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.

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 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.

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.

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.

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 Dynamical behavior of alternate base expansions

2021 ◽  
pp. 1-34
Author(s):  
ÉMILIE CHARLIER ◽  
CÉLIA CISTERNINO ◽  
KARMA DAJANI
Keyword(s):  
Dynamical System ◽  
Invariant Measure ◽  
Lebesgue Measure ◽  
Dynamical Behavior ◽  
Base Case ◽  
Real Numbers ◽  
Absolutely Continuous ◽  
Dynamical Behaviors ◽  
Unique Measure ◽  
Dynamical Properties

Abstract We generalize the greedy and lazy $\beta $ -transformations for a real base $\beta $ to the setting of alternate bases ${\boldsymbol {\beta }}=(\beta _0,\ldots ,\beta _{p-1})$ , which were recently introduced by the first and second authors as a particular case of Cantor bases. As in the real base case, these new transformations, denoted $T_{{\boldsymbol {\beta }}}$ and $L_{{\boldsymbol {\beta }}}$ respectively, can be iterated in order to generate the digits of the greedy and lazy ${\boldsymbol {\beta }}$ -expansions of real numbers. The aim of this paper is to describe the measure-theoretical dynamical behaviors of $T_{{\boldsymbol {\beta }}}$ and $L_{{\boldsymbol {\beta }}}$ . We first prove the existence of a unique absolutely continuous (with respect to an extended Lebesgue measure, called the p-Lebesgue measure) $T_{{\boldsymbol {\beta }}}$ -invariant measure. We then show that this unique measure is in fact equivalent to the p-Lebesgue measure and that the corresponding dynamical system is ergodic and has entropy $({1}/{p})\log (\beta _{p-1}\cdots \beta _0)$ . We give an explicit expression of the density function of this invariant measure and compute the frequencies of letters in the greedy ${\boldsymbol {\beta }}$ -expansions. The dynamical properties of $L_{{\boldsymbol {\beta }}}$ are obtained by showing that the lazy dynamical system is isomorphic to the greedy one. We also provide an isomorphism with a suitable extension of the $\beta $ -shift. Finally, we show that the ${\boldsymbol {\beta }}$ -expansions can be seen as $(\beta _{p-1}\cdots \beta _0)$ -representations over general digit sets and we compare both frameworks.

Non-ergodicity for C1 expanding maps and g-measures

1996 ◽  
Vol 16 (3) ◽  
pp. 531-543 ◽  
Author(s):  
Anthony N. Quasf
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Probability Measures ◽  
Continuous Invariant Measure ◽  
Expanding Maps ◽  
Absolutely Continuous ◽  
Expanding Map ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

AbstractWe introduce a procedure for finding C1 Lebesgue measure-preserving maps of the circle isomorphic to one-sided shifts equipped with certain invariant probability measures. We use this to construct a C1 expanding map of the circle which preserves Lebesgue measure, but for which Lebesgue measure is non-ergodic (that is there is more than one absolutely continuous invariant measure). This is in contrast with results for C1+e maps. We also show that this example answers in the negative a question of Keane's on uniqueness of g-measures, which in turn is based on a question raised by an incomplete proof of Karlin's dating back to 1953.

scholarly journals Maximal absolutely continuous invariant measures for piecewise linear Markov transformations

1990 ◽  
Vol 10 (4) ◽  
pp. 645-656 ◽  
Author(s):  
W. Byers ◽  
P. Góra ◽  
A. Boyarsky
Keyword(s):  
Invariant Measure ◽  
Invariant Measures ◽  
Piecewise Linear ◽  
Maximal Entropy ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measures ◽  
Absolutely Continuous Invariant Measure ◽  
Necessary And Sufficient ◽  
Absolutely Continuous Invariant

AbstractLet be an irreducible 0–1 matrix such that the non-zero entries in each row are consecutive. Let be the class of piecewise linear Markov transformations τ on [0, 1] into [0, 1] induced by for which the absolutely continuous invariant measure has maximal entropy. The main result presents necessary and sufficient slope conditions on τ which guarantee that τ ∈ .

scholarly journals Möbius disjointness along ergodic sequences for uniquely ergodic actions

2018 ◽  
Vol 39 (10) ◽  
pp. 2793-2826 ◽  
Author(s):  
JOANNA KUŁAGA-PRZYMUS ◽  
MARIUSZ LEMAŃCZYK
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Metric Space ◽  
Irrational Rotation ◽  
Compact Metric Space ◽  
Irrational Rotations ◽  
Ergodic Flow ◽  
Weakly Mixing ◽  
Image Position ◽  
Uniquely Ergodic

We show that there is an irrational rotation $Tx=x+\unicode[STIX]{x1D6FC}$ on the circle $\mathbb{T}$ and a continuous $\unicode[STIX]{x1D711}:\mathbb{T}\rightarrow \mathbb{R}$ such that for each (continuous) uniquely ergodic flow ${\mathcal{S}}=(S_{t})_{t\in \mathbb{R}}$ acting on a compact metric space $Y$, the automorphism $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}$ acting on $(X\times Y,\unicode[STIX]{x1D707}\otimes \unicode[STIX]{x1D708})$ by the formula $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}(x,y)=(Tx,S_{\unicode[STIX]{x1D711}(x)}(y))$, where $\unicode[STIX]{x1D707}$ stands for the Lebesgue measure on $\mathbb{T}$ and $\unicode[STIX]{x1D708}$ denotes the unique ${\mathcal{S}}$-invariant measure, has the property of asymptotically orthogonal powers. This gives a class of relatively weakly mixing extensions of irrational rotations for which Sarnak’s conjecture on the Möbius disjointness holds for all uniquely ergodic models of $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}$. Moreover, we obtain a class of ‘random’ ergodic sequences $(c_{n})\subset \mathbb{Z}$ such that if $\boldsymbol{\unicode[STIX]{x1D707}}$ denotes the Möbius function, then $$\begin{eqnarray}\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n\leq N}g(S_{c_{n}}y)\boldsymbol{\unicode[STIX]{x1D707}}(n)=0\end{eqnarray}$$ for all (continuous) uniquely ergodic flows ${\mathcal{S}}$, all $g\in C(Y)$ and $y\in Y$.

scholarly journals On some symmetric multidimensional continued fraction algorithms

2017 ◽  
Vol 38 (5) ◽  
pp. 1601-1626 ◽  
Author(s):  
PIERRE ARNOUX ◽  
SÉBASTIEN LABBÉ
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Positive Measure ◽  
Continued Fraction ◽  
Natural Extension ◽  
Fractal Boundary ◽  
Absolutely Continuous ◽  
Invariant Domain ◽  
Multidimensional Continued Fraction ◽  
Unique Invariant Measure

We compute explicitly the density of the invariant measure for the reverse algorithm which is absolutely continuous with respect to Lebesgue measure, using a method proposed by Arnoux and Nogueira. We also apply the same method on the unsorted version of the Brun algorithm and Cassaigne algorithm. We illustrate some experimentations on the domain of the natural extension of those algorithms. For some other algorithms, which are known to have a unique invariant measure absolutely continuous with respect to Lebesgue measure, the invariant domain found by this method seems to have a fractal boundary, and it is unclear whether it is of positive measure.

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