Variational principles for BS dimension under amenable group actions

In this manuscript, we focus on the investigation of the BS dimension and BS packing dimension under amenable group actions. Firstly, we obtain a Bowen’s equation which illustrate the relation of BS packing dimension to the packing topological pressure under amenable group actions. Moreover, we establish the variational principle and inverse variational principle for BS dimension and BS packing dimension under amenable group actions. Finally, we also get an analogue of Billingsley’s type theorem for BS packing dimension under amenable group actions.