A Note on the Conjugacy Between Two Critical Circle Maps

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

Complex bounds for renormalization of critical circle maps

10.1017/s0143385799120947 ◽

1999 ◽

Vol 19 (1) ◽

pp. 227-257 ◽

Cited By ~ 12

Author(s):

MICHAEL YAMPOLSKY

Keyword(s):

Rotation Number ◽

A Priori ◽

Julia Sets ◽

Irrational Rotation ◽

Local Connectivity ◽

Circle Maps ◽

Critical Circle ◽

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.

Quasisymmetric orbit-flexibility of multicritical circle maps

10.1017/etds.2021.104 ◽

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).

Cherry Maps with Different Critical Exponents: Bifurcation of Geometry

Nelineinaya Dinamika ◽

10.20537/nd200409 ◽

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.

Limit drift

10.1017/etds.2016.10 ◽

2016 ◽

Vol 37 (8) ◽

pp. 2643-2670 ◽

Cited By ~ 2

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.

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

Vestnik Udmurtskogo Universiteta Matematika Mekhanika Komp yuternye Nauki ◽

10.35634/vm200301 ◽

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 ◽

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 $.

Non-rigidity for circle homeomorphisms with several break points

10.1017/etds.2017.121 ◽

2017 ◽

Vol 39 (9) ◽

pp. 2305-2331

Author(s):

ABDELHAMID ADOUANI ◽

HABIB MARZOUGUI

Keyword(s):

Lebesgue Measure ◽

Rotation Number ◽

Break Point ◽

Irrational Rotation ◽

Singular Points ◽

Singular Function ◽

Absolutely Continuous ◽

Almost Everywhere ◽

Break Points ◽

In this work, we consider two class $P$-homeomorphisms, $f$ and $g$, of the circle with break point singularities, that are differentiable maps except at some singular points where the derivative has a jump. Assume that they have the same irrational rotation number of bounded type and that the derivatives $\text{Df}$ and $\text{Dg}$ are absolutely continuous on every continuity interval of $\text{Df}$ and $\text{Dg}$, respectively. We show that if $f$ and $g$ are not break-equivalent, then any topological conjugating $h$ between $f$ and $g$ is a singular function, i.e., it is continuous on the circle, but $\text{Dh}(x)=0$ almost everywhere (a.e.) with respect to the Lebesgue measure. In particular, this result holds under some combinatorial assumptions on the jumps at break points. It also generalizes previous results obtained for one and two break points and complements that of Cunha–Smania which was established for break equivalence.

A simple proof of Denjoy's theorem

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100064860 ◽

1988 ◽

Vol 103 (2) ◽

pp. 299-303 ◽

Cited By ~ 11

Author(s):

R. S. MacKay

Keyword(s):

Rotation Number ◽

Simple Proof ◽

Irrational Rotation ◽

Topological Conjugacy ◽

Uniform Rotation ◽

Circle Maps ◽

AbstractA simple proof of Denjoy's theorem on topological conjugacy of circle maps with irrational rotation number to uniform rotation is presented.

On conjugations of circle homeomorphisms with two break points

10.1017/etds.2012.159 ◽

2012 ◽

Vol 34 (3) ◽

pp. 725-741 ◽

Cited By ~ 7

Author(s):

HABIBULLA AKHADKULOV ◽

AKHTAM DZHALILOV ◽

DIETER MAYER

Keyword(s):

Lebesgue Measure ◽

Rotation Number ◽

Invariant Measures ◽

Irrational Rotation ◽

Singular Function ◽

Almost Everywhere ◽

Break Points ◽

AbstractLetfi∈C2+α(S1∖{ai,bi}),α>0,i=1,2, be circle homeomorphisms with two break pointsai,bi, that is, discontinuities in the derivativeDfi, with identical irrational rotation numberρandμ1([a1,b1])=μ2([a2,b2]), whereμiare the invariant measures offi,i=1,2. Suppose that the products of the jump ratios ofDf1andDf2do not coincide, that is,Df1(a1−0)/Df1(a1+0)⋅Df1(b1−0)/Df1(b1+0)≠Df2(a2−0)/Df2(a2+0)⋅Df2(b2−0)/Df2(b2+0) . Then the mapψconjugatingf1andf2is a singular function, that is, it is continuous onS1, butDψ(x)=0 almost everywhere with respect to Lebesgue measure.

A Necessary Condition of Nonminimal Aubry–Mather Sets for Monotone Recurrence Relations

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414500126 ◽

2014 ◽

Vol 24 (01) ◽

pp. 1450012 ◽

Cited By ~ 1

Author(s):

Ya-Nan Wang ◽

Wen-Xin Qin

Keyword(s):

Rotation Number ◽

Recurrence Relations ◽

Irrational Rotation ◽

Necessary Condition ◽

In this paper, we show that a necessary condition for nonminimal Aubry–Mather sets of monotone recurrence relations is that the set of all Birkhoff minimizers with some irrational rotation number does not constitute a foliation, i.e. the gaps of the minimal Aubry–Mather set are not filled up with Birkhoff minimizers.

On the structure of the family of Cherry fields on the torus

10.1017/s014338570000273x ◽

1985 ◽

Vol 5 (1) ◽

pp. 27-46 ◽

Cited By ~ 21

Author(s):

Colin Boyd

Keyword(s):

Vector Field ◽

Hausdorff Dimension ◽

Rotation Number ◽

Vector Fields ◽

Cantor Set ◽

Irrational Rotation ◽

Codimension One ◽

The Family ◽

AbstractA class of vector fields on the 2-torus, which includes Cherry fields, is studied. Natural paths through this class are defined and it is shown that the parameters for which the vector field is unstable is the closure ofhas irrational rotation number}, where ƒ is a certain map of the circle andRtis rotation throught. This is shown to be a Cantor set of zero Hausdorff dimension. The Cherry fields are shown to form a family of codimension one submanifolds of the set of vector fields. The natural paths are shown to be stable paths.