Universal projection theorems with applications to multifractal analysis and the dimension of every ergodic measure on self-conformal sets simultaneously
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2024
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| author | Bárány, Balázs Simon, Károly Śpiewak, Adam |
| author_facet | Bárány, Balázs Simon, Károly Śpiewak, Adam |
| contents | We prove a universal projection theorem, giving conditions on a parametrized family of maps $Π_λ: X \to \mathbb{R}^d$ and a collection M of measures on X under which for almost every $λ$ equality $\dim_H Π_λμ= \min\{d, \dim_H μ\}$ holds for all measures $μ\in M$ simultaneously (i.e. on a full measure set of $λ$'s independent of $μ$). We require family $Π_λ$ to satisfy a transversality condition and collection M to satisfy a new condition called relative dimension separability. Under the same assumptions, we also prove that if the Assouad dimension of X is smaller than d, then for almost every $λ$, projection $Π_λ$ is nearly bi-Lipschitz (i.e. with pointwise $α$-Hölder inverse for every $α\in (0,1)$) at $μ$-a.e. x, for all $μ\in M$ simultaneously. Our setting encompasses families of orthogonal projections, natural projections corresponding to conformal iterated function systems, and non-autonomous or random IFS.
As applications, we provide novel results on the multifractal analysis, giving formula for the Hausdorff dimension of a level set of the local dimension for a typical (w.r.t the translation parameter) self-similar measure on the line, valid for the full range spectrum (including the decreasing part of the spectrum; previous results were covering only the increasing part).
Among another applications, we prove that given a parametrized contracting conformal IFS satisfying the transversality condition, for almost every parameter the dimension formula holds for all ergodic shift-invariant measures simultaneously. We also prove that the dimension part of the Marstrand's projection theorem holds simultaneously for the collection of all ergodic measures on a strongly separated self-conformal set and for the collection of all Gibbs measures on a self-conformal set (without any separation). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_03529 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Universal projection theorems with applications to multifractal analysis and the dimension of every ergodic measure on self-conformal sets simultaneously Bárány, Balázs Simon, Károly Śpiewak, Adam Dynamical Systems Classical Analysis and ODEs 37E05, 28A80, 28A75 We prove a universal projection theorem, giving conditions on a parametrized family of maps $Π_λ: X \to \mathbb{R}^d$ and a collection M of measures on X under which for almost every $λ$ equality $\dim_H Π_λμ= \min\{d, \dim_H μ\}$ holds for all measures $μ\in M$ simultaneously (i.e. on a full measure set of $λ$'s independent of $μ$). We require family $Π_λ$ to satisfy a transversality condition and collection M to satisfy a new condition called relative dimension separability. Under the same assumptions, we also prove that if the Assouad dimension of X is smaller than d, then for almost every $λ$, projection $Π_λ$ is nearly bi-Lipschitz (i.e. with pointwise $α$-Hölder inverse for every $α\in (0,1)$) at $μ$-a.e. x, for all $μ\in M$ simultaneously. Our setting encompasses families of orthogonal projections, natural projections corresponding to conformal iterated function systems, and non-autonomous or random IFS. As applications, we provide novel results on the multifractal analysis, giving formula for the Hausdorff dimension of a level set of the local dimension for a typical (w.r.t the translation parameter) self-similar measure on the line, valid for the full range spectrum (including the decreasing part of the spectrum; previous results were covering only the increasing part). Among another applications, we prove that given a parametrized contracting conformal IFS satisfying the transversality condition, for almost every parameter the dimension formula holds for all ergodic shift-invariant measures simultaneously. We also prove that the dimension part of the Marstrand's projection theorem holds simultaneously for the collection of all ergodic measures on a strongly separated self-conformal set and for the collection of all Gibbs measures on a self-conformal set (without any separation). |
| title | Universal projection theorems with applications to multifractal analysis and the dimension of every ergodic measure on self-conformal sets simultaneously |
| topic | Dynamical Systems Classical Analysis and ODEs 37E05, 28A80, 28A75 |
| url | https://arxiv.org/abs/2412.03529 |