How chaotic is an almost mean equicontinuous system? (Q1661195)

In this paper, the chaotic degree of an almost mean equicontinuous system is discussed. The following are the main results in this exposition. Theorem 1. Every dynamical system \((X,T)\) can be embedded into a shift subsystem \((Z_X,\sigma)\) of \((([0,1]^N)^N,\sigma)\), such that i) \((Z_X,\sigma)\) is almost mean equicontinuous ii) \((Z_X,\sigma)\) is an almost one-to-one extension of a mean equicontinuous system iii) \(h_{top}(X,T)=h_{top}(Z_X,\sigma)\). Theorem 2. There exists a transitive dynamical system \((X,T)\), such that i) \((X,T)\) is Devaney chaotic ii) \((X,T)\) is topologically \(K\) iii) \((X,T)\) is almost mean equicontinuous. Further aspects occasioned by these developments are also discussed.