Bedford–McMullen carpets meet nonlinear curves

Qiaonan Gu; Kan Jiang; Lifeng Xi

doi:10.5802/crmath.752

Research article - Dynamical systems

Bedford–McMullen carpets meet nonlinear curves

Qiaonan Gu ¹; Kan Jiang ¹; Lifeng Xi ¹

¹ Department of Mathematics, Ningbo University, Ningbo, People’s Republic of China

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 845-851

Abstract (Original language)
Résumé

Let $f$ be a regular function (the precise definition will be provided in the paper). Then, for any $\alpha \in (0,2)$ and any $\epsilon > 0$, there exists a Bedford–McMullen carpet $K$ with convex hull $[0,1]^2$ such that

\[ \alpha - \epsilon < \dim _{H}(K) < \alpha \qquad \text{and} \qquad \bigl \lbrace (x,y) : y=f(x)\bigr \rbrace \cap K = \bigl \lbrace (0,1)\bigr \rbrace , \]

where $\dim _{H}$ denotes the Hausdorff dimension. In particular, this result holds for $y=\cos (x)$. Unlike classical methods for analyzing slicing sets, our proofs rely exclusively on mathematical induction.

Soit $f$ une fonction régulière (la définition précise sera donnée dans l’article). Alors, pour tout $\alpha \in (0,2)$ et tout $\epsilon > 0$, il existe un tapis de Bedford–McMullen $K$ d’enveloppe convexe $[0,1]^2$ tel que

\[ \alpha - \epsilon < \dim _{H}(K) < \alpha \qquad \text{et} \qquad \bigl \lbrace (x,y) : y=f(x)\bigr \rbrace \cap K = \bigl \lbrace (0,1)\bigr \rbrace , \]

où $\dim _{H}$ désigne la dimension de Hausdorff. En particulier, ce résultat est valable pour $y=\cos (x)$. Contrairement aux méthodes classiques d’analyse des ensembles de coupe, notre démonstration repose exclusivement sur un raisonnement par récurrence.

Received: 2025-02-13
Revised: 2025-04-09
Accepted: 2025-04-18
Published online: 2025-07-15

DOI: 10.5802/crmath.752

Classification: 28A80, 28A78

Author's affiliations:

Qiaonan Gu ¹; Kan Jiang ¹; Lifeng Xi ¹

¹ Department of Mathematics, Ningbo University, Ningbo, People’s Republic of China

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Qiaonan Gu and Kan Jiang and Lifeng Xi},
     title = {Bedford{\textendash}McMullen carpets meet nonlinear curves},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {845--851},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.752},
     language = {en},
}

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Qiaonan Gu; Kan Jiang; Lifeng Xi. Bedford–McMullen carpets meet nonlinear curves. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 845-851. doi: 10.5802/crmath.752

References
Cited by

[1] John E. Hutchinson Fractals and self-similarity, Indiana Univ. Math. J., Volume 30 (1981) no. 5, pp. 713-747 | DOI | MR | Zbl

[2] Pertti Mattila Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability., Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, 1999, xii+343 pages | Zbl

[3] Curt McMullen The Hausdorff dimension of general Sierpiński carpets, Nagoya Math. J., Volume 96 (1984), pp. 1-9 | DOI | MR | Zbl

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