Invariant densities for generalized<i>β</i>-maps

Góra, Paweł

doi:10.1017/s0143385707000053

Cited by 36 publications

(50 citation statements)

References 15 publications

(22 reference statements)

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“…Similar to Lemma 10 in [7], we get that h * β,E is non-negative. Since ψ β,E (1/β) = β · 1 0 h * β,E dm > 0, we get the conclusion.…”

Section: Where F (β E) Denotes the Normalizing Factor For The Invarisupporting

confidence: 60%

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Artin-Mazur Zeta Functions of Generalized Beta-Transformations

Suzuki¹

2017

Kyushu J. Math.

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Abstract. In this paper, we study the Artin-Mazur zeta function of a generalization of the well-known β-transformation introduced by Góra [Invariant densities for generalized β-maps. Ergodic Theory Dynam. Systems 27 (2007), [1583][1584][1585][1586][1587][1588][1589][1590][1591][1592][1593][1594][1595][1596][1597][1598]. We show that the ArtinMazur zeta function can be extended to a meromorphic function via an expansion of 1 defined by using the transformation. As an application, we relate its analytic properties to the algebraic properties of β.

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“…Similar to Lemma 10 in [7], we get that h * β,E is non-negative. Since ψ β,E (1/β) = β · 1 0 h * β,E dm > 0, we get the conclusion.…”

Section: Where F (β E) Denotes the Normalizing Factor For The Invarisupporting

confidence: 60%

“…In [7] Góra showed that every map τ β,E has a unique absolutely continuous invariant probability measure μ β,E and its density function h β,E can be expressed by using the coefficients of the τ β,E -expansion of 1. Precisely, the density function h β,E is expressed as n+1 .…”

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confidence: 99%

Artin-Mazur Zeta Functions of Generalized Beta-Transformations

Suzuki¹

2017

Kyushu J. Math.

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“…The curve α = 0 is a trivial example of such a curve for the fixed point x = 0. In section 4, we study the generalized β-transformations introduced by Góra [7]. A generalized β-transformation is similar to a β-transformation, but each lap is replaced by an increasing or decreasing lap of constant slope β according to a sequence of signs.…”

Section: B Faller and C-e Pfistermentioning

confidence: 99%

“…Introduced by Góra in [7], they have only one critical orbit like β-transformations, but they admit increasing and decreasing laps. A family {T β } β>1 of generalized β-transformations is defined by k ≥ 2 and a sequence s = (s n ) 0≤n<k with s i ∈ {−1, 1}.…”

Section: Normality In Generalized β-Transformationsmentioning

confidence: 99%

“…, k − 1. Góra constructed the unique measure µ β absolutely continuous with respect to Lebesgue measure (Theorem 6 and Proposition 8 in [7]). Using the same argument as Hofbauer in [9], we deduce that a measure of maximal entropy is always absolutely continuous with respect to Lebesgue measure, hence the measure µ β is the unique measure of maximal entropy.…”

Section: Normality In Generalized β-Transformationsmentioning

confidence: 99%

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A point is normal for almost all mapsβx+αmod 1 or generalizedβ-transformations

Faller¹,

Pfister²

2009

Ergod. Th. Dynam. Sys.

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Abstract. We consider the map T α,β (x) := βx + α mod 1, which admits a unique probability measure of maximal entropy µ α,β . For x ∈ [0, 1], we show that the orbit of x is µ α,β -normal for almost all (α, β) ∈ [0, 1) × (1, ∞) (Lebesgue measure). Nevertheless we construct analytic curves in [0, 1) × (1, ∞) along them the orbit of x = 0 is at most at one point µ α,β -normal. These curves are disjoint and they fill the set [0, 1) × (1, ∞). We also study the generalized β-transformations (in particular the tent map). We show that the critical orbit x = 1 is normal with respect to the measure of maximal entropy for almost all β. IntroductionIn this paper, we consider a dynamical system (X, d, T ) where (X, d) is a compact metric space endowed with its Borel σ-algebra B and T : X → X is a measurable map. Let C(X) denote the set of all continuous functions from X into R. The set M (X) of all Borel probability measures is equipped with the weak * -topology.is the subset of all T -invariant probability measures. For µ ∈ M (X, T ), let h(µ) denote the measure-theoretic entropy of µ. For all x ∈ X and n ≥ 1, the empirical measure of order n at x iswhere δ x is the Dirac mass at x. Let V T (x) ⊂ M (X, T ) denote the set of all cluster points of {E n (x)} n≥1 in the weak * -topology.Prepared using etds.cls [Version: 1999/07/21 v1.0]

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Invariant Measure of Rotational Beta Expansion and Tarski’s Plank Problem

Akiyama

Caalim

2016

Discrete Comput Geom

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We study invariant measures of a piecewise expanding map in R m defined by an expanding similitude modulo lattice. Using the result of Bang [5] on the plank problem of Tarski, we show that when the similarity ratio is not less than m + 1, it has an absolutely continuous invariant measure equivalent to the mdimensional Lebesgue measure, under some mild assumption on the fundamental domain. Applying the method to the case m = 2, we obtain an alternative proof of the result in [1] together with some improvement.

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Invariant densities for generalizedβ-maps

Cited by 36 publications

References 15 publications

Artin-Mazur Zeta Functions of Generalized Beta-Transformations

Artin-Mazur Zeta Functions of Generalized Beta-Transformations

A point is normal for almost all mapsβx+αmod 1 or generalizedβ-transformations

Invariant Measure of Rotational Beta Expansion and Tarski’s Plank Problem

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