Some results on entropy and sequence entropy

In this paper we study some formulas involving metric and topological entro py and sequence entropy. We summarize some classical formulas satisfied by metric and topological entropy and ask the question whether the same or sim ilar results hold for sequence entropy. In general the answer is negative; still some questions involving these formulas remain open. We make a specia l emphasis on the commutativity formula for topological entropy h(f circle g) = h(g circle f) recently proved by Kolyada and Snoha. We give a new elem entary proof and use similar ideas to prove commutativity formulas for metr ic entropy and other topological invariants. Finally we prove a Misiurewicz -Szlenk type inequality for topological sequence entropy for piecewise mono tone maps on the interval I = [0, 1]. For this purpose we introduce the not ion of upper entropy.