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Analysis on hybrid fractals

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    * Corresponding author 

The first author is partially supported by a Feodor Lynen Fellowship from the Humboldt Foundation

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  • We introduce hybrid fractals as a class of fractals constructed by gluing several fractal pieces in a specific manner and study energy forms and Laplacians on them. We consider in particular a hybrid based on the $ 3 $-level Sierpinski gasket, for which we construct explicitly an energy form with the property that it does not "capture" the $ 3 $-level Sierpinski gasket structure. This characteristic type of energy forms that "miss" parts of the structure of the underlying space is investigated in the more general framework of finitely ramified cell structures. The spectrum of the associated Laplacian and its asymptotic behavior in two different hybrids is analyzed theoretically and numerically. A website with further numerical data analysis is available at http://www.math.cornell.edu/~harry970804/.

    Mathematics Subject Classification: Primary: 28A80, 35P20, 31C25; Secondary: 31C20.

    Citation:

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  • 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

    Table 1.  Bottom part of the Dirichlet spectrum, $ r = a = 1/6 $, levels $ 1-7 $. Eigenvalues are shown with their multiplicity and listed in increasing order

    level 1 level 2 level 3 level 4 level 5 level 6 level 7
    43.20, 1 33.98, 1 33.74, 1 33.69, 1 33.68, 1 33.68, 1 33.68, 1
    57.19, 2 47.86, 2 49.44, 2 49.79, 2 49.88, 2 49.90, 2 49.91, 2
    135.55, 2 104.23, 2 117.75, 2 121.25, 2 122.12, 2 122.34, 2 122.40, 2
    149.54, 1 126.28, 1 184.79, 1 202.29, 1 206.86, 1 208.02, 1 208.31, 1
    257.04, 1 207.30, 1 217.81, 1 220.21, 1 220.80, 1 220.95, 1
    265.30, 2 289.42, 2 331.89, 2 341.94, 2 344.43, 2 345.05, 2
    1881.37, 2 397.92, 2 474.25, 2 495.84, 2 501.41, 2 502.81, 2
    1881.41, 1 466.78, 1 584.16, 1 615.23, 1 623.24, 1 625.26, 1
    3103.35, 2 476.77, 1 700.76, 1 768.64, 1 786.02, 1 790.39, 1
    3103.74, 1 514.87, 2 787.80, 2 869.46, 2 890.54, 2 895.85, 2
    3259.84, 1 1607.46, 2 1089.04, 2 1264.93, 2 1309.12, 2 1320.11, 2
    3260.17, 2 1607.46, 1 1130.98, 1 1328.95, 1 1379.76, 1 1392.49, 1
    4883.03, 2 2150.98, 1 1455.32, 1 1653.59, 1 1662.94, 1 1664.78, 1
    4883.03, 1 2150.99, 2 1481.31, 2 1679.55, 2 1686.86, 2 1688.66, 2
    4889.90, 1 2314.95, 2 1666.05, 2 1861.84, 1 1948.37, 1 1970.24, 1
    4889.90, 2 2314.96, 1 1675.70, 1 1870.76, 2 1961.04, 2 1983.08, 2
    5383.38, 3 4046.57, 2 1812.72, 2 2405.81, 1 2492.48, 1 2510.96, 1
    4046.57, 1 1814.01, 1 2447.98, 2 2565.41, 2 2591.25, 2
    6047.56, 1 2042.00, 1 2740.18, 2 2847.39, 2 2876.06, 2
    6047.56, 2 2042.44, 2 2949.41, 1 3148.07, 1 3178.77, 1
    6264.63, 2 2557.42, 1 3012.62, 1 3148.76, 1 3203.72, 1
    6264.63, 1 2557.63, 2 3256.00, 2 3438.70, 2 3485.04, 2
    9253.61, 3 2967.71, 2 3571.97, 2 3954.97, 2 4055.68, 2
    9551.29, 3 2967.75, 1 3674.09, 1 4019.24, 1 4108.50, 1
    9552.10, 3 5053.44, 3 4260.92, 1 4843.37, 2 4972.61, 2
    67729.76, 3 7462.73, 3 4270.27, 2 4923.03, 1 5092.29, 1
    83789.93, 3 7514.45, 3 4962.44, 1 5503.63, 1 5592.27, 1
    83790.87, 3 9156.79, 3 5022.27, 2 5738.82, 2 5866.46, 2
    111725.04, 3 11142.55, 3 5527.03, 2 6202.97, 2 6437.89, 2
     | Show Table
    DownLoad: CSV

    Table 2.  Convergence of the Dirichlet spectrum, with the estimated convergence order 1

    level 1 level 2 level 3 level 4 level 5 level 6 level 7
    Eigenvalue 43.2000 33.9771 33.7412 33.6913 33.6794 33.6764 33.6756
    Difference 0.2359 0.0499 0.0119 0.0030 0.0008
    Eigenvalue 57.1907 47.8622 49.4401 49.7929 49.8791 49.9005 49.9058
    Difference 1.5779 0.3528 0.0862 0.0214 0.0053
    Eigenvalue 135.5477 104.2339 117.748 121.248 122.1248 122.3441 122.3989
    Difference 13.5141 3.5000 0.8768 0.2193 0.0548
    Eigenvalue 149.5385 126.2839 184.7948 202.2888 206.8625 208.0185 208.3083
    Difference 58.5109 17.4940 4.5737 1.156 0.2853
    Eigenvalue 257.0447 207.2981 217.8053 220.2108 220.8016 220.9487
    Difference 49.7466 10.5072 2.4055 0.5908 0.1471
    Eigenvalue 265.302 289.4171 331.8853 341.9429 344.4302 345.0504
    Difference 42.4682 10.0576 2.4873 0.6202
     | Show Table
    DownLoad: CSV

    Table 3.  (cont.) Top of the spectrum, eigenvalues in increasing order, $ r = a = 1/6 $, level 7

    level 7
    Dirichlet Neumann
    113759984105.32153, 3 25411878638.230247, 3
    114102841006, 6 114102841006, 6
    114788072698, 18 114788072698, 18
    116839632421, 54 116839632421, 54
    122950383156,162 122950383156,162
    140734901210,243 140734901210,243
    140736477695,243 140736477626,243
    187655171187, 3 186882957201, 3
    187659146198, 6 187659146198, 6
    187667196655, 18 187667196655, 18
    187692182839, 54 187692182839, 54
    187775569901,162 187775569901,162
    188131568230,243 188131568230,243
    188147174195, 3 188147026223, 3
    188147176156, 6 188147176156, 6
    188147180228, 18 188147180228, 18
    188147193778, 54 188147193778, 54
    188147252150,162 188147252150,162
    197109853834,243 197109853834,243
    197130271198,243 197130271198,243
    197130271465,243 197130271465,243
    295258340787, 3 294738682923, 3
    295259492723, 6 295259492723, 6
    295261805573, 18 295261805573, 18
    295268816917, 54 295268816917, 54
    295290531946,162 295290531946,162
    295362581347,243 295362581347,243
    295362609295,243 295362609295,243
    295673500703,243 295673500703,243
    295673630427,486 295673630427,486
    325512681630,729 325512681630,729
     | Show Table
    DownLoad: CSV

    Table 4.  Bottom of spectrum for quantum graph compared to the spectrum of the discrete level 6 graph approximation of the Hanoi attractor. Ev. = eigenvalue, Renorm. ev. = renormalized eigenvalue, Mult. = multiplicity

    Level 0(Q) Level 1(Q) Level 2(Q) Hanoi Attractor
    Ev. Renorm. ev. Ev. Renorm. ev. Ev. Renorm. ev. Ev. Mult.
    10.247 44.402 8.578 37.173 7.896 34.216 33.676 1
    13.627 59.051 12.266 53.153 11.424 49.506 49.906 2
    41.306 178.992 32.951 142.786 30.030 130.132 122.399 2
    59.750 258.918 54.613 236.657 51.955 225.139 208.308 1
    75.686 327.975 57.438 248.897 52.592 227.897 220.949 1
    107.259 464.788 89.685 388.635 83.999 363.995 345.050 2
    156.406 677.761 132.033 572.143 122.324 530.069 502.813 2
    213.693 926.002 172.604 747.953 156.876 679.794 625.255 1
    217.180 941.113 192.661 834.863 186.323 807.398 790.386 1
    280.562 1215.767 232.571 1007.807 218.448 946.610 895.853 2
    358.903 1555.247 320.370 1388.268 1320.110 2
    400.372 1734.945 343.876 1490.131 1392.494 1
     | Show Table
    DownLoad: CSV
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