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For a one-to-one self-conformal contractive system ${w_j}_{j=1}^m$ on ${ℝ}^d$ with attractor K and conformality dimension α, Peres et al. showed that the open set condition and strong open set condition are both equivalent to $0<{\mathscr{H}}^{\alpha }\left(K\right)<\infty$. We give a simple proof of this result as well as discuss some further properties related to the separation condition.

The Ruelle operator and the associated Perron-Frobenius property have been extensively studied in dynamical systems. Recently the theory has been adapted to iterated function systems (IFS) $(X,{w_j}_{j=1}^m,{p_j}_{j=1}^m)$, where the $w_j$’s are contractive self-maps on a compact subset $X\subseteq {ℝ}^d$ and the $p_j$’s are positive Dini functions on X [FL]. In this paper we consider Ruelle operators defined by weakly contractive IFS and nonexpansive IFS. It is known that in such cases, positive bounded eigenfunctions may not exist in general. Our theorems...