Abstract
Abstract
Let A, B be matrices in
S
L
2
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 pair of 2 × 2 matrices with nonnegative integer entries and determinant 1.
Funder
Ministero dell’Istruzione, dell’Università e della Ricerca
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics
Reference27 articles.
1. Bounded semigroups of matrices;Berger;Linear Algebr. Appl.,1992
2. An elementary counterexample to the finiteness conjecture;Blondel;SIAM J. Matrix Anal. Appl.,2003
3. Ergodic optimization of Birkhoff averages and Lyapunov exponents;Bochi,2018
4. Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture;Bousch;J. Am. Math. Soc.,2002