We can define a subclass of the regular languages. Fix an alphabet Sigma. Define the "circular" languages (actually, the name already exists to denote a different thing it seems, used in the field of DNA computing. AFAICT, that's a different class of languages). 

A language L is circular iff. for all words w in Sigma*, we have:

w belongs to L if and only if, for all integers k > 0, w^k belongs to L.

Is this class of languages known? I am interested in:

- a name for it

- decidability of the problem, given an automaton (in particular: a DFA), whether the accepted language obeys to the above definition

- a "nice" characterization (e.g. equational?) of the definition.