the set of periods of periodic orbits for continuous maps of Y = (z e C: z3 e [0,1]} into itself having zero as a fixed point. We also obtain new proofs of some known results for maps of an interval into itself.

Action and periodic orbits on annulus

2021

We consider the classical problem of area-preserving maps on annulus A = S × [0, 1] . Let Mf be the set of all invariant probability measures of an area-preserving, orientation preserving diffeomorphism f on A. Given any μ1 and μ2 in Mf , Franks [2][3], generalizing Poincaré’s last geometric theorem (Birkhoff [1]), showed that if their rotation numbers are different, then f has infinitely many periodic orbits. In this paper, we show that if μ1 and μ2 have different actions, even if they have the same rotation number, then f has infinitely many periodic orbits. In particular, if the action difference is larger than one, then f has at least two fixed points. The same result is also true for area-preserving diffeomorphisms on unit disk, where no rotation number is available.

Rotation sets and almost periodic sequences

Mathematische Zeitschrift, 2016

We study the rotational behaviour on minimal sets of torus homeomorphisms and show that the associated rotation sets can be any type of line segments as well as non-convex and even plane-separating continua. This shows that restrictions which hold for rotation set on the whole torus are not valid on minimal sets. The proof uses a construction of rotational horseshoes by Kwapisz to transfer the problem to a symbolic level, where the desired rotational behaviour is implemented by means of suitable irregular Toeplitz sequences.

Dynamics close to a non semi-simple 1:-1 resonant periodic orbit

Discrete and Continuous Dynamical Systems - Series B, 2005

In this work, our target is to analyze the dynamics around the 1 : −1 resonance which appears when a family of periodic orbits of a real analytic three-degree of freedom Hamiltonian system changes its stability from elliptic to a complex hyperbolic saddle passing through degenerate elliptic. Our analytical approach consists of computing, up to some given arbitrary order, the normal form around that resonant (or critical) periodic orbit.