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.

Mixing properties of (α,β)-expansions

2009 ◽  
Vol 29 (4) ◽  
pp. 1119-1140 ◽  
Author(s):  
KARMA DAJANI ◽  
YUSUF HARTONO ◽  
COR KRAAIKAMP
Keyword(s):  
Dynamical System ◽  
Invariant Measure ◽  
Lebesgue Measure ◽  
Natural Extension ◽  
Mixing Properties ◽  
Special Values ◽  
Image Position ◽  
Relationship Of ◽  
The Relationship

AbstractLet 0<α<1 andβ>1. We show that everyx∈[0,1] has an expansion of the formwherehi=hi(x)∈{0,α/β}, andpi=pi(x)∈{0,1}. We study the dynamical system underlying this expansion and give the density of the invariant measure that is equivalent to the Lebesgue measure. We prove that the system is weakly Bernoulli, and we give a version of the natural extension. For special values ofα, we give the relationship of this expansion with the greedyβ-expansion.

$\beta$-transformation, natural extension and invariant measure

2000 ◽  
Vol 20 (5) ◽  
pp. 1271-1285 ◽  
Author(s):  
GAVIN BROWN ◽  
QINGHE YIN
Keyword(s):  
Invariant Measure ◽  
Lebesgue Measure ◽  
Natural Extension ◽  
Connected Region ◽  
Two Dimensional ◽  
Constant Multiple ◽  
Unit Square ◽  
Simply Connected Region ◽  
Simply Connected ◽  
Dimensional Lebesgue Measure

For $\beta>1$, consider the $\beta$-transformation $T_\beta$. When $\beta$ is an integer, the natural extension of $T_\beta$ can be represented explicitly as a map on the unit square with an invariant measure: the corresponding two-dimensional Lebesgue measure. We show that, under certain conditions on $\beta$, the natural extension is defined on a simply connected region and an invariant measure is a constant multiple of the Lebesgue measure.We characterize those $\beta$ in terms of the $\beta$-expansion of one, and study the structure and size of the set of all such $\beta$.

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 Applications of (a,b)-continued fraction transformations

2011 ◽  
Vol 32 (2) ◽  
pp. 739-761 ◽  
Author(s):  
SVETLANA KATOK ◽  
ILIE UGARCOVICI
Keyword(s):  
Continued Fraction ◽  
Natural Extension ◽  
Absolutely Continuous ◽  
Coding Sequences ◽  
One Dimensional ◽  
Sofic Shift ◽  
Absolutely Continuous Invariant Measure ◽  
Special Cases ◽  
The One ◽  
Absolutely Continuous Invariant

AbstractWe describe a general method of arithmetic coding of geodesics on the modular surface based on the study of one-dimensional Gauss-like maps associated to a two-parameter family of continued fractions introduced in [Katok and Ugarcovici. Structure of attractors for (a,b)-continued fraction transformations.J. Modern Dynamics4(2010), 637–691]. The finite rectangular structure of the attractors of the natural extension maps and the corresponding ‘reduction theory’ play an essential role. In special cases, when an (a,b)-expansion admits a so-called ‘dual’, the coding sequences are obtained by juxtaposition of the boundary expansions of the fixed points, and the set of coding sequences is a countable sofic shift. We also prove that the natural extension maps are Bernoulli shifts and compute the density of the absolutely continuous invariant measure and the measure-theoretic entropy of the one-dimensional map.

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 Another proof of Jakobson's Theorem and related results

1988 ◽  
Vol 8 (1) ◽  
pp. 93-109 ◽  
Author(s):  
Marek Ryszard Rychlik
Keyword(s):  
Invariant Measure ◽  
Positive Measure ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
Absolutely Continuous Invariant Measure ◽  
Absolutely Continuous Invariant

AbstractThe author shows that any family C2-close to fα(x) = 1 − αx2(2 − ε ≤ α ≤ 2) satisfies Jakobson's theorem: For a positive measure set of α the transformation fα has an absolutely continuous invariant measure. He also indicates some generalizations.

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 Positive Liapunov exponents and absolute continuity for maps of the interval

1983 ◽  
Vol 3 (1) ◽  
pp. 13-46 ◽  
Author(s):  
P. Collet ◽  
J.-P. Eckmann
Keyword(s):  
Invariant Measure ◽  
Critical Point ◽  
Lebesgue Measure ◽  
Absolute Continuity ◽  
Sufficient Condition ◽  
Liapunov Exponent ◽  
Absolutely Continuous ◽  
Liapunov Exponents ◽  
Unimodal Map ◽  
Weak Regularity

AbstractWe give a sufficient condition for a unimodal map of the interval to have an invariant measure absolutely continuous with respect to the Lebesgue measure. Apart from some weak regularity assumptions, the condition requires positivity of the forward and backward Liapunov exponent of the critical point.

Exactness of the Euclidean algorithm and of the Rauzy induction on the space of interval exchange transformations

2011 ◽  
Vol 33 (1) ◽  
pp. 221-246 ◽  
Author(s):  
TOMASZ MIERNOWSKI ◽  
ARNALDO NOGUEIRA
Keyword(s):  
Lebesgue Measure ◽  
Continued Fraction ◽  
Euclidean Algorithm ◽  
Interval Exchange ◽  
Two Dimensional ◽  
Interval Exchange Transformations ◽  
Multidimensional Continued Fraction ◽  
Rauzy Induction ◽  
Definition Of

AbstractThe two-dimensional homogeneous Euclidean algorithm is the central motivation for the definition of the classical multidimensional continued fraction algorithms, such as Jacobi–Perron, Poincaré, Brun and Selmer algorithms. The Rauzy induction, a generalization of the Euclidean algorithm, is a key tool in the study of interval exchange transformations. Both maps are known to be dissipative and ergodic with respect to Lebesgue measure. Here we prove that they are exact.

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