Abstract
For an integer
γ
>
1
$\gamma>1$
gamma greater than 1
, in the systems
T
1
:
[
0
,
1
]
→
[
0
,
1
]
,
x
↦
γ
x
mod
1
and
T
2
:
[
0
,
1
]
→
[
0
,
1
]
,
x
↦
(
1
/
x
)
mod
1
,
$T_1: [0,1]\to [0,1], x\mapsto \gamma x \text{ mod } 1\ \text{and}\ T_2:[0,1]\to [0,1], x\mapsto ({1}/{x}) \text{ mod } 1,$
upper T 1 colon left bracket 0 comma 1 right bracket right arrow left bracket 0 comma 1 right bracket comma x right arrow from bar gamma x mod 1 and upper T 2 colon left bracket 0 comma 1 right bracket right arrow left bracket 0 comma 1 right bracket comma x right arrow from bar left parenthesis 1 divided by x right parenthesis mod 1 comma
given a sequence of functions
(
f
n
:
[
0
,
1
]
→
[
0
,
1
]
)
n
=
1
∞
$(f_n:[0,1]\to [0,1])_{n=1}^\infty $
left parenthesis f Subscript n Baseline colon left bracket 0 comma 1 right bracket right arrow left bracket 0 comma 1 right bracket right parenthesis Subscript n equals 1 Superscript infinity
, consider the set
BA
(
(
f
n
)
n
)
:=
{
x
∈
X
:
lim inf
n
→
∞
dist
(
T
i
n
x
,
f
n
(
x
)
)
>
0
}
,
i
∈
{
1
,
2
}
,
$\mathfrak{BA}((f_n)_n):=\{x\in X:\, \liminf_{n\to\infty}{\mathrm{dist}}(T_i^n x,f_n(x))>0\}, i\in \{1,2\},$
German upper B upper A left parenthesis left parenthesis f Subscript n Baseline right parenthesis Subscript n Baseline right parenthesis colon equals StartSet x element of upper X colon limit inf Underscript n right arrow infinity Endscripts dist left parenthesis upper T Subscript i Superscript n Baseline x comma f Subscript n Baseline left parenthesis x right parenthesis right parenthesis greater than 0 EndSet comma i element of StartSet 1 comma 2 EndSet comma
which generalizes the non-recurrent set (by taking
f
n
$f_n$
f Subscript n
to be identity functions) and the non-shrinking-target set (by taking the
f
n
$f_n$
f Subscript n
s to be constant functions, which generalizes the set of badly approximable numbers). It has been shown that if
(
f
n
)
n
$(f_n)_n$
left parenthesis f Subscript n Baseline right parenthesis Subscript n
is a sequence of the same constant functions, then
BA
(
(
f
n
)
n
)
$\mathfrak {BA}((f_n)_n)$
German upper B upper A left parenthesis left parenthesis f Subscript n Baseline right parenthesis Subscript n Baseline right parenthesis
is winning in the sense of McMullen’s game, which implies it is dense and has full Hausdorff dimension everywhere. In this paper, we strengthen this result by extending
(
f
n
)
n
$(f_n)_n$
left parenthesis f Subscript n Baseline right parenthesis Subscript n
to be any equicontinuous sequence.