Supercyclic dynamics of translations on weighted Orlicz spaces

Let G be a locally compact group and T_a,u be a weighted translation on weighted Orlicz space L^Φ _ω (G). In this paper, we first concerned with the equivalent characterizations for T_a,u to be supercyclic. We show that T_a,u is ℝ^+ -supercyclic if and only if T_a,u is ℂ -supercyclic and σ _p(T_a,u^*)=∅ . Based on this, the sufficient and necessary conditions for powers of weighted translations to be disjoint supercyclic and positively d-supercyclic are obtained. In this article, we also provide the equivalent conditions for d-supercyclicity of weighted translations with the same aperiodic element, and unweighted translations generated by different aperiodic elements.