Proof of a counterexample to the finiteness conjecture in the spirit of the theory of dynamical systems
- Kozyakin, Victor
2010 Mathematics Subject Classification
- 15A60 26E25 37E10 37E45 39A11
- Infinite matrix products; Generalized spectral radius; Joint spectral radius; Extremal norms; Irreducibility; Discontinuous circle maps; Rotation number
In 1995 J.C. Lagarias and Y. Wang conjectured that the generalized spectral radius of a finite set of square matrices can be attained on a finite product of matrices. The first counterexample to this Finiteness Conjecture was given in 2002 by T. Bousch and J. Mairesse and their proof was based on measure-theoretical ideas. In 2003 V.D. Blondel, J. Theys and A.A. Vladimirov proposed another proof of a counterexample to the Finiteness Conjecture which extensively exploited combinatorial properties of permutations of products of positive matrices. In this paper, it is proposed one more proof of a counterexample of the Finiteness Conjecture fulfilled in a rather traditional manner of the theory of dynamical systems. It is presented description of the structure of trajectories with the maximal growing rate in terms of extremal norms and associated with them so called extremal trajectories. The construction of the counterexample is based on a detailed analysis of properties of extremal norms of two-dimensional positive matrices in which the technique of the Gram symbols is essentially used. At last, notions and properties of the rotation number for discontinuous orientation preserving circle maps play significant role in the proof.