2023/10/03 by Simon Baker, Baker, Simon, George Bender +1
Mathematics · #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2310.01902
openalex publication_date 2023/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The graphs of Okamoto's functions, denoted by Kq, are self-affine fractal curves contained in [0,1]2, parameterised by q ∈ (1,2). In this paper we consider the cardinality and dimension of the intersection of these curves with horizontal lines. Our first theorem proves that if q is sufficiently close to 2, then Kq admits a horizontal slice with exactly three elements. Our second theorem proves that if a horizontal slice of Kq contains an uncountable number of elements then it has positive Hausdorff dimension provided q is in a certain subset of (1,2). Finally, we prove that if q is a k-Bonacci number for some k ∈ ℕ≥ 3, then the set of y ∈ [0,1] such that the horizontal slice at height y has (2m+1) elements has positive Hausdorff dimension for any m ∈ ℕ. We also show that, under the same assumption on q, there is some horizontal slice whose cardinality is countably infinite.