On the attainability of an estimate of a number of Perron exponents for discrete linear diagonal systems with bounded coefficients

The Lyapunov, Perron and Bohl exponents are the most important numerical characteristics of dynamic systems used in control theory. Properties of the first one are well investigated. Properties of the Perron and Bohl exponents are much less described in the literature, for example the questions about the structure of the sets of values of these quantities for general discrete-time systems are still open. In this paper we give an example of a two-dimensional system with bounded coefficients that have three different Perron exponents. In addition, we prove that the s-dimensional diagonal discrete-time system with bounded coefficients can have any number of different Perron exponents between 1 and 2^s - 1.