Absolutely winning property for equicontinuously-twisted badly approximable points in continued fractions and beta expansions

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 →