Using ergodic theory, in this paper we present a Gel'fand-type spectral radius formula which states that the joint spectral radius is equal to the generalized spectral radius for a matrix multiplicative semigroup S + restricted to a subset that need not carry the algebraic structure of S + . This generalizes the Berger-Wang formula. Using it as a tool, we study the absolute exponential stability of a linear switched system driven by a compact subshift of the one-sided Markov shift associated to S. transformation x → S w x induced by any preassigned vector norm · on C d ; that is to say,The joint spectral radius of S (free of constraints) is introduced by G.-C. Rota and G. Strang in [37] as follows:Since log supfor all ℓ, m ≥ 1, i.e., the subadditivity holds, the above limit always exists. On the other hand, the generalized spectral radius of S (free of constraints) is defined by I. Daubechies and J.C. Lagarias in [13] aswhere ρ(A) denotes the usual spectral radius of the matrix A ∈ C d×d . Then, the so-called generalized Gel'fand spectral-radius formula, due to M.A. Berger and Y. Wang [2] and conjectured by I. Daubechies and J.C. Lagarias [13], can be stated as follows:The Berger-Wang Formula 1.1 (See [2]). If S = {S i } i∈I is a bounded subset of C d×d , then there holds the equality ρ(S) =ρ(S).This formula was proved by using different approaches, for example, in [2,15,39,8,4,9]. Recently, this formula has been generalized to sets of precompact linear operators constraint-free acting on a Banach space by Ian D. Morris in [33] using ergodic theory.The above Gel'fand-type spectral-radius formula is an important tool in a number of research areas, such as in the theory of control and stability of unforced systems, see [1,25,20,12] for example; in coding theory, see [32]; in wavelet regularity, see [13,14,22,31]; and in the study of numerical solutions to ordinary differential equations, see, e.g., [19].However, in many real-world situations, constraints on allowable switching signals often arise naturally as a result of physical requirements on a system. One often needs to consider some switching constraints imposed by some kind of uncertainty about the model or about environment in which the object operates, see [41,27,28,29,6] and so on. Consider in the control theory, for example, a proper subset Λ of Σ + I as the set of admissible switching signals, such as