SOME VARIANTS OF ORPONEN'S THEOREM ON VISIBLE PARTS OF FRACTAL SETS

Carlos Matheus (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications

It was recently established by T. Orponen that the visible parts from almost every direction of a compact subset of R n have Hausdorff dimension at most n − 1 50n. In this note, we refine Orponen's argument in order to show that the visible parts from almost every direction of a compact subset of R n have Hausdorff dimension at most n − min{ 1 5 , 1 n+2 }. Moreover, we also show that some classes of dynamically defined Cantor sets K ⊂ R n with Hausdorff dimension d > max{ √ 3, (n−1)+ √ (n−1)(n+3) 2 } have visible parts of Hausdorff dimension at most max{ 3d+3 d+3 , (n+1)d+(n−1) d+2 } from almost every direction.

Domaines: Mathématiques [math]; Analyse classique [math.CA]; Systèmes dynamiques [math.DS]

Carlos Matheus. SOME VARIANTS OF ORPONEN'S THEOREM ON VISIBLE PARTS OF FRACTAL SETS. 2020. ⟨hal-02500421⟩