Analysis on hybrid fractals

Figure 1.  The Hanoi attractor or stretched Sierpinski gasket

Figure 2.  3-level Sierpinski gasket, interval and regular inverted Sierpinski gasket

Figure 3.  The $ 3 $-level Sierpinski gasket

Figure 4.  Line segments and one inverted $ \operatorname{SG} $ join the six triangular $ 1 $-cells of the first level approximation of $ \operatorname{SG}_3 $

Figure 5.  Generators of the hybrid fractal $ \rm H $ based on the the upright triangle

Figure 6.  Graph subdivision and corresponding resistance scaling factors

Figure 7.  As networks, each triangular cell must be equivalent to its (n + 1)th-level subdivision

Figure 8.  The bonds $ \operatorname{I}\in\{K_j\}_{j\in B} $

Figure 9.  Graph subdivision and corresponding resistance scaling factors at level n

Figure 10.  Approximating graphs $ \Gamma_0 $, $ \Gamma_1 $ and $ \Gamma_2 $

Figure 11.  Localized D-N eigenfunctions, see [12] for further examples

Figure 12.  Eigenvalue counting functions of discrete Dirichlet Laplacians, see [12] for further examples

Figure 13.  '$ y $-$ \log x $' plots of Weyl ratios corresponding to different choices of $ a $ and $ r $ at level 6, see [12] for further examples

Figure 14.  Restriction of the eigenfunction corresponding to the lowest eigenvalue to one edge, $ r = a = \frac{1}{6} $, levels $ 1 $ through $ 6 $

Figure 15.  Restriction of the eigenfunction corresponding to the lowest eigenvalue to one edge, $ r = a = \frac{1}{10} $, levels $ 1 $ through $ 6 $

Figure 17.  Eigenfunction corresponding to the lowest eigenvalue, $ a = \frac{1}{6} $, level 6, increasing $ r $

Figure 18.  Eigenfunction corresponding to the lowest eigenvalue, $ r = \frac{1}{3} $, level $ 6 $, increasing $ a $, see [12] for further examples

Figure 19.  Plot of $ N_{Q_m}(\sqrt{x}) $ for level 1 (blue) and level 2 (yellow) quantum hybrid SG-based graph with scaling factor $ r = \frac{1}{6} $

Figure 20.  Eigenfunctions for level 1 quantum graph approximation of $ \rm H $, see [12] for further examples

Figure 16.  Eigenfunction corresponding to the lowest eigenvalue, $ r = a = \frac{1}{6} $, see [12] for further examples

Figure 21.  Contraction property of the resistance

Figure 22.  Contraction property of measure

Figure 23.  Level 4 eigenfunction, see [12] for further examples

Figure 25.  Directed graph associated with the hybrid $ \rm H $ with base $ \operatorname{SG}_3 $ from Section 2