Finiteness Property of a Bounded Set of Matrices with Uniformly Sub-Peripheral Spectrum

In the paper, a simple condition guaranteing the finiteness property for a bounded set of matrices is presented. Given a bounded set S of real or complex matrices, it is shown that existence of a sequence of matrix products such that the spectrum of each matrix in this sequence is uniformly sub-peripheral and tends to the joint spectral radius of S, guarantees the spectral finiteness property for S.