Variational Principle for Non-additive Neutralized Bowen Topological Pressure

Ovadia and Rodriguez-Hertz (Neutralized local entropy and dimension bounds for invariant measures. arXiv:2302.10874v2) defined the neutralized Bowen open ball as \(B_n(x,e^{-n\varepsilon })=\{y\in X:d(T^j(x),T^j(y))<e^{-n\varepsilon },\forall 0\le j\le n-1\}.\) $$B_n(x,e^{-n\epsilon })=\{y\in X:d(T^j(x),T^j(y))<e^{-n\epsilon },\forall 0\le j\le n-1\}.$$ Yang et al. (Variational principle for neutralized Bowen topological entropy, arXiv:2303.01738v1) introduced the notion of neutralized Bowen topological entropy of subsets by replacing the usual Bowen ball by neutralized Bowen open ball. And they established variational principles for this notion. In this note, we extend this notion to the non-additive neutralized Bowen topological pressure and establish the variational principle for non-additive potentials with tempered distortion. Besides, we establish a Billingsley type theorem for non-additive neutralized Bowen topological pressure.