Topology of the set of univoque bases

Vries, Martijn de; Komornik, Vilmos; Loreti, Paola

doi:10.1016/j.topol.2016.01.023

Cited by 44 publications

(65 citation statements)

References 15 publications

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“…The topological properties of U , U and V were investigated in [12]. They proved that the difference sets U \U and V \U are both countable.…”

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

“…Furthermore, U \ U is dense in U , and V \ U is discrete and dense in V . Indeed, the set V is the union of a Cantor set U and a countable set V \ U , see [12]. By [12,Lemma 3.5] it follows that the smallest base of V is the generalized golden ratio…”

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

“…Erdős and Joó proved that U has Lebesgue measure zero [14], Daróczy and Kátai [10] later showed that U has full Hausdorff dimension (see also, [18,Theorem 1.6]). Recently, de Vries et al [12,Theorem 1.2] proved that its closure U is a Cantor set, i.e., a nonempty compact set with empty interior and no isolated points. They also obtained (1, M + 1] \ U = (q 0 , q * 0 ).…”

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confidence: 99%

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

Barrera

Baker

Kong

2018

Trans. Amer. Math. Soc.

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Abstract. Let M be a positive integer and q ∈ (1, M + 1]. We consider expansions of real numbers in base q over the alphabet {0, . . . , M }. In particular, we study the set Uq of real numbers with a unique q-expansion, and the set Uq of corresponding sequences.It was shown in [18, Theorem 1.7] that the function H, which associates to each q ∈ (1, M + 1] the topological entropy of Uq, is a Devil's staircase. In this paper we explicitly determine the plateaus of H, and characterize the bifurcation set E of q's where the function H is not locally constant. Moreover, we show that E is a Cantor set of full Hausdorff dimension. We also investigate the topological transitivity of a naturally occurring subshift (Vq, σ), which has a close connection with open dynamical systems. Finally, we prove that the Hausdorff dimension and box dimension of Uq coincide for all q ∈ (1, M + 1].

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“…The topological properties of U , U and V were investigated in [12]. They proved that the difference sets U \U and V \U are both countable.…”

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

Section: Preliminaries and Main Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

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

Barrera

Baker

Kong

2018

Trans. Amer. Math. Soc.

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“…Finally, we recall some topological properties of U and U which were essentially established in [8,18] (see also, [9]). …”

Section: Lemma 24 Let Q ∈ (1 M + 1) Then Q ∈ U If and Only Ifmentioning

confidence: 99%

“…Recently, Komornik and Loreti showed in [18] that its closure U is a Cantor set (see also, [9]), i.e., a nonempty closed set having neither isolated nor interior points. Writing the open set (1, M + 1]\U = (1, M + 1)\U as the disjoint union of its connected components, i.e.,…”

Section: Introductionmentioning

confidence: 99%