Path-dependent shrinking target problems in beta-dynamical systems

Abstract

This paper is concerned with the path-dependent shrinking target problems in beta-dynamical systems. For β &gt; 1 let T β be the β-transformation on [ 0 , 1 ] . Let 1 &lt; β 1 ⩽ β 2 , and f , g be two positive continuous functions. For any x 0 , y 0 ∈ [ 0 , 1 ] , we show that the set <?CDATA \begin{align*}E\left(T_{\beta_1},T_{\beta_2},f,g\right) = \left\{\left(x,y\right)\in\left[0,1\right]^2: \begin{aligned} &|T_{\beta_1}^{\,n}x-x_0| < e^{-S_n^{\beta_1}f\left(x\right)}\\ &|T_{\beta_2}^{\,n}y-y_0| E T β 1 , T β 2 , f , g = x , y ∈ 0 , 1 2 : | T β 1 n x − x 0 | &lt; e − S n β 1 f x | T β 2 n y − y 0 | &lt; e − S n β 2 g y for infinitely many n ∈ N has large intersection property under the condition β 1 e min f ( x ) ≥ β 2 , where S n β h ( x ) denotes the ergodic sum ∑ j = 0 n − 1 h ( T β j x ) . Consequently, we can determine the Hausdorff dimension of E ( T β 1 , T β 2 , f , g ) , which is the solution to some modified pressure function defined through singular value function.