Regular sequences and the joint spectral radius

We classify the growth of a <i>k</i>-regular sequence based on information from its <i>k</i>-kernel. In order to provide such a classification, we introduce the notion of a growth exponent for <i>k</i>-regular sequences and show that this exponent is equal to the base-k logarithm of the joint spectral radius of any set of a special class of matrices determined by the <i>k</i>-kernel