Higher-Dimensional Multifractal Analysis on C¹ Non-Conformal Repellers

Let J be a repeller of an expanding and topologically mixing C¹-map g, and Φ=(φ₁,…,φd): Jᵈ, Ψ=(ψ₁, …,ψd): J→(R⁺)ᵈ be continuous functions. For a given compact (and possibly connected) subset C, we denote by EQ∘(Φ×Ψ)(C) and EQ∘(Φ×Ψ)equ(C) the two distinct level sets defined in (1.2) and (1.3), respectively. In this paper, we investigate the complexity of EQ∘(Φ×Ψ)(C) and EQ∘(Φ×Ψ)equ(C). For average conformal repellers, we study the generalized topological pressure and the Hausdorff dimension of EQ∘(Φ×Ψ)(C). For non-conformal repellers, we analyze EQ∘(Φ×Ψ)equ(C) via the topological pressure. Consequently, applying the result to the average conformal case yields a variational principle for the Hausdorff dimension of EQ∘(Φ×Ψ)equ(C).