Changing rotation intervals of endomorphisms of the circle (original) (raw)

Rotation intervals of endomorphisms of the circle

Ergodic Theory and Dynamical Systems, 1984

The rotation number of a diffeomorphism f: S1 → S1 with lift is defined as . We investigate the case where f is an endomorphism. Then this limit may not exist and may depend on x. We investigate the set of limit points of , as a function of x.

On conjugacy of homeomorphisms of the circle possessing periodic points

Journal of Mathematical Analysis and Applications, 2007

We give a necessary and sufficient condition for topological conjugacy of homeomorphisms of the circle having periodic points. As an application we get the following theorem on the representation of homeomorphisms. The homeomorphism F : S 1 → S 1 has a periodic point of period n iff there exist a positive integer q < n relatively prime to n and a homeomorphism Φ : S 1 → S 1 such that the lift of Φ −1 • F • Φ restricted to [0, 1] has the form

On the Regularity of the Displacement Sequence of an Orientation Preserving Circle Homeomorphism

2015

We investigate the regularity properties of the displacement sequence () () () (), 2 exp , 1 mod 1 ix z x x z n n n π = Φ − Φ = η − where R R → Φ : is a lift of an orientation preserving circle homeomorphism. If the rotation number () q p = ϕ is rational, then () z n η is asymptotically periodic with semi-period q. This WACŁAW MARZANTOWICZ and JUSTYNA SIGNERSKA 12 convergence to a periodic sequence is uniform in z if we admit that some points are iterated backward instead of taking only forward iterations for all z. This leads to the notion of an basins'-ε edge, which we illustrate by the numerical example. If () , Q ∈ / ϕ then some classical results in topological dynamics yield that the displacement sequence also exhibits some regularity properties, which we define and prove in the second part of the paper.

On the Problem of Classification of Polynomial Endomorphisms of the Plane

Mathematics of the USSR-Sbornik, 1969

The paper is a continuation of the author's paper [5] (Math. USSR Sbornik 6 (1968), 97-114). §1 concerns the iterations of a polynomial P(z) of degree d > 1 on a singular set J. It is assumed that the critical points of Ρ (ζ) lie either in the domains of attraction of finite attracting cycles or at infinity. The theorems of [5] (Theorem 1 concerning the topological isomorphism of the transformation P(z)/j and of a shift on the space of one-sided d-ary sequences with a finite number of identifications; Theorem 2-P/J ~ Ρ /3) are generalized for the case of a disconnected J. In § 2 the author investigates the iterations of P(z) on the entire plane 77. He shows (Theorem 3) that the dynamical systems Ρ/π and Ρ /π are topologically isomorphic for sufficiently small |f in the case of polynomials satisfying one of the hypotheses of * § 1 and a certain "coarse" condition of "nonconjugacy" of the iterations of distinct critical points. Hypothesis: the set of structurally stable mappings ζ-> P{z) investigated in the paper is everywhere dense in the space of coefficients. Nine figures; bibliography of eight titles. We note that the notion of a structurally stable transformation subsumes the notion of the smallness (closeness to identity) of the homeomorphism φ: J-> J establishing conjugacy.

Rotation sets for orbits of degree one circle maps

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 2002

Let F be the lifting of a circle map of degree one. In [Bamón et al., 1984] a notion of F-rotation interval of a point was given. In this paper we define and study a new notion of a rotation set of point which preserves more of the dynamical information contained in the sequences than the one preserved from [Bamón et al., 1984]. In particular, we characterize dynamically the endpoints of these sets and we obtain an analogous version of the Main Theorem of [Bamón et al., 1984] in our settings.

Rotation numbers of discontinuous orientation-preserving circle maps revisited

The theory of circle homeomorphisms has a great number of deep results. However, sometimes continuity or single-valuedness of a circle map may be restrictive in theoretical constructions or applications. In this paper it is shown that some principal properties of circle homeomorphisms are inherited by the class of orientation-preserving circle maps. The latter class is rather broad and contains not only circle homeomorphisms but also a variety of non continuous maps arising in applications. Of course, even in cases when a property remains to be valid for orientation-preserving circle maps, absence of continuity sometimes results in noticeable changes of related proofs.

On some rational piecewise linear rotations

arXiv (Cornell University), 2023

We study the dynamics of the piecewise planar rotations F λ pzq " λpz´Hpzqq, with z P C, Hpzq " 1 if Impzq ě 0, Hpzq "´1 if Impzq ă 0, and λ " e iα P C, being α a rational multiple of π. Our main results establish the dynamics in the so called regular set, which is the complementary of the closure of the set formed by the preimages of the discontinuity line. We prove that any connected component of this set is open, bounded and periodic under the action of F λ , with a period ℓ, that depends on the connected component. Furthermore, F ℓ λ restricted to each component acts as a rotation with a period which also depends on the connected component. As a consequence, any point in the regular set is periodic. Among other results, we also prove that for any connected component of the regular set, its boundary is a convex polygon with certain maximum number of sides.

Characterization of rotation words generated by homeomorphisms on a circle

Discrete Applied Mathematics, 2015

In this note we study the words generated by orientation preserving homeomorphisms on a circle. We show that the word is eventually periodic if the rotation number is rational, otherwise either the word is identically 0 except for one place or it is a linear rotation word. We also discuss the unique generation of irrational linear rotation words.