autocommutativity degree for topological groups

Let g be a compact hausdorff topological group and suppose that aut(g) bethe group of topological automorphisms of g, which itself is a compact hausdorff topologicalgroup. in this paper, we will define the notion of autocommutativity degree for the groupg, which generalizes the concept of autocommutativity degree, in the case of finite groups.we will prove some properties of the autocommutativity degree for topological groups. inparticular, we will state an upper bound for the autocommutativity degree of non-abeliangroups and investigate the structure of groups that attain this upper bound. also we provide some examples of inifinite topological groups and their autocommutativity degree.