The finiteness conjecture holds in SL(2,Z>=0)^2

Abstract

GIOVANNI PANTI AND DAVIDE SCLOSA A. Let A, B be matrices in SL2 R having trace greater than or equal to 2. Assume the pair A, B is coherently oriented, that is, can be conjugated to a pair having nonnegative entries. Assume also that either A, B −1 is coherently oriented as well, or A, B have integer entries. Then the Lagarias-Wang finiteness conjecture holds for the set {A, B}, with optimal product in {A, B, AB, A 2 B, AB 2 }. In particular, it holds for every matrix pair in SL2 Z ≥0 . 2020 Math. Subj. Class.: 05A05; 15A60; 37F32. The first author is partially supported by the MIUR Grant E83C18000100006 Regular and stochastic behaviour in dynamical systems.