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The notion of Lyapunov regularity for a dynamics with discrete time defined by a bounded sequence of matrices can be characterized in many ways, highlighting different aspects of this important property introduced by Lyapunov. In strong contrast to the case of bounded sequences, not all these properties are equivalent to regularity for unbounded sequences. We first show that certain properties remain equivalent for unbounded sequences of matrices. We then show that unlike for bounded sequences and, more generally, tempered sequences, certain properties related to the existence of limits for the Lyapunov exponents on the diagonal are no longer equivalent to regularity for unbounded sequences.

Lyapunov regularity and triangularization for unbounded sequences