Tubular dimension: Leaf-Wise Asymptotic Local Product Structure, and Entropy and Volume Growth
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917586792349696 |
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| author | Ovadia, Snir Ben |
| author_facet | Ovadia, Snir Ben |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02496 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tubular dimension: Leaf-Wise Asymptotic Local Product Structure, and Entropy and Volume Growth Ovadia, Snir Ben Dynamical Systems 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. |
| title | Tubular dimension: Leaf-Wise Asymptotic Local Product Structure, and Entropy and Volume Growth |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2402.02496 |