Decoding Rauzy induction: An answer to Bufetov's general question

Given a typical interval exchange transformation, we may naturally associate to it an infinite sequence of matrices through Rauzy induction. These matrices encode visitations of the induced interval exchange transformations within the original. In 2010, W. A. Veech showed that these matrices suffice to recover the original interval exchange transformation, unique up to topological conjugacy, answering a question of A. Bufetov. In this work, we show that interval exchange transformation may be recovered and is unique modulo conjugacy when we instead only know consecutive products of these matrices. This answers another question of A. Bufetov. We also extend this result to any inductive scheme that produces square visitation matrices.