On the conformal preimage decay exponent of the Julia sets of rational graph-directed Markov systems
Tadashi Arimitsu
TL;DR
The paper studies a conformal preimage decay exponent $e_{\delta}(R)$ for Julia sets of rational graph-directed Markov systems by linking it to the entropy and pressure of the associated rational skew product $\tilde f$. It proves a spectral-entropy relation $h_{\mathrm{top}}(\tilde f)=\log \rho(M)$ and a pressure gap $e_{\delta}(R)= h_{\mathrm{top}}(\tilde f)-\overline{\mathcal{CP}}(\tilde f|_{J(\tilde f)},\delta\tilde{\varphi},\widetilde{\mathsf{HP}}(R))$, with a further equality $\overline{\mathcal{CP}}(\tilde f,\delta\tilde{\varphi},\widetilde{\mathsf{HP}}(R))=\overline{\mathrm{P}^{\mathrm{geom}}}(\delta,R)$ under the vertex-wise separation condition. An explicit expression $e_{\delta}(R)=\mathcal{P}(\sigma,\psi)-\overline{\mathrm{P}^{\mathrm{geom}}}(\delta,R)$ is derived, and positivity of the decay exponent is established under the standard hypotheses (VSC, expansion, and topological transitivity). The results tie the decay rate to the entropy-geometry gap and provide a quantitative framework for open dynamical systems in rational GDMS, connecting fractal geometry of Julia sets with symbolic and geometric pressures.
Abstract
We define and investigate the conformal preimage decay exponent of the Julia sets of rational graph-directed Markov systems. We show that this exponent coincides with the difference between the topological entropy and upper sequential capacity topological pressure for the rational skew product map associated with the system $S$. Here, upper sequential capacity topological pressure is a slight generalisation of upper capacity topological pressure given in \cite{MR969568}.