Variational Principles for $r$-Neutralized Pfister-Sullivan Entropy

In this paper, we study $r$-neutralized Pfister–Sullivan entropy, defined by measuring the complexity of orbit segments whose empirical measures lie near a prescribed invariant measure. We mainly obtain the following three results: (i) We disprove a variational conjecture posed by C.~Dong and Q.~Qiao [On $r$-neutralized entropy: entropy formula and existence of measures attaining the supremum, Comm. Math. Phys. 406 (2025), Article No. 74.] (ii)] We obtain variational formulas over probability measures for both upper and lower $r$-neutralized topological entropies. We also introduce new measure-theoretic quantities through PS localization and establish some variational principles for upper $r$-neutralized topological entropy (iii) We obtain upper bounds for $r$-neutralized PS entropy under different smoothness assumptions and lower bounds for $C^{1+\alpha }$ diffeomorphisms, in terms of entropy, dimension and Lyapunov exponents. For $C^{\mathrm{\infty }}$ diffeomorphisms and $C^1$ Anosov diffeomorphisms, the $r$-neutralized PS entropies equal $h_{\mu }(f)+r\dim M$ for every ergodic measure $\mu$. We also establish exact formulas on mixing surface hyperbolic basic sets and show that the whole-manifold measure formula can fail for $C^2$ surface diffeomorphisms.