Entropy, topological transitivity, and dimensional properties of unique 𝑞-expansions

Barrera, Rafael Alcaraz; Baker, Simon; Kong, Derong

doi:10.1090/tran/7370

Cited by 23 publications

(2 citation statements)

References 27 publications

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“…For β > β c the authors in [3] calculated the dimension of U β only for β being a multinacci number. In view of the work by Komornik et al [24] and Alcaraz Barrera et al [25] we ask the following analogous question.…”

Section: Open Questionsmentioning

confidence: 99%

Critical base for the unique codings of fat Sierpinski gasket

Kong

2020

Nonlinearity

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Given β ∈ (1, 2) the fat Sierpinski gasket S β is the self-similar set in R 2 generated by the iterated function system (IFS) f β,d (x) = x+d β , d ∈ A := {(0, 0), (1, 0), (0, 1)}. Then for each point P ∈ S β there exists a sequenceand the in nite sequence (d i ) is called a coding of P. In general, a point in S β may have multiple codings since the overlap region O β := c,d∈A,c =d f β,c (∆ β ) ∩ f β,d (∆ β ) has non-empty interior, where ∆ β is the convex hull of S β . In this paper we are interested in the invariant set U β := ∞ i=1 d i β i ∈ S β : ∞ i=1 d n+i β i / ∈ O β ∀n 0 . Then each point in U β has a unique coding. We show that there is a transcendental number β c ≈ 1.552 63 related to the Thue-Morse sequence, such that U β has positive Hausdorff dimension if and only if β > β c . Furthermore, for β = β c the set U β is uncountable but has zero Hausdorff dimension, and for β < β c the set U β is at most countable. Consequently, we also answer a conjecture of Sidorov (2007).Our strategy is using combinatorics on words based on the lexicographical characterization of U β .

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Section: Open Questionsmentioning

confidence: 99%

Critical base for the unique codings of fat Sierpinski gasket

Kong

2020

Nonlinearity

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“…One of the main results in [3] claims that for any β ∈ (G(m), m + 1] ∩ U m , the symmetric β-shift X m,β is a topologically transitive subshift if and only if the quasi-greedy β-expansion of 1 is an irreducible sequence, or β = β T .…”

Section: It Is Easy To Check That For Anymentioning

confidence: 99%

The Ruelle operator for symmetric $\beta$-shifts

Lopes¹,

Vargas²

2020

Publicacions Matemàtiques

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Consider m ∈ N and β ∈ (1, m + 1]. Assume that a ∈ R can be represented in base β using a development in series a = ∞ n=1 x(n)β −n , where the sequence x = (x(n)) n∈N takes values in the alphabet Am := {0, . . . , m}. The above expression is called the β-expansion of a and it is not necessarily unique. We are interested in sequences x = (x(n)) n∈N ∈ A N m which are associated to all possible values a which have a unique expansion. We denote the set of such x (with some more technical restrictions) by X m,β ⊂ A N m . The space X m,β is called the symmetric β-shift associated to the pair (m, β). It is invariant by the shift map but in general it is not a subshift of finite type. Given a Hölder continuous potential A : X m,β → R, we consider the Ruelle operator L A and we show the existence of a positive eigenfunction ψ A and an eigenmeasure ρ A for some values of m and β. We also consider a variational principle of pressure. Moreover, we prove that the family of entropies (h(µ tA )) t>0 converges, when t → ∞, to the maximal value among the set of all possible values of entropy of all A-maximizing probabilities.

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Non-uniform Expansions of Real Numbers

Neunhäuserer

2021

Mediterr. J. Math.

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Entropy, topological transitivity, and dimensional properties of unique 𝑞-expansions

Cited by 23 publications

References 27 publications

Critical base for the unique codings of fat Sierpinski gasket

Critical base for the unique codings of fat Sierpinski gasket

The Ruelle operator for symmetric $\beta$-shifts

Non-uniform Expansions of Real Numbers

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