Multifractal Analysis of Unstable Polynomial Entropies in Partially Hyperbolic Systems

We investigate the multifractal structure of unstable local polynomial entropies for arbitrary Borel probability measures in partially hyperbolic dynamical systems. We introduce the notions of Bowen unstable polynomial entropy, unstable local polynomial entropy, and a measure-theoretic polynomial entropy in the unstable direction and study their fundamental properties. Our main result establishes a multifractal relation for the level sets determined by unstable local polynomial entropies, showing that the Bowen unstable polynomial entropy of these sets is described by a Legendre-type formula involving the corresponding measure-theoretic polynomial entropy.