Multifractal Analysis of F-exponents for Finitely Irreducible Conformal Graph Directed Markov Systems

Let $\mathrm{\Phi }=\{{\phi }_e{\}}_{e\in E}$ be a finitely irreducible conformal graph directed Markov system (CGDMS) with symbolic representation $E_A^{\mathrm{\infty }}$ and limit set $J$. Under a mild condition on the system, we give a multifractal analysis of level sets of Birkhoff averages with respect to Hausdorff dimension for a large family of functions. We then apply these results to a few examples in the case of both $E$ finite and $E$ countably infinite.