The \emph{Orbit problem} is defined as follows: Given a matrix
$A\in \Q ^{n\times n}$ and vectors $\x,\y\in \Q ^n$, 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 in
\cite{KL1986}. In this paper we place the problem in the logspace
counting hierarchy $\GapLH$. We also show that the problem is hard
for $\CeqL$ with respect to logspace many-one reductions.

The \emph{Orbit problem} is defined as follows: Given a matrix $A\in
\Q ^{n\times n}$ and vectors $\x,\y\in \Q ^n$, 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 in
\cite{KL1986}. In this paper we place the problem in the logspace
counting hierarchy $\GapLH$. We also show that the problem is hard
for $\L$ under $\NCone$-many-one reductions.