A classification of pseudo-Anosov homeomorphisms via geometric Markov partitions

We continue with the ideas of Bonatti-Langevin-Jeandenans towards a constructive and algorithmic classification of pseudo-Anosov homeomorphisms (possibly with spines), up to topological conjugacy. We begin by indicating how to assign to every pseudo-Anosov homeomorphism an abstract geometric type through a Markov partition whose rectangles have been endowed with a vertical direction; these are known as geometric Markov partitions. Such assignment is not unique, as it depends on the specific geometric Markov partition, and not every abstract geometric type is realized by a pseudo-Anosov homeomorphism. This poses interesting problems related to the algorithmic computability of the classification. The article contains three main results: $i)$ the geometric type is a total invariant of conjugacy, $(ii)$ a pictorial and combinatorial criterion for determining when a given geometric type is realizable by a pseudo-Anosov homeomorphism is provided, and $(iii)$ we describe an algorithmic procedure for determining when two geometric types correspond to topologically conjugate pseudo-Anosov homeomorphisms.