Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hausdorff dimension of the Wedding cake type surfaces

Balázs Bárány, Manuj Verma

TL;DR

This work analyzes the Hausdorff dimension of wedding cake fractal interpolation surfaces, realized as attractors of self-affine IFS in $\mathbb{R}^3$ derived from fractal interpolation over triangular domains. It advances dimension theory beyond the strongly irreducible/proximal regime by combining affinity-dimension arguments with Furstenberg measures, proving $\dim_H(K)=t_0$ outside a zero-measure parameter set and providing explicit results for a broad family of surfaces (Massopust and Geronimo–Hardin). For Massopust's surfaces, the authors establish a general sufficient condition via the Furstenberg measure, then show that the Hausdorff dimension equals the affinity dimension for almost every scaling and, in a concrete $N=3$ case, for all parameters in certain regions, including the uniform-scaling scenario. In the Geronimo–Hardin setting, similar almost-sure and “every-type” results are obtained through a graph-directed Furstenberg framework and overlap analysis. Overall, the paper delivers both typical and parameter-complete dimension results for wedding cake-type FISs, combining rigorous IFS theory with explicit geometric constructions to quantify fractal regularity in three dimensions.

Abstract

In this paper, we study the Hausdorff dimension of fractal interpolation surfaces (FISs) over a triangular domain. These FISs are known as `wedding cake surfaces'. These surfaces are the attractor of some deterministic self-affine iterated function systems (IFS) on $\mathbb{R}^3$ generated by a fractal interpolation algorithm. Due to the recent seminal result of Rapaport (Adv. Math. 449 (2024) 109734), the dimension theory of self-affine IFS on $\mathbb{R}^3$ is known whenever the IFS is strongly irreducible and proximal. However, the self-affine IFSs associated with FIS are not strongly irreducible. We prove that the Hausdorff dimension of the self-affine set (or FIS) is the same as the affinity dimension outside a set of scaling parameters with zero Lebesgue measure. Lastly, by computing the overlapping number for the associated Furstenberg IFS, we determine the Hausdorff dimension for every type of scaling parameter in a certain range of parameters.

Hausdorff dimension of the Wedding cake type surfaces

TL;DR

This work analyzes the Hausdorff dimension of wedding cake fractal interpolation surfaces, realized as attractors of self-affine IFS in derived from fractal interpolation over triangular domains. It advances dimension theory beyond the strongly irreducible/proximal regime by combining affinity-dimension arguments with Furstenberg measures, proving outside a zero-measure parameter set and providing explicit results for a broad family of surfaces (Massopust and Geronimo–Hardin). For Massopust's surfaces, the authors establish a general sufficient condition via the Furstenberg measure, then show that the Hausdorff dimension equals the affinity dimension for almost every scaling and, in a concrete case, for all parameters in certain regions, including the uniform-scaling scenario. In the Geronimo–Hardin setting, similar almost-sure and “every-type” results are obtained through a graph-directed Furstenberg framework and overlap analysis. Overall, the paper delivers both typical and parameter-complete dimension results for wedding cake-type FISs, combining rigorous IFS theory with explicit geometric constructions to quantify fractal regularity in three dimensions.

Abstract

In this paper, we study the Hausdorff dimension of fractal interpolation surfaces (FISs) over a triangular domain. These FISs are known as `wedding cake surfaces'. These surfaces are the attractor of some deterministic self-affine iterated function systems (IFS) on generated by a fractal interpolation algorithm. Due to the recent seminal result of Rapaport (Adv. Math. 449 (2024) 109734), the dimension theory of self-affine IFS on is known whenever the IFS is strongly irreducible and proximal. However, the self-affine IFSs associated with FIS are not strongly irreducible. We prove that the Hausdorff dimension of the self-affine set (or FIS) is the same as the affinity dimension outside a set of scaling parameters with zero Lebesgue measure. Lastly, by computing the overlapping number for the associated Furstenberg IFS, we determine the Hausdorff dimension for every type of scaling parameter in a certain range of parameters.

Paper Structure

This paper contains 17 sections, 20 theorems, 173 equations, 8 figures.

Table of Contents

  1. Introduction and Statements
  2. Historical background.
  3. Massopust's Fractal Interpolation Surfaces.
  4. Geronimo-Hardin FISs
  5. Preliminaries
  6. Dimension concepts
  7. Symbolic space
  8. Affinity dimension and Furstenberg measure
  9. Dimension theory of some self-similar IFS having a common fixed points structure and some negative contraction parameters on line
  10. Hausdorff dimension of Massopust's surfaces
  11. The Furstenberg measure and a sufficient condition
  12. Dimension for almost every parameter
  13. Hausdorff dimension of the FIS in the case of $N=3$ for every parameter
  14. Uniform scaling factors
  15. Geronimo-Hardin surfaces
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\mathcal{I}:=\{W_i,i\in \{1,2,\dots,N^2\}\}$ be a self-affine IFS on $\mathbb{R}^3$ defined as above. Let $G(f^*)$ be the attractor of the IFS $\mathcal{I}$. For each $i\in \{1,2,\dots,N^2\}$, we consider $a_{1}^{i}\ne a_{2}^{i}$ if $a_{1}^{i}$ and $a_{2}^{i}$ both are not on the boundary of or for Lebesgue all most every scaling parameters $\underline{s}\in (\frac{1}{N},1)^{\#\mathcal{A}_1}\

Figures (8)

  • Figure 1: Triangularization of the equilateral triangle for $N=3$. To visualise the sign of the change of the data set on each horizontal edge, we coloured it blue where the data set decreases from left to right, and otherwise, we coloured it red.
  • Figure 2: Graph of the fractal interpolation surface (top & side view) with parameters $N=3$, $s_i=s=0.75$ and $a=1$.
  • Figure 3: Triangularization in the Geronimo-Hardin construction.
  • Figure 4: Graph of the fractal surfaces (top & aerial view) with parameters $s=0.82$ and $a=1$
  • Figure 5: Visualisation of the configuration in \ref{['eq3.1']}.
  • ...and 3 more figures

Theorems & Definitions (43)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Rapaport
  • Proposition 2.4
  • proof
  • ...and 33 more