Combinatorial lemmas and applications to dynamics

The well-known combinatorial lemma of Karpovsky, Milman and Alon and a very recent one of Kerr and Li are extended. The obtained lemmas are applied to study the maximal pattern entropy introduced in the paper. It turns out that the maximal pattern entropy is equal to the supremum of sequence entropies over all sequences both in topological and measure-theoretical settings. Moreover, it is shown the maximal pattern entropy of any topological system is logk for some k∈N∪{∞} with k the maximal length of intrinsic sequence entropy tuples; and a zero-dimensional system has zero sequence entropy for any sequence if and only if the maximal pattern with respect to any open cover is of polynomial order.