Measures of intermediate entropies and homogeneous dynamics

Abstract

Consider the homogeneous dynamical system induced by the left translation of on the homogeneous space , where G is a connected semisimple Lie group and a cocompact lattice. We study the properties of the system related to almost weak specification. Based on the work of Quas and Soo, we prove Katok’s conjecture on intermediate metric entropies for the system for almost all (with respect to the Haar measure on G), i.e. the setE(G/Γ,g):={hμ(g):μ is a g–ergodic measure on G/Γ}is equal to . Moreover, we show that for a topological dynamical system which satisfies almost weak specification and asymptotically entropy expansiveness, the set defined as above is dense in .