Weak mixing for dynamical systems with ${\bm{\mathscr M}}_{\bm\alpha}$-shadowing

Abstract

This paper studies weak mixing for two classes of dynamical systems with the \(\mathscr{M}_{\alpha}\)-shadowing property using the generic factor characterization of weak scattering. First, for every \(\alpha\in[0,1)\), every \(M\)-system with the \(\mathscr{M}_{\alpha}\)-shadowing property is weakly mixing, thereby answering [P. Oprocha et. al., On partial shadowing of complete pseudo-orbits, J. Math. Anal. Appl., Question~1] negatively. Second, for every \(\alpha\in(0,1)\), the \(\mathscr{M}_{\alpha}\)-shadowing property implies weak mixing for \(E\)-systems and, in particular, for dynamical systems whose measure center is the whole space.Finally, an \(E\)-system with the \(\mathscr{M}_{0}\)-shadowing property which is not weakly mixing is constructed, showing that the parameter range \(\alpha\in(0,1)\) in the \(E\)-system result is optimal.