Box dimension of the graphs of recurrent fractal interpolation functions

TL;DR

The paper advances the theory of box dimension for graphs of generalized affine recurrent fractal interpolation functions by constructing restricted vertical scaling matrices from underlying local IFSs. It establishes monotonicity of the spectral radii and irreducibility under positivity, enabling limits that bound the box dimension and, in favorable cases, yield an explicit formula for dim_B Γf as the maximum over components of $d_r^*=1+\frac{\log \rho_r}{\log T_r}$ and the baseline 1. The framework connects local combinatorial structure (basic intervals and SCCs) with spectral data to quantify fractal geometry, and the provided example demonstrates the practicality and accuracy of the approach. Overall, the work offers a rigorous method to compute or bound the box dimension of RFIF graphs via restricted vertical scaling matrices and their spectral radii. It has potential implications for modeling fractal data and assessing dimensional properties of recurrent fractal interpolants in applications.

Abstract

Let $f$ be a generalized affine recurrent fractal interpolation function with vertical scaling functions. In this paper, by introducing underlying local iterated function systems of $f$, we define restricted vertical scaling matrices. Then we prove the monotonicity of spectral radii of these matrices without additional conditions. We also prove the irreducibility of these matrices under the assumption that vertical scaling functions are positive. With these results, we estimate the upper and lower box dimensions of the graphs of $f$ by the limits of spectral radii of restricted vertical scaling matrices. In particular, we obtain an explicit formula of the box dimension of the graph of $f$ under certain constraint conditions.

Figures (1)

• Figure 1: Graph of $f$.

Theorems & Definitions (50)

• Theorem 1.1: BEH89RXY21
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• proof : Proof of Theorem \ref{['thm:irr']}