scholarly journals On the Hausdorff dimension faithfulness and the Cantor series expansion

2020 ◽  
Vol 26 (4) ◽  
pp. 298-310
Author(s):  
S. Albeverio ◽  
Ganna Ivanenko ◽  
Mykola Lebid ◽  
Grygoriy Torbin
Keyword(s):  
Hausdorff Dimension ◽  
Series Expansion ◽  
Cantor Series

scholarly journals SHRINKING TARGETS FOR NONAUTONOMOUS DYNAMICAL SYSTEMS CORRESPONDING TO CANTOR SERIES EXPANSIONS

2015 ◽  
Vol 92 (2) ◽  
pp. 205-213 ◽  
Author(s):  
LIOR FISHMAN ◽  
BILL MANCE ◽  
DAVID SIMMONS ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Wide Class ◽  
Series Expansions ◽  
Closed Formula ◽  
Nonautonomous Dynamical Systems ◽  
Nonautonomous Dynamical System ◽  
Cantor Series

We provide a closed formula of Bowen type for the Hausdorff dimension of a very general shrinking target scheme generated by the nonautonomous dynamical system on the interval$[0,1)$, viewed as$\mathbb{R}/\mathbb{Z}$, corresponding to a given method of Cantor series expansion. We also examine a wide class of examples utilising our theorem. In particular, we give a Diophantine approximation interpretation of our scheme.

scholarly journals BETA CANTOR SERIES EXPANSION AND ADMISSIBLE SEQUENCES

2020 ◽  
Vol 60 (3) ◽  
pp. 214-224
Author(s):  
Jonathan Caalim ◽  
Shiela Demegillo
Keyword(s):  
Real Number ◽  
Series Expansion ◽  
Numeration System ◽  
Integer Sequence ◽  
Cantor Series ◽  
Admissible Sequences

We introduce a numeration system, called the <em>beta Cantor series expansion</em>, that generalizes the classical positive and negative beta expansions by allowing non-integer bases in the Q-Cantor series expansion. In particular, we show that for a fix $\gamma \in \mathbb{R}$ and a sequence $B=\{\beta_i\}$ of real number bases, every element of the interval $x \in [\gamma,\gamma+1)$ has a <em>beta Cantor series expansion</em> with respect to B where the digits are integers in some alphabet $\mathcal{A}(B)$. We give a criterion in determining whether an integer sequence is admissible when $B$ satisfies some condition. We provide a description of the reference strings, namely the expansion of $\gamma$ and $\gamma+1$, used in the admissibility criterion.

Hausdorff dimension of some kind of alternating Oppenheim series expansion

2007 ◽  
Vol 23 (2) ◽  
pp. 171-179
Author(s):  
Luming Shen ◽  
Chao Ma ◽  
Yuehua Liu
Keyword(s):  
Hausdorff Dimension ◽  
Series Expansion
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