Invertible measure preserving transformations and pointwise convergence

An investigation of pointwise convergence of sequences $\{ \sum\nolimits_{j = - \infty }^\infty {a_j^kf({T^{ - j}}x):k = 1,2, \cdots } \}$ where $f$ lies in the space ${L^1}([0,1])$ of Lebesgue integrable functions on the unit interval, $T$ is an invertible measure preserving transformation on $[0,1]$, and the sequence of polynomials $\{ \sum\nolimits_{j = - \infty }^\infty {a_j^k{z^{ - j}}:k = 1,2, \cdots } \}$ is uniformly bounded and pointwise convergent for all $z$ such that $\vert z\vert = 1$.