Elementary fractal geometry. 6. The dynamical interior of self-similar sets

TL;DR

The paper addresses how the interior of self-similar sets under the Open Set Condition (OSC) can be understood beyond Hausdorff dimension by introducing the dynamical interior $I(A)$ and a discretized magnification flow. It develops two automata-based tools—the neighbor graph for boundary structure and the neighborhood graph $G_N$ for interior structure—along with a constructive algorithm that derives interior neighborhoods from the boundary automaton, aided by a substitution matrix $S$ and stationary distribution $p$. The key contributions are precise definitions of $B(A)$ and $I(A)$, a practical method to compute and classify interior pieces, and a Markov-chain interpretation that yields neighborhood frequencies, enabling visualization and analysis of finite-type attractors. This framework provides a computational fractal-geometry toolbox for visualizing, classifying, and extending the study of self-similar sets, with potential applications to higher dimensions and more general IFSs.

Abstract

On the one hand, the dynamical interior of a self-similar set with open set condition is the complement of the dynamical boundary. On the other hand, the dynamical interior is the recurrent set of the magnification flow. For a finite type self-similar set, both boundary and interior are described by finite automata. The neighbor graph defines the boundary. The neighborhood graph, based on work by Thurston, Lalley, Ngai and Wang, defines the interior. If local views are considered up to similarity, the interior obtains a discrete manifold structure, and the magnification flow is discretized by a Markov chain. This leads to new methods for the visualization and description of finite type attractors.

Figures (9)

• Figure 1: The self-similar set A with three magnifications. In the second one, pieces are colored with respect to level 12. The IFS data are in Table \ref{['tab1']}.
• Figure 2: The self-similar set B with a greater variety of magnifications.
• Figure 3: The self-similar set C contains a lot of structural details.
• Figure 4: The chair, generated by $f_1(z)=\frac{z}{2},\ f_2(z)=\frac{1}{2} (z-1),\ f_3(z)=\frac{i}{2} (z-1),$ and $f_4(z)=\frac{-i}{2} (z-1).$ Left: Level 1, with five boundary sets indicated. Right: Level 2, with three other boundary sets.
• Figure 5: The positions of neighboring tiles of the chair, denoted by the neighbor maps. The boundary set $Q$ corresponds to three neighbor maps $q=p^{-1}, m^{-1},$ and $n^{-1}.$ These neighborhoods appear in every tiling and will be called $N^1,N^{4a}$ and $N^{3b}$ in Section \ref{['inte']}.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2