TY - JOUR

T1 - INTERMEDIATE β-SHIFTS AS GREEDY β-SHIFTS WITH A HOLE

AU - Langeveld, Niels

AU - Samuel, Tony

PY - 2023/5/25

Y1 - 2023/5/25

N2 - We show that every intermediate \mbox{$\beta$-transformation} is topologically conjugate to a greedy \mbox{$\beta$-transformation} with a hole at zero, and provide a counterexample illustrating that the correspondence is not \mbox{one-to-one}. This characterisation is employed to (1) build a Krieger embedding theorem for intermediate \mbox{$\beta$-transformation}, complementing the result of Li, Sahlsten, Samuel and Steiner [2019], and (2) obtain new metric and topological results on survivor sets of intermediate \mbox{$\beta$-transformations} with a hole at zero, extending the work of Kalle, Kong, Langeveld and Li [2020]. Further, we derive a method to calculate the Hausdorff dimension of such survivor sets as well as results on certain bifurcation sets. Moreover, by taking unions of survivor sets of intermediate \mbox{$\beta$-transformations} one obtains an important class of sets arising in metric number theory, namely sets of badly approximable numbers in non-integer bases. We prove, under the assumption that the underlying symbolic space is of finite type, that these sets of badly approximable numbers are winning in the sense of Schmidt games, and hence have the countable intersection property, extending the results of Hu and Yu [2014], Tseng [2009] and F\"{a}rm, Persson and Schmeling [2010].

AB - We show that every intermediate \mbox{$\beta$-transformation} is topologically conjugate to a greedy \mbox{$\beta$-transformation} with a hole at zero, and provide a counterexample illustrating that the correspondence is not \mbox{one-to-one}. This characterisation is employed to (1) build a Krieger embedding theorem for intermediate \mbox{$\beta$-transformation}, complementing the result of Li, Sahlsten, Samuel and Steiner [2019], and (2) obtain new metric and topological results on survivor sets of intermediate \mbox{$\beta$-transformations} with a hole at zero, extending the work of Kalle, Kong, Langeveld and Li [2020]. Further, we derive a method to calculate the Hausdorff dimension of such survivor sets as well as results on certain bifurcation sets. Moreover, by taking unions of survivor sets of intermediate \mbox{$\beta$-transformations} one obtains an important class of sets arising in metric number theory, namely sets of badly approximable numbers in non-integer bases. We prove, under the assumption that the underlying symbolic space is of finite type, that these sets of badly approximable numbers are winning in the sense of Schmidt games, and hence have the countable intersection property, extending the results of Hu and Yu [2014], Tseng [2009] and F\"{a}rm, Persson and Schmeling [2010].

KW - intermediate beta fransformations

KW - alpha winning

KW - embeddings

U2 - 10.1007/s10474-023-01337-3

DO - 10.1007/s10474-023-01337-3

M3 - Article

VL - 170.2023

SP - 269

EP - 301

JO - Acta mathematica Hungarica

JF - Acta mathematica Hungarica

SN - 0236-5294

IS - 1

ER -