Dimension spectrum of asymptotically additive potentials for \(C^1\) average conformal repellers (Q2852517)

Suppose that \(f:U\to M\) is a \(C^r\)-mapping on an open subset \(U\subseteq M\) of a Riemannian \(C^\infty\)-manifold \(M\). Furthermore, let \(\Lambda\subset U\) be a compact invariant set on which \(f\) is expanding and assume \(\varphi_n,\psi_n:\Lambda\to{\mathbb R}\) are potential functions. This paper investigates the dimension spectrum NEWLINE\[NEWLINE D(\alpha)=\text{dim}_H\left\{x\in\Lambda:\,\lim_{n\to\infty}\frac{\varphi_n(x)}{\psi_n(x)}=\alpha\right\} NEWLINE\]NEWLINE of asymptotically additive potentials for \(C^1\)-average conformal repellers. Beyond corresponding dimension estimates for subsets for \(C^1\)-average conformal repellers, additionally the pointwise dimension of invariant measures is investigated. These results are illustrated by giving an affimative answer to a problem of \textit{L. Olsen} [J. Math. Pures Appl. (9) 82, No. 12, 1591--1649 (2003; Zbl 1035.37025)] concerning general types of level sets. The dimension spectrum of weak Gibbs measures for continuous potentials on repellers of the above type is given.