The orbit problem is in the GapL hierarchy

The Orbit problem is defined as follows: Given a matrix A∈ℚ ^n×n and vectors x,y∈ℚ ⁿ , does there exist a non-negative integer i such that A ⁱ x=y. This problem was shown to be in deterministic polynomial time by Kannan and Lipton (J. ACM 33(4):808–821, 1986). In this paper we place the problem in the logspace counting hierarchy GapLH. We also show that the problem is hard for C₌L with respect to logspace many-one reductions.