2 results
Shrinking parallelepiped targets for
$\beta $-dynamical systems
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- Ergodic Theory and Dynamical Systems , First View
- Published online by Cambridge University Press:
- 19 November 2024, pp. 1-16
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For
$ \beta>1 $, let 
$ T_\beta $ be the 
$\beta $-transformation on 
$ [0,1) $. Let 
$ \beta _1,\ldots ,\beta _d>1 $ and let 
$ \mathcal P=\{P_n\}_{n\ge 1} $ be a sequence of parallelepipeds in 
$ [0,1)^d $. Define When each
$$ \begin{align*}W(\mathcal P)=\{\mathbf{x}\in[0,1)^d:(T_{\beta_1}\times\cdots \times T_{\beta_d})^n(\mathbf{x})\in P_n\text{ infinitely often}\}.\end{align*} $$
$ P_n $ is a hyperrectangle with sides parallel to the axes, the ‘rectangle to rectangle’ mass transference principle by Wang and Wu [Mass transference principle from rectangles to rectangles in Diophantine approximation. Math. Ann. 381 (2021) 243–317] is usually employed to derive the lower bound for 
$\dim _{\mathrm {H}} W(\mathcal P)$, where 
$\dim _{\mathrm {H}}$ denotes the Hausdorff dimension. However, in the case where 
$ P_n $ is still a hyperrectangle but with rotation, this principle, while still applicable, often fails to yield the desired lower bound. In this paper, we determine the optimal cover of parallelepipeds, thereby obtaining 
$\dim _{\mathrm {H}} W(\mathcal P)$. We also provide several examples to illustrate how the rotations of hyperrectangles affect 
$\dim _{\mathrm {H}} W(\mathcal P)$.
TOPOLOGICAL FULL GROUPS OF
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}C^*$-ALGEBRAS ARISING FROM 
$\beta $-EXPANSIONS
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- Journal:
- Journal of the Australian Mathematical Society / Volume 97 / Issue 2 / October 2014
- Published online by Cambridge University Press:
- 17 July 2014, pp. 257-287
- Print publication:
- October 2014
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We introduce a family of infinite nonamenable discrete groups as an interpolation of the Higman–Thompson groups by using the topological full groups of the groupoids defined by
$\beta $-expansions of real numbers. They are regarded as full groups of certain interpolated Cuntz algebras, and realized as groups of piecewise-linear functions on the unit interval in the real line if the 
$\beta $-expansion of 
$1$ is finite or ultimately periodic. We also classify them by a number-theoretical property of 
$\beta $.
