ON POINTS WITH POSITIVE DENSITY OF THE DIGIT SEQUENCE IN INFINITE ITERATED FUNCTION SYSTEMS
2016 ◽
Vol 102
(3)
◽
pp. 435-443
Keyword(s):
Let $\{f_{n}\}_{n\geq 1}$ be an infinite iterated function system on $[0,1]$ and let $\unicode[STIX]{x1D6EC}$ be its attractor. Then, for any $x\in \unicode[STIX]{x1D6EC}$, it corresponds to a sequence of integers $\{a_{n}(x)\}_{n\geq 1}$, called the digit sequence of $x$, in the sense that $$\begin{eqnarray}x=\lim _{n\rightarrow \infty }f_{a_{1}(x)}\circ \cdots \circ f_{a_{n}(x)}(1).\end{eqnarray}$$ In this note, we investigate the size of the points whose digit sequences are strictly increasing and of upper Banach density one, which improves the work of Tong and Wang and Zhang and Cao.
2018 ◽
Vol 7
(3.31)
◽
pp. 126
2013 ◽
Vol 59
(2)
◽
pp. 281-298
2013 ◽
Vol 06
(04)
◽
pp. 1350055
◽