Simulation of Turning Machines by a Left-Linear Rewrite Rule

We prove in this paper that for every Turing machine there exists a left-linear, variable preserving and non-overlapping rewrite rule that simulates its behaviour. The main corollary is the undecidability of the termination for such a rule. If we suppose that the left-hand side can be unified with an only subterm of the right-hand side, then termination is decidable.