Separation conditions for iterated function systems with overlaps on Riemannian manifolds

TL;DR

This paper extends separation conditions and Hausdorff dimension formulas for conformal iterated function systems on Riemannian manifolds with nonnegative Ricci curvature, generalizing previous Euclidean space results.

Contribution

It introduces weak and finite type separation conditions on Riemannian manifolds and derives Hausdorff dimension formulas for self-similar and graph self-similar sets under these conditions.

Findings

Formulated weak separation condition for conformal IFS on Riemannian manifolds.

Derived Hausdorff dimension formula for self-similar sets satisfying finite type condition.

Extended dimension results to graph self-similar sets with graph finite type condition.

Abstract

We formulate the weak separation condition and the finite type condition for conformal iterated function systems on Riemannian manifolds with nonnegative Ricci curvature, and generalize the main theorems by Lau \textit{et al.} in [Monatsch. Math. 156 (2009), 325-355]. We also obtain a formula for the Hausdorff dimension of a self-similar set defined by an iterated function system satisfying the finite type condition, generalizing a corresponding result by Jin-Yau [Comm. Anal. Geom. 13 (2005), 821--843] and Lau-Ngai [Adv. Math. 208 (2007), 647-671] on Euclidean spaces. Moreover, we obtain a formula for the Hausdorff dimension of a graph self-similar set generated by a graph-directed iterated function system satisfying the graph finite type condition, extending a result by Ngai \textit{et al.} in [Nonlinearity 23 (2010), 2333--2350].