scholarly journals On conjugations of circle homeomorphisms with two break points

2012 ◽  
Vol 34 (3) ◽  
pp. 725-741 ◽  
Author(s):  
HABIBULLA AKHADKULOV ◽  
AKHTAM DZHALILOV ◽  
DIETER MAYER
Keyword(s):  
Lebesgue Measure ◽  
Rotation Number ◽  
Invariant Measures ◽  
Irrational Rotation ◽  
Singular Function ◽  
Almost Everywhere ◽  
Break Points ◽  
Irrational Rotation Number

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.

scholarly journals Non-rigidity for circle homeomorphisms with several break points

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 ◽  
Irrational Rotation Number

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.

Conjugation between circle maps with several break points

2015 ◽  
Vol 36 (8) ◽  
pp. 2351-2383 ◽  
Author(s):  
ABDELHAMID ADOUANI
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Break Point ◽  
Irrational Rotation ◽  
Singular Function ◽  
Absolutely Continuous ◽  
Circle Maps ◽  
Almost Everywhere ◽  
Rotation Numbers ◽  
Break Points

Let$f$and$g$be two class$P$-homeomorphisms of the circle$S^{1}$with break point singularities, which are differentiable maps except at some singular points where the derivative has a jump. Assume that$f$and$g$have irrational rotation numbers and the derivatives$\text{Df}$and$\text{Dg}$are absolutely continuous on every continuity interval of$\text{Df}$and$\text{Dg}$, respectively. We prove that if the product of the$f$-jumps along all break points of$f$is distinct from that of$g$then the homeomorphism$h$conjugating$f$and$g$is a singular function, i.e. it is continuous on$S^{1}$, but$\text{Dh}(x)=0$ almost everywhere with respect to the Lebesgue measure. This result generalizes previous results for one and two break points obtained by Dzhalilov, Akin and Temir, and Akhadkulov, Dzhalilov and Mayer. As a consequence, we get in particular Dzhalilov–Mayer–Safarov’s theorem: if the product of the$f$-jumps along all break points of$f$is distinct from$1$, then the invariant measure$\unicode[STIX]{x1D707}_{f}$is singular with respect to the Lebesgue measure.

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.

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

Singular measures for classP-circle homeomorphisms with several break points

2012 ◽  
Vol 34 (2) ◽  
pp. 423-456 ◽  
Author(s):  
ABDELHAMID ADOUANI ◽  
HABIB MARZOUGUI
Keyword(s):  
Invariant Measure ◽  
Rotation Number ◽  
Haar Measure ◽  
Break Point ◽  
Irrational Rotation ◽  
Singular Points ◽  
Absolutely Continuous ◽  
Singular Measures ◽  
Break Points ◽  
Irrational Rotation Number

AbstractLetfbe a classP-homeomorphism of the circle with break point singularities, that is, differentiable except at some singular points where the derivative has a jump. Letfhave irrational rotation number andDfbe absolutely continuous on every continuity interval ofDf. We prove that if the product of thef-jumps along any subset of break points is distinct from 1 then the invariant measureμfis singular with respect to the Haar measure. This result generalizes previous results obtained by Dzhalilov and Khanin, Dzhalilov, Akhadkulov, Dzhalilov–Liousse and Mayer. Moreover, we prove that if the rotation numberρ(f) is irrational of bounded type then (a) if the product of thef-jumps on some orbit is distinct from 1 then the invariant measureμfis singular with respect to the Haar measurem, and (b) if the product of thef-jumps on each orbit is equal to 1 andD2f∈Lp(S1) for somep>1 thenμfis equivalent to the Haar measure.

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

2014 ◽  
Vol 24 (01) ◽  
pp. 1450012 ◽  
Author(s):  
Ya-Nan Wang ◽  
Wen-Xin Qin
Keyword(s):  
Rotation Number ◽  
Recurrence Relations ◽  
Irrational Rotation ◽  
Necessary Condition ◽  
Irrational Rotation Number

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.

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

1985 ◽  
Vol 5 (1) ◽  
pp. 27-46 ◽  
Author(s):  
Colin Boyd
Keyword(s):  
Vector Field ◽  
Hausdorff Dimension ◽  
Rotation Number ◽  
Vector Fields ◽  
Cantor Set ◽  
Irrational Rotation ◽  
Codimension One ◽  
The Family ◽  
Irrational Rotation Number

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.

scholarly journals Attractors with irrational rotation number

2011 ◽  
Vol 153 (1) ◽  
pp. 59-77 ◽  
Author(s):  
LUIS HERNÁNDEZ-CORBATO ◽  
RAFAEL ORTEGA ◽  
FRANCISCO R. RUIZ DEL PORTAL
Keyword(s):  
Fixed Point ◽  
Rotation Number ◽  
Irrational Rotation ◽  
Asymptotically Stable ◽  
Stable Fixed Point ◽  
Region Of Attraction ◽  
Prime Ends ◽  
Irrational Rotation Number

AbstractLet h: 2 → 2 be a dissipative and orientation preserving homeomorphism having an asymptotically stable fixed point. Let U be the region of attraction and assume that it is proper and unbounded. Using Carathéodory's prime ends theory one can associate a rotation number, ρ, to h|U. We prove that any map in the above conditions and with ρ ∉ induces a Denjoy homeomorphism in the circle of prime ends. We also present some explicit examples of maps in this class and we show that, if the infinity point is accessible by an arc in U, ρ ∉ if and only if Per(h) = Fix(h) = {p}.

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 On the invariant distributions of $$C^2$$ circle diffeomorphisms of irrational rotation number

2012 ◽  
Vol 274 (1-2) ◽  
pp. 315-321 ◽  
Author(s):  
Andrés Navas ◽  
Michele Triestino
Keyword(s):  
Rotation Number ◽  
Irrational Rotation ◽  
Invariant Distributions ◽  
Irrational Rotation Number ◽  
Circle Diffeomorphisms
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