Parameter spaces in complex dynamics and related topics

27 May 2024 - 31 May 2024

Aims

The goal for this workshop is to share ideas and techniques on how to study parameter spaces of holomorphic maps, seen from the dynamical point of view. It specifically aims at bringing together researchers working in one dimensional complex dynamics and in higher dimensional complex dynamics.

Complex dynamics is a small yet vibrant field that investigates the iteration of a holomorphic map on a complex manifold. Key questions involve studying which points behave in a stable way (that is, their iterative behaviour is similar to the behaviour of nearby points) and points which behave in a chaotic way (arbitrarily close points will have very different asymptotic behaviours). Quite different tools are used depending on whether the manifold is one-dimensional (usually the complex plane $\mathbb{C}$) or higher dimensional- we will especially think of $\mathbb{C}^2$.

Instead of restricting to a single map, one can consider families of holomorphic maps related to each other in a natural way, for example, the family of all polynomials of degree 2, or rational maps of degree d, or transcendental maps with similar function theoretical properties. If chosen wisely, these families can be associated to finite dimensional manifolds whose points corresponds to maps in the families (parameter spaces). The key questions are to understand stable components (maximal connected domains in parameter space on which all maps have similar dynamics) and the set of parameters at which there is a sudden change in the dynamics (bifurcation locus). The geometric structures, density, Hausdorff dimension, local connectivity, and much more, of such sets are object of investigation.

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