scholarly journals On Critical Circle Homeomorphisms with Infinite Number of Break Points

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Akhtam Dzhalilov ◽  
Mohd Salmi Md Noorani ◽  
Sokhobiddin Akhatkulov
Keyword(s):  
Unit Circle ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Infinite Number ◽  
Circle Homeomorphism ◽  
Critical Circle ◽  
Break Points

We prove that a critical circle homeomorphism with infinite number of break points without periodic orbits is conjugated to the linear rotation by a quasisymmetric map if and only if its rotation number is of bounded type. And we also prove that any two adjacent atoms of dynamical partition of a unit circle are comparable.

scholarly journals Quasisymmetric orbit-flexibility of multicritical circle maps

2021 ◽  
pp. 1-40
Author(s):  
EDSON DE FARIA ◽  
PABLO GUARINO
Keyword(s):  
Rotation Number ◽  
A Priori ◽  
Gauss Map ◽  
Circle Map ◽  
A Priori Bounds ◽  
Circle Maps ◽  
Circle Homeomorphism ◽  
Critical Circle ◽  
Full Lebesgue Measure ◽  
Topological Conjugacies

Abstract Two given orbits of a minimal circle homeomorphism f are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with f. By a well-known theorem due to Herman and Yoccoz, if f is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. It follows from the a priori bounds of Herman and Świątek, that the same holds if f is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if f is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in $(0,1)$ , then the number of equivalence classes is uncountable (Theorem 1.1). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. As a by-product of our techniques, we construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and we show that this phenomenon is abundant, both from the topological and measure-theoretical viewpoints (Theorems 1.6 and 1.8).

scholarly journals Limit drift

2016 ◽  
Vol 37 (8) ◽  
pp. 2643-2670 ◽  
Author(s):  
GENADI LEVIN ◽  
GRZEGORZ ŚWIA̧TEK
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Critical Point ◽  
Rotation Number ◽  
Sharp Contrast ◽  
Contour Integral ◽  
Unimodal Maps ◽  
Golden Mean ◽  
Circle Homeomorphism ◽  
Critical Circle

We study the problem of the existence of wild attractors for critical circle coverings with Fibonacci dynamics. This is known to be related to the drift for the corresponding fixed points of renormalization. The fixed point depends only on the order of the critical point$\ell$and its drift is a number$\unicode[STIX]{x1D717}(\ell )$which is finite for each finite$\ell$. We show that the limit$\unicode[STIX]{x1D717}(\infty ):=\lim _{\ell \rightarrow \infty }\unicode[STIX]{x1D717}(\ell )$exists and is finite. The finiteness of the limit is in a sharp contrast with the case of Fibonacci unimodal maps. Furthermore,$\unicode[STIX]{x1D717}(\infty )$is expressed as a contour integral in terms of the limit of the fixed points of renormalization when$\ell \rightarrow \infty$. There is a certain paradox here, since this dynamical limit is a circle homeomorphism with the golden mean rotation number whose own drift is$\infty$for topological reasons.

scholarly journals A Note on the Conjugacy Between Two Critical Circle Maps

2021 ◽  
pp. 287-300
Author(s):  
Utkir A. Safarov
Keyword(s):  
Critical Point ◽  
Rotation Number ◽  
Irrational Rotation ◽  
Singular Function ◽  
Circle Maps ◽  
Circle Homeomorphism ◽  
Critical Circle ◽  
Irrational Rotation Number

We study a conjugacy between two critical circle homeomorphisms with irrational rotation number. Let fi, i = 1, 2 be a C3 circle homeomorphisms with critical point x(i) cr of the order 2mi + 1. We prove that if 2m1 + 1 ̸= 2m2 + 1, then conjugating between f1 and f2 is a singular function. Keywords: circle homeomorphism, critical point, conjugating map, rotation number, singular function

scholarly journals Numerical detection of unstable periodic orbits in continuous-time dynamical systems with chaotic behaviors

2007 ◽  
Vol 14 (5) ◽  
pp. 615-620 ◽  
Author(s):  
Y. Saiki
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Infinite Number ◽  
Continuous Time ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Complex Phenomenon ◽  
Unstable Periodic Orbits ◽  
Newton Raphson ◽  
The Lorenz System

Abstract. An infinite number of unstable periodic orbits (UPOs) are embedded in a chaotic system which models some complex phenomenon. Several algorithms which extract UPOs numerically from continuous-time chaotic systems have been proposed. In this article the damped Newton-Raphson-Mees algorithm is reviewed, and some important techniques and remarks concerning the practical numerical computations are exemplified by employing the Lorenz system.

scholarly journals Twist Periodic Orbits for Continuous Maps of the Eight Space

2013 ◽  
Vol 6 ◽  
pp. 4-8
Author(s):  
Iftichar Mudhar Talb Al-Shraa
Keyword(s):  
Fixed Point ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Continuous Map ◽  
Continuous Maps

Let g be a continuous map from 8 to itself has a fixed point at (0,0), we prove that g has a twist periodic orbit if there is a rational rotation number.

Knotted periodic orbits in suspensions of annulus maps

1987 ◽  
Vol 411 (1841) ◽  
pp. 351-378 ◽  
Keyword(s):  
Periodic Orbits ◽  
Rotation Number ◽  
Existence And Uniqueness ◽  
Torus Knots ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Rotation Numbers ◽  
Higher Genus

We consider a class of suspensions of diffeomorphisms of the annulus as flows in the orientable 3-manifold T 2 x I. Using results of Birman & Williams ( Topology 22, 47‒82 (1983); Contemp. Math . 20, 1‒60 (1983)), we construct a knotholder or template that carries the set of periodic orbits of the flow. We define rotation numbers and show that any orbit of period q and rotation number p / q can be arranged as a positive braid on p strands. This yields existence and uniqueness results for families of resonant torus knots ( p -braids that are ( p , q )-torus knots of period q > p which correspond to order-preserving (Birkhoff-) periodic orbits of the diffeomorphism. We show that all other q -periodic p -braids have higher genus, and we establish bounds on the genera of such knots. We obtain existence and uniqueness results for a number of other, non-resonant, torus knots, including non-order-preserving ( q + s , q )-torus knots of rotation number 1.

scholarly journals Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point

1985 ◽  
Vol 5 (4) ◽  
pp. 501-517 ◽  
Author(s):  
Lluís Alsedà ◽  
Jaume Llibre ◽  
Michał Misiurewicz ◽  
Carles Simó
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Periodic Orbit ◽  
Topological Entropy ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Rotation Interval ◽  
Continuous Map ◽  
Continuous Maps

AbstractLet f be a continuous map from the circle into itself of degree one, having a periodic orbit of rotation number p/q ≠ 0. If (p, q) = 1 then we prove that f has a twist periodic orbit of period q and rotation number p/q (i.e. a periodic orbit which behaves as a rotation of the circle with angle 2πp/q). Also, for this map we give the best lower bound of the topological entropy as a function of the rotation interval if one of the endpoints of the interval is an integer.

Topological conjugacy of 𝑛-multiple Cartesian products of circle rough transformations

2021 ◽  
Vol 29 (6) ◽  
pp. 851-862
Author(s):  
Iuliana Golikova ◽  
◽  
Svetlana Zinina ◽  
◽  
Keyword(s):  
Conjugacy Class ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Topological Invariant ◽  
Topological Invariants ◽  
Topological Conjugacy ◽  
Cartesian Products ◽  
Torus Surface

It is known from the 1939 work of A. G. Mayer that rough transformations of the circle are limited to the diffeomorphisms of Morse – Smale. A topological conjugacy class of orientation-preserving diffeomorphism is entirely determined by its rotation number and the number of its periodic orbits, while for orientation-changing diffeomorphism the topological invariant will be only the number of periodic orbits. Thus, the purpose of this study is to find topological invariants of n-fold Cartesian products of diffeomorphisms of a circle. Methods. This paper explores the rough Morse – Smale diffeomorphisms on the n-torus surface. To prove the main result, additional constructions and formation of subsets of considered sets were used. Results. In this paper, a numerical topological invariant is introduced for n-fold Cartesian products of rough circle transformations. Conclusion.The criterion of topological conjugacy of n-fold Cartesian products of rough transformations of a circle is formulated.

scholarly journals A circle diffeomorphism with breaks that is absolutely continuously linearizable

2016 ◽  
Vol 38 (1) ◽  
pp. 371-383 ◽  
Author(s):  
ALEXEY TEPLINSKY
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Piecewise Linear ◽  
Irrational Rotation ◽  
Rigid Rotation ◽  
Absolutely Continuous ◽  
Circle Homeomorphism ◽  
Break Points ◽  
Circle Diffeomorphism ◽  
Irrational Rotation Number

In this paper we answer positively to a question of whether it is possible for a circle diffeomorphism with breaks to be smoothly conjugate to a rigid rotation in the case where its breaks are lying on pairwise distinct trajectories. An example constructed is a piecewise linear circle homeomorphism that has four break points lying on distinct trajectories and whose invariant measure is absolutely continuous with respect to the Lebesgue measure. The irrational rotation number for our example can be chosen to be a Roth number, but not of bounded type.

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