A thermodynamic path metric for complex H\'enon maps

Abstract

We construct a Hermitian covariance form on hyperbolic components in parameter spaces of complex H\'enon maps, associated to the full complex unstable derivative cocycle. The form measures infinitesimal variations in the marked complex unstable multiplier spectrum. Using a recent multiplier rigidity theorem by Cantat--Dujardin, we prove that it induces a distance on every hyperbolic component. Motivated by Sullivan's dictionary and by the thermodynamic interpretation of the Weil--Petersson metric, our result gives a first higher-dimensional holomorphic-dynamical counterpart of pressure-type metric structures. On the other hand, the construction differs from the one-dimensional theory in an essential way: it replaces the real geometric potential measuring unstable expansion by the full complex unstable derivative cocycle. This also suggests a complex derivative cocycle counterpart to pressure-type metric structures in Teichm\"uller theory and Anosov representation theory.