Measure preserving transformations similar to Markov shifts

Similarity, that is, the existence of joint common extensions, defines an interesting equivalence relation for infinite measure preserving transformations T. We provide a sufficient condition, given in terms of return processes to reference sets of finite measure, for T to be similar to a Markov shift. This is then shown to apply to various piecewise smooth dynamical systems, including weakly hyperbolic transformations with indifferent periodic points or flat critical points, and discrete random walks driven by (weakly) hyperbolic maps.