Tubular dimension: Leaf-Wise Asymptotic Local Product Structure, and Entropy and Volume Growth

Abstract

We introduce the notion of tubular dimension, and give a formula for it. As an application we show that every invariant measure of a $C^{1+γ}$ diffeomorphism of a closed Riemannian manifold admits an asymptotic local product structure for conditional measures on intermediate foliations of unstable leaves. As a second application, we prove a bound on the gap between any two consecutive conditional entropies, in the form of volume growth. As a third application, for certain $C^\infty$ maps we compute all conditional entropies for the measure of maximal entropy; And in particular as a consequence, in a follow-up paper we compute the Hausdorff dimension of the equilibrium measure of holomorphic endomorphisms of $\mathbb{C}\mathbb{P}^k$, $k\geq 1$, giving a solution to the Binder-DeMarco conjecture, and answering a question of Fornæss and Sibony.

Figures (1)

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Theorems & Definitions (34)

• Definition 2.1: Pesin blocks
• Definition 2.2: Conditional entropy and dimension LedrappierYoungII
• Theorem 2.3: LedrappierYoungILedrappierYoungII
• Definition 2.4: Transverse structure
• Definition 2.5: Pucks, intermediate entropy, and scaling parameters
• Lemma 3.1: Puck covers
• proof
• Lemma 3.2: Puck differentiation
• proof
• Theorem 3.3