April  2017, 10(2): 213-240. doi: 10.3934/dcdss.2017011

A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy

1. 

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

2. 

Department of Mathematics and Statistics, California State Polytechnic University, Pomona, CA 91768, USA

3. 

Department of Mathematics, University of California, Riverside, CA 92521, USA

* Corresponding author:John A. Rock.

Received  October 2015 Revised  November 2016 Published  January 2017

Fund Project: The authors Dettmers, Giza, and Rock were supported by NSF grant DMS-1247679

The theory of complex dimensions of fractal strings developed by Lapidus and van Frankenhuijsen has proven to be a powerful tool for the study of Minkowski measurability of fractal subsets of the real line. In a very general setting, the Minkowski measurability of such sets is characterized by the structure of corresponding complex dimensions. Also, this tool is particularly effective in the setting of self-similar fractal subsets of $\mathbb{R}$ which have been shown to be Minkowski measurable if and only if they are nonlattice. This paper features a survey on the pertinent results of Lapidus and van Frankenhuijsen and a preliminary extension of the theory of complex dimensions to subsets of Euclidean space, with an emphasis on self-similar sets that satisfy various separation conditions. This extension is developed in the context of box-counting measurability, an analog of Minkowski measurability, which is shown to be characterized by complex dimensions under certain mild conditions.

Citation: Kristin Dettmers, Robert Giza, Rafael Morales, John A. Rock, Christina Knox. A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 213-240. doi: 10.3934/dcdss.2017011
References:
[1]

M. Barnsley, Fractals Everywhere -Mathematical Foundations and Applications, $2^{nd}$ edition, Ⅲ. Academic Press Professional, Boston, MA, 1993.

[2]

K. J. Falconer, On the Minkowski measurability of fractals, Proc. Amer. Math. Soc. , 123 (1995), 1115-1124. doi: 10.1090/S0002-9939-1995-1224615-4.

[3]

K. Falconer, Fractal Geometry -Mathematical Foundations and Applications, $3^{rd}$ edition, John Wiley, Chichester, 2014.

[4]

H. Federer, Geometric Measure Theory Springer-Verlag, New York, 1969.

[5]

P. M. Fitzpatrick and H. L. Royden, Real Analysis $4^{th}$ edition, Pearson, 2010.

[6]

D. Gatzouras, Lacunarity of self-similar and stochastically self-similar sets, Trans. Amer. Math. Soc., 352 (2000), 1953-1983. doi: 10.1090/S0002-9947-99-02539-8.

[7]

J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.

[8]

S. Kombrink, Fractal Curvature Measures and Minkowski Content for Limit Sets of Conformal Function Systems Ph. D. thesis, Universität Bremen, 2011.

[9]

S. P. Lalley, The packing and covering functions of some self-similar fractals, Indiana Univ. Math. J., 37 (1988), 699-710. doi: 10.1512/iumj.1988.37.37034.

[10]

S. P. Lalley, Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits, Acta Math., 163 (1989), 1-55. doi: 10.1007/BF02392732.

[11]

M. L. Lapidus and H. Maier, The Riemann hypothesis and inverse spectral problems for fractal strings, J. London Math. Soc. 52 (1995), 15-34. doi: 10.1112/jlms/52.1.15.

[12]

M. L. Lapidus, E. P. J. Pearse and S. Winter, Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators, Adv. in Math. , 227 (2011), 1349-1398. (Also: E-print, arXiv: 1006.3807v3, [math. MG], 2011. ) doi: 10.1016/j.aim.2011.03.004.

[13]

M. L. Lapidus, E. P. J. Pearse and S. Winter, Minkowski measurability results for self-similar tilings and fractals with monophase generators, in Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics (eds. D. Carfi, M. L. Lapidus, E. P. J. Pearse, and M. van Frankenhuijsen), Contemporary Mathematics, Amer. Math. Soc. , 600 (2013), 185-203. doi: 10.1090/conm/600/11951.

[14]

M. L. Lapidus and C. Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums, Proc. London Math. Soc. (3), 66 (1993), 41-69. doi: 10.1112/plms/s3-66.1.41.

[15]

M. L. LapidusG. Radunović and D. Žubrinić, Fractal zeta functions and complex dimensions of relative fractal drums, J. Fixed Point Theory Appl., 15 (2014), 321-378. doi: 10.1007/s11784-014-0207-y.

[16]

M. L. Lapidus, J. A. Rock and D. Žubrinić, Box-counting fractal strings, zeta functions, and equivalent forms of Minkowski dimension, in Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics (eds. D. Carfi, M. L. Lapidus, E. P. J. Pearse, and M. van Frankenhuijsen), Contemporary Mathematics, Amer. Math. Soc. , 600 (2013), 239-271. doi: 10.1090/conm/600/11929.

[17]

M. L. Lapidus and M. van Frankenhuijsen, Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings $2^{nd}$ edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-2176-4.

[18]

R. Morales, Complex Dimensions and Measurability Master's thesis, California State Polytechnic University, Pomona, 2014.

[19]

J. Rataj and S. Winter, Characterization of Minkowski measurability in terms of surface area, J. Math. Anal. Appl., 400 (2013), 120-132. doi: 10.1016/j.jmaa.2012.10.059.

[20]

C. Sargent, Box Counting Zeta Functions of Self-Similar Sets Master's thesis, California State Polytechnic University, Pomona, 2014.

[21]

A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc., 122 (1994), 111-115. doi: 10.1090/S0002-9939-1994-1191872-1.

[22]

J. -P. Serre, A Course in Arithmetic English translation, Springer-Verlag, Berlin, 1973.

show all references

References:
[1]

M. Barnsley, Fractals Everywhere -Mathematical Foundations and Applications, $2^{nd}$ edition, Ⅲ. Academic Press Professional, Boston, MA, 1993.

[2]

K. J. Falconer, On the Minkowski measurability of fractals, Proc. Amer. Math. Soc. , 123 (1995), 1115-1124. doi: 10.1090/S0002-9939-1995-1224615-4.

[3]

K. Falconer, Fractal Geometry -Mathematical Foundations and Applications, $3^{rd}$ edition, John Wiley, Chichester, 2014.

[4]

H. Federer, Geometric Measure Theory Springer-Verlag, New York, 1969.

[5]

P. M. Fitzpatrick and H. L. Royden, Real Analysis $4^{th}$ edition, Pearson, 2010.

[6]

D. Gatzouras, Lacunarity of self-similar and stochastically self-similar sets, Trans. Amer. Math. Soc., 352 (2000), 1953-1983. doi: 10.1090/S0002-9947-99-02539-8.

[7]

J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.

[8]

S. Kombrink, Fractal Curvature Measures and Minkowski Content for Limit Sets of Conformal Function Systems Ph. D. thesis, Universität Bremen, 2011.

[9]

S. P. Lalley, The packing and covering functions of some self-similar fractals, Indiana Univ. Math. J., 37 (1988), 699-710. doi: 10.1512/iumj.1988.37.37034.

[10]

S. P. Lalley, Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits, Acta Math., 163 (1989), 1-55. doi: 10.1007/BF02392732.

[11]

M. L. Lapidus and H. Maier, The Riemann hypothesis and inverse spectral problems for fractal strings, J. London Math. Soc. 52 (1995), 15-34. doi: 10.1112/jlms/52.1.15.

[12]

M. L. Lapidus, E. P. J. Pearse and S. Winter, Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators, Adv. in Math. , 227 (2011), 1349-1398. (Also: E-print, arXiv: 1006.3807v3, [math. MG], 2011. ) doi: 10.1016/j.aim.2011.03.004.

[13]

M. L. Lapidus, E. P. J. Pearse and S. Winter, Minkowski measurability results for self-similar tilings and fractals with monophase generators, in Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics (eds. D. Carfi, M. L. Lapidus, E. P. J. Pearse, and M. van Frankenhuijsen), Contemporary Mathematics, Amer. Math. Soc. , 600 (2013), 185-203. doi: 10.1090/conm/600/11951.

[14]

M. L. Lapidus and C. Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums, Proc. London Math. Soc. (3), 66 (1993), 41-69. doi: 10.1112/plms/s3-66.1.41.

[15]

M. L. LapidusG. Radunović and D. Žubrinić, Fractal zeta functions and complex dimensions of relative fractal drums, J. Fixed Point Theory Appl., 15 (2014), 321-378. doi: 10.1007/s11784-014-0207-y.

[16]

M. L. Lapidus, J. A. Rock and D. Žubrinić, Box-counting fractal strings, zeta functions, and equivalent forms of Minkowski dimension, in Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics (eds. D. Carfi, M. L. Lapidus, E. P. J. Pearse, and M. van Frankenhuijsen), Contemporary Mathematics, Amer. Math. Soc. , 600 (2013), 239-271. doi: 10.1090/conm/600/11929.

[17]

M. L. Lapidus and M. van Frankenhuijsen, Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings $2^{nd}$ edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-2176-4.

[18]

R. Morales, Complex Dimensions and Measurability Master's thesis, California State Polytechnic University, Pomona, 2014.

[19]

J. Rataj and S. Winter, Characterization of Minkowski measurability in terms of surface area, J. Math. Anal. Appl., 400 (2013), 120-132. doi: 10.1016/j.jmaa.2012.10.059.

[20]

C. Sargent, Box Counting Zeta Functions of Self-Similar Sets Master's thesis, California State Polytechnic University, Pomona, 2014.

[21]

A. Schief, Separation properties for self-similar sets, Proc. Amer. Math. Soc., 122 (1994), 111-115. doi: 10.1090/S0002-9939-1994-1191872-1.

[22]

J. -P. Serre, A Course in Arithmetic English translation, Springer-Verlag, Berlin, 1973.

Figure 1.  On the left, an approximation of the Sierpiński gasket $S_G$ discussed in Example 2.14. On the right, an approximation of the Quarter Fractal $Q$ discussed in Example 2.15.
Figure 3.  A lattice approximation of $\mathcal{D}_{\mathcal{L}_\phi}$, the complex dimensions of the Golden string $\Omega_\phi$ with lengths $\mathcal{L}_\phi$. The plots show the complex dimensions $\mathcal{D}_M=\{z \in{\mathbb{C}}: 2^{-z} + 2^{-z\phi_M} = 1\}$ where $\phi_M=f_{M+1}/f_M$ approximates $\phi$ for $M=2,\ldots,9$. In each case, the point $D$ denotes the Minkowski dimension of the approximating attractor, and the figure repeats with period $\mathbf{p}$. Note, however, that $\mathcal{D}_{\mathcal{L}_\phi}$ itself is not periodic. See Examples 2.13, 2.32, and 3.23 as well as Theorem 3.25 and Remark 3.28.
Figure 2.  The geometric oscillations of the Cantor string $\Omega_{CS}$ seen in the plot and semilog plot of $N_{CS}(x)/x^{D}$, on the left and right, respectively. Here, $N_{CS}$ is the geometric counting function of the Cantor string and $D=\dim_BC=\log_32$. See Example 3.18. (The function is discontinuous, the vertical line segments are an artifact of the program used to generate the images.)
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