scholarly journals Quasi-symmetric conjugacy for circle maps with a flat interval

2017 ◽  
Vol 39 (2) ◽  
pp. 425-445 ◽  
Author(s):  
LIVIANA PALMISANO
Keyword(s):  
Rotation Number ◽  
Circle Maps ◽  
Bounded Geometry ◽  
Circle Homeomorphism ◽  
Dynamic Methods ◽  
Proof Techniques

In this paper we study quasi-symmetric conjugations of ${\mathcal{C}}^{2}$ weakly order-preserving circle maps with a flat interval. Under the assumption that the maps have the same rotation number of bounded type and that bounded geometry holds, we construct a quasi-symmetric conjugation between their non-wandering sets. Further, this conjugation is extended to a quasi-symmetric circle homeomorphism. Our proof techniques hinge on real-dynamic methods, allowing us to construct the conjugation under general and natural assumptions.

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 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 The thermodynamic formalism and exponents of singularity of invariant measure of circle maps with a single break

2020 ◽  
Vol 30 (3) ◽  
pp. 343-366
Author(s):  
A.A. Dzhalilov ◽  
J.J. Karimov
Keyword(s):  
Invariant Measure ◽  
Rotation Number ◽  
Continued Fraction ◽  
Thermodynamic Formalism ◽  
Measurable Subset ◽  
Break Point ◽  
Golden Mean ◽  
Circle Maps ◽  
Circle Homeomorphism ◽  
Irrational Rotation Number

Let $T \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0 $, be a circle homeomorphism with one break point $x_{b}$, at which $ T'(x) $ has a discontinuity of the first kind and both one-sided derivatives at the point $x_{b} $ are strictly positive. Assume that the rotation number $\rho_{T}$ is irrational and its decomposition into a continued fraction beginning from a certain place coincides with the golden mean, i.e., $\rho_{T}=[m_{1}, m_{2}, \ldots, m_{l}, \, m_{l + 1}, \ldots] $, $ m_{s} = 1$, $s> l> 0$. Since the rotation number is irrational, the map $ T $ is strictly ergodic, that is, possesses a unique probability invariant measure $\mu_{T}$. A.A. Dzhalilov and K.M. Khanin proved that the probability invariant measure $ \mu_{G} $ of any circle homeomorphism $ G \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0$, with one break point $ x_{b} $ and the irrational rotation number $ \rho_{G} $ is singular with respect to the Lebesgue measure $ \lambda $ on the circle, i.e., there is a measurable subset of $ A \subset S^{1} $ such that $ \mu_ {G} (A) = 1 $ and $ \lambda (A) = 0$. We will construct a thermodynamic formalism for homeomorphisms $ T_{b} \in C^{2+ \varepsilon} (S^{1} \setminus \{x_{b} \})$, $\varepsilon> 0 $, with one break at the point $ x_{b} $ and rotation number equal to the golden mean, i.e., $ \rho_{T}:= \frac {\sqrt{5} -1}{2} $. Using the constructed thermodynamic formalism, we study the exponents of singularity of the invariant measure $ \mu_{T} $ of homeomorphism $ T $.

OBSERVED ROTATION NUMBERS IN FAMILIES OF CIRCLE MAPS

2001 ◽  
Vol 11 (01) ◽  
pp. 73-89 ◽  
Author(s):  
MICHAEL A. SAUM ◽  
TODD R. YOUNG
Keyword(s):  
Lebesgue Measure ◽  
Rotation Number ◽  
Narrow Band ◽  
Initial Conditions ◽  
Positive Probability ◽  
Circle Maps ◽  
Numerical Evidence ◽  
Rotation Interval ◽  
Large Numbers ◽  
Rotation Numbers

Noninvertible circle maps may have a rotation interval instead of a unique rotation number. One may ask which of the numbers or sets of numbers within this rotation interval may be observed with positive probability in term of Lebesgue measure on the circle. We study this question numerically for families of circle maps. Both the interval and "observed" rotation numbers are computed for large numbers of initial conditions. The numerical evidence suggests that within the rotation interval only a very narrow band or even a unique rotation number is observed. These observed rotation numbers appear to be either locally constant or vary wildly as the parameter is changed. Closer examination reveals that intervals with wild variation contain many subintervals where the observed rotation numbers are locally constant. We discuss the formation of these intervals. We prove that such intervals occur whenever one of the endpoints of the rotation interval changes. We also examine the effects of various types of saddle-node bifurcations on the observed rotation numbers.

Complex bounds for renormalization of critical circle maps

1999 ◽  
Vol 19 (1) ◽  
pp. 227-257 ◽  
Author(s):  
MICHAEL YAMPOLSKY
Keyword(s):  
Rotation Number ◽  
A Priori ◽  
Julia Sets ◽  
Irrational Rotation ◽  
Local Connectivity ◽  
Circle Maps ◽  
Critical Circle ◽  
Irrational Rotation Number

We use the methods developed with Lyubich for proving complex bounds for real quadratics to extend de Faria's complex a priori bounds to all critical circle maps with an irrational rotation number. The contracting property for renormalizations of critical circle maps follows.As another application of our methods we present a new proof of a theorem of Petersen on local connectivity of some Siegel Julia sets.

scholarly journals Cherry Maps with Different Critical Exponents: Bifurcation of Geometry

2020 ◽  
Vol 16 (4) ◽  
pp. 651-672
Author(s):  
B. Ndawa Tangue ◽  
Keyword(s):  
Hausdorff Dimension ◽  
Rotation Number ◽  
Critical Exponents ◽  
Irrational Rotation ◽  
Circle Maps ◽  
Irrational Rotation Number ◽  
Nonwandering Set ◽  
Flat Piece

We consider order-preserving $C^3$ circle maps with a flat piece, irrational rotation number and critical exponents $(l_1, l_2)$. We detect a change in the geometry of the system. For $(l_1, l_2) \in [1, 2]^2$ the geometry is degenerate and becomes bounded for $(l_1, l_2) \in [2, \infty)^2 \backslash \{(2, 2)\}$. When the rotation number is of the form $[abab \ldots]$; for some $a, b \in \mathbb{N}^*$, the geometry is bounded for $(l_1, l_2)$ belonging above a curve defined on $]1, +\infty[^2$. As a consequence, we estimate the Hausdorff dimension of the nonwandering set $K_f=\mathcal{S}^1\backslash \bigcup^\infty_{i=0}f^{-i}(U)$. Precisely, the Hausdorff dimension of this set is equal to zero when the geometry is degenerate and it is strictly positive when the geometry is bounded.

scholarly journals The behavior of renormalizations of circle maps with rational rotation numbers and with Zygmund conditions

2020 ◽  
Vol 16 (4) ◽  
pp. 444-449
Author(s):  
Habibulla Akhadkulov ◽  
Abdumajid Begmatov ◽  
Teh Yuan Ying
Keyword(s):  
Rotation Number ◽  
Break Point ◽  
Parameter Family ◽  
Jump Discontinuity ◽  
Linear Transformations ◽  
Circle Maps ◽  
Large Rank ◽  
Rotation Numbers

Let  be one-parameter family of circle homeomorphisms with a break point, that is, the derivative  has jump discontinuity at this point. Suppose  satisfies a certain Zygmund condition which is dependent on parameter . We prove that the renormalizations of circle homeomorphisms from this family with rational rotation number of sufficiently large rank are approximated by piecewise fractional linear transformations in  and -norms, depending on the values of the parameter   and , respectively.

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 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 Some remarks on the geometry of circle maps with a break point

Filomat ◽  
2018 ◽  
Vol 32 (16) ◽  
pp. 5549-5563
Author(s):  
Necip Simsek ◽  
Akhtam Dzhalilov ◽  
Emilio Musso
Keyword(s):  
Asymptotic Behaviour ◽  
Rotation Number ◽  
Continued Fraction ◽  
Break Point ◽  
Jump Discontinuity ◽  
Circle Maps ◽  
Renormalization Transformation ◽  
Study Circle ◽  
Single Break

We study circle homeomorphisms f ? C2(S1\{xb}) whose rotation number ?f is irrational, with a single break point xb at which f' has a jump discontinuity. We prove that the behavior of the ratios of the lengths of any two adjacent intervals of the dynamical partition depends on the size of break and on the continued fraction decomposition of ?f. We also prove a result analogous to Yoccoz?s lemma on the asymptotic behaviour of the lengths of the intervals of trajectories of the renormalization transformation Rn(f).

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