Markovian actions are Anosov-like

Motivated by the introduction of Anosov-like actions by Thomas Barthelmé, Steven Frankel, and Kathryn Mann, in this paper we study group actions on the plane that preserve a pair of transverse singular foliations, as well as a strong Markovian family, namely, a collection of rectangles serving as an analogue of a Markov partition for group actions. We prove that any such action is Anosov-like. In particular, the existence of a strong Markovian family constrains the behavior of the invariant bifoliation and endows the action with characteristic features of hyperbolic dynamics, such as the existence of infinitely many elements with fixed points.