relative autocommutativity degree for topological groups

Let g be a compact hausdorff topological group with automorphism group aut(g), which itself is a compact hausdorff topological group. suppose that h is a closed subgroup of g. in this paper, using haar measure, we define the concept of relative autocommutativity degree pr_a(h) for subgroup h which generalizes this concept for finite groups. we obtain some properties of pr_a(h) which are analogous to the case of finite groups. in particular, we provide an upper bound for the relative autocommutativity degree of a non-abelian subgroup h and characterize the structural properties of h when this bound is attained.