Box dimension of the graphs of the generalized Weierstrass-type functions

2022/10/22 by Haojie Ren, Ren, Haojie
Mathematics · #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2210.12434

openalex publication_date 2022/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a Lipschitz ℤ-periodic function ϕ:ℝ→ ℝ² satisfied that ℝ²∖\ϕ(x):x∈ℝ\ is not connected, an integer b≥ 2 and λ∈ (c/b^\frac12,1), we prove the following for the generalized Weierstrass-type function W(x)=∑_n=0^∞λⁿϕ(bⁿx): the box dimension of its graph is equal to 3+2log_bλ, where c is a constant depending on ϕ.

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