Entropy And Preimage Entropy Of Semigroup Actions

In this paper, we define the entropy and preimage entropy of free semigroup actions ina new method. Based on these definitions, we get some relations between topological entropyand measure entropy, and the relations among kinds of preimage entropies. The main results ofthis paper are as follows:1. The topological entropy is invariant under equi-conjugacy;2. thepower rule for the measure-theoretic entropy holds;3. the measure-theoretic entropy has theaffine property;4. all kinds of preimage entropies are all invariants,i.e.,h_p（G₁）=h_p（G₂）, h_m（G₁）=h_m（G₂）,h_i（G₁）=h_i（G₂）,h_r（G₁）=h_r（G₂） We also get relations among these entropy invariants:1)h_p（G₁）≤h_m（G₁）≤（G₁）2)h_i（G₁）≤h_r（G₁）,3)h（G₁）≤h_i（G₁）+h_m（G₁）;Further more, in the study of two special semigroup actions, we get that for the positivelyexpansive systems, we have h_i（G₁）=h_r（G₁）.We can also find one kind of zero preimagebranch entropy system.