HIGHER-DIMENSIONAL MULTIFRACTAL ANALYSIS ON $ C^1 $ NON-CONFORMAL REPELLERS

Abstract

Let $ J $ be a repeller of an expanding and topologically mixing $ C^1 $-map $ g $, and $ \Phi=(\phi_1,\ldots,\phi_d): J\to\mathbb{R}^d $, $ \Psi=(\psi_1, \ldots,\psi_d): J\to(\mathbb{R}^+)^d $ be continuous functions. For a given compact (and possibly connected) subset $ C $, we denote by $ E_{Q\circ(\Phi\times\Psi)}(C) $ and $ E_{Q\circ(\Phi\times\Psi)}^{equ}(C) $ the two distinct level sets defined in (1.2) and (1.3), respectively. In this paper, we investigate the complexity of $ E_{Q\circ(\Phi\times\Psi)}(C) $ and $ E_{Q\circ(\Phi\times\Psi)}^{equ}(C) $. For average conformal repellers, we study the generalized topological pressure and the Hausdorff dimension of $ E_{Q\circ(\Phi\times\Psi)}(C) $. For non-conformal repellers, we analyze $ E_{Q\circ(\Phi\times\Psi)}^{equ}(C) $ via the topological pressure. Consequently, applying the result to the average conformal case yields a variational principle for the Hausdorff dimension of $ E_{Q\circ(\Phi\times\Psi)}^{equ}(C) $.