Non-autonomous conformal graph directed Markov systems

In this paper we introduce and develop the theory of non-autonomous graph directed Markov systems which is a generalization of the theory of conformal graph directed Markov systems of Mauldin and Urba\'nski, first presented in their book, and the theory of non-autonomous conformal iterated function systems set forth by Rempe-Gillen and Urba\'nski. We exhibit several large classes of functions for which Bowen's formula for Hausdorff dimension holds. In particular we consider weakly balanced finite systems, where we have some control over the growth of the derivatives, and ascending systems. Our results, particularly for ascending systems, generalize and go well beyond what is currently known for autonomous graph directed Markov systems and non-autonomous iterated function systems. We also provide an application to non-autonomous conformal dynamics by estimating the Hausdorff dimension of the Julia set of non-autonomous affine perturbations of an elliptic function from below.