Hausdorff dimension of invariant measure of circle diffeomorphisms with a break point

2017 ◽  
Vol 39 (5) ◽  
pp. 1331-1339
Author(s):  
KONSTANTIN KHANIN ◽  
SAŠA KOCIĆ
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Rotation Number ◽  
Break Point ◽  
Irrational Rotation ◽  
Rotation Numbers ◽  
Circle Diffeomorphism ◽  
Almost All ◽  
Circle Diffeomorphisms

We prove that, for almost all irrational $\unicode[STIX]{x1D70C}\in (0,1)$, the Hausdorff dimension of the invariant measure of a $C^{2+\unicode[STIX]{x1D6FC}}$-smooth $(\unicode[STIX]{x1D6FC}\in (0,1))$ circle diffeomorphism with a break of size $c\in \mathbb{R}_{+}\backslash \{1\}$, with rotation number $\unicode[STIX]{x1D70C}$, is zero. This result cannot be extended to all irrational rotation numbers.

scholarly journals Dimensional characteristics of invariant measures for circle diffeomorphisms

2009 ◽  
Vol 29 (6) ◽  
pp. 1979-1992 ◽  
Author(s):  
VICTORIA SADOVSKAYA
Keyword(s):  
Invariant Measure ◽  
Rotation Number ◽  
Invariant Measures ◽  
Liouville Number ◽  
Hausdorff Dimensions ◽  
Rotation Numbers ◽  
Circle Diffeomorphism ◽  
Unique Invariant Measure ◽  
Circle Diffeomorphisms ◽  
Box Dimensions

AbstractWe consider pointwise, box, and Hausdorff dimensions of invariant measures for circle diffeomorphisms. We discuss the cases of rational, Diophantine, and Liouville rotation numbers. Our main result is that for any Liouville number τ there exists a C∞ circle diffeomorphism with rotation number τ such that the pointwise and box dimensions of its unique invariant measure do not exist. Moreover, the lower pointwise and lower box dimensions can equal any value 0≤β≤1.

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.

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.

Spectral Theory of Schrödinger Operators over Circle Diffeomorphisms

2021 ◽  
Author(s):  
Svetlana Jitomirskaya ◽  
Saša Kocić
Keyword(s):  
Type Theorem ◽  
Schrödinger Operators ◽  
Dynamical Analysis ◽  
Schrodinger Operators ◽  
Hölder Continuous ◽  
Rotation Numbers ◽  
Continuous Potential ◽  
Circle Diffeomorphism ◽  
Almost All ◽  
Circle Diffeomorphisms

Abstract We initiate the study of Schrödinger operators with ergodic potentials defined over circle map dynamics, in particular over circle diffeomorphisms. For analytic circle diffeomorphisms and a set of rotation numbers satisfying Yoccoz’s ${{\mathcal{H}}}$ arithmetic condition, we discuss an extension of Avila’s global theory. We also give an abstract version and a short proof of a sharp Gordon-type theorem on the absence of eigenvalues for general potentials with repetitions. Coupled with the dynamical analysis, we obtain that, for every $C^{1+BV}$ circle diffeomorphism, with a super Liouville rotation number and an invariant measure $\mu $, and for $\mu $-almost all $x\in{{\mathbb{T}}}^1$, the corresponding Schrödinger operator has purely continuous spectrum for every Hölder continuous potential $V$.

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 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 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.

Sign in / Sign up

Export Citation Format

Share Document