Relative entropy, topological pressure and variational principle for locally compact sofic group actions

For a locally compact sofic group continuously acting on a compact metric space, we first study the relative sofic entropy and prove an inequality relating sofic entropy and relative sofic entropy. Moreover, it is shown that the relative variational principle remains valid in this paper. Secondly, the topological pressure for locally compact sofic group actions is investigated and the variational principle for topological pressure in this sofic context is established. As an application, we show a sufficient condition for a signed measure to be a $ G $-invariant measure. These contributions generalize the classical results for countable sofic groups on such spaces.