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September  2017, 37(9): 4611-4623. doi: 10.3934/dcds.2017198

Polynomial approximation of self-similar measures and the spectrum of the transfer operator

Christoph Bandt and  Helena PeÑa

Institute of Mathematics, University of Greifswald, Germany

Received  November 2016 Published  June 2017

We consider self-similar measures on $\mathbb{R}.$ The Hutchinson operator $H$ acts on measures and is the dual of the transfer operator $T$ which acts on continuous functions. We determine polynomial eigenfunctions of $T.$ As a consequence, we obtain eigenvalues of $H$ and natural polynomial approximations of the self-similar measure. Bernoulli convolutions are studied as an example.

Keywords: Self-similar measure, iterated function system, transfer operator, Bernoulli convolution, spectrum of Hutchinson operator.
Mathematics Subject Classification: Primary: 28A80; Secondary: 37D20, 41A10.
Citation: Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198
References:
[1]

D. H. Bailey, J. M. Borwein, R. E. Crandall and M. G. Rose, Expectations on fractal sets, Appl. Math. Comput., 220 (2013), 695-721.  doi: 10.1016/j.amc.2013.06.078.

[2]

K. Barański, Dimension of the graphs of the weierstrass-type functions, in Fractal Geometry and Stochastics V (eds. C. Bandt, K. Falconer and M. Zähle), vol. 70 of Progress in Probability, Springer International Publishing, 2015, 77-91. doi: 10.1007/978-3-319-18660-3_5.

[3]

M. Barnsley and S. Demko, Iterated function systems and the global construction of fractals, Proc. R. Soc. Lond., 399 (1985), 243-275.  doi: 10.1098/rspa.1985.0057.

[4]

B. Beckermann, The condition number of real Vandermonde, Krylov and positive definite Hankel matrices, Numerische Mathematik, 85 (2000), 553-577.  doi: 10.1007/PL00005392.

[5]

M.-D. Choi, Tricks or treats with the Hilbert matrix, Amer. Math. Monthly, 90 (1983), 301-312.  doi: 10.2307/2975779.

[6]

D. J. Driebe, The bernoulli map, in Fully Chaotic Maps and Broken Time Symmetry, vol. 4 of Nonlinear Phenomena and Complex Systems, Springer Netherlands, 1999, 19-43.

[7]

K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications 3rd edition, J. Wiley & sons, 2014.

[8]

D. Hilbert, Ein Beitrag zur Theorie des Legendre'schen Polynoms, Acta Mathematica, 18 (1894), 155-159.  doi: 10.1007/BF02418278.

[9]

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

[10]

T. Kato, Perturbation Theory for Linear Operators Classics in Mathematics, Springer Berlin Heidelberg, 1995.

[11]

P. Shmerkin, On the exceptional set for absolute continuity of Bernoulli convolutions, Geometric and Functional Analysis, 24 (2014), 946-958.  doi: 10.1007/s00039-014-0285-4.

[12]

J. Shohat and J. Tamarkin, The Problem of Moments Mathematical surveys and monographs, American Mathematical Society, 1943.

[13]

B. Solomyak, Notes on Bernoulli convolutions, in Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 1, vol. 72 of Proceedings of Symposia in Pure Mathematics, AMS, 2004,207-230.

[14]

B. Solomyak, Y. Peres and W. Schlag, Sixty years of Bernoulli convolutions, Progress in Probability, 46 (2000), 39-65. 

[15]

T. Tao, An Epsilon of Room I Graduate Studies in Mathematics, American Mathematical Society, 2010. doi: 10.1090/gsm/117.

show all references

References:
[1]

D. H. Bailey, J. M. Borwein, R. E. Crandall and M. G. Rose, Expectations on fractal sets, Appl. Math. Comput., 220 (2013), 695-721.  doi: 10.1016/j.amc.2013.06.078.

[2]

K. Barański, Dimension of the graphs of the weierstrass-type functions, in Fractal Geometry and Stochastics V (eds. C. Bandt, K. Falconer and M. Zähle), vol. 70 of Progress in Probability, Springer International Publishing, 2015, 77-91. doi: 10.1007/978-3-319-18660-3_5.

[3]

M. Barnsley and S. Demko, Iterated function systems and the global construction of fractals, Proc. R. Soc. Lond., 399 (1985), 243-275.  doi: 10.1098/rspa.1985.0057.

[4]

B. Beckermann, The condition number of real Vandermonde, Krylov and positive definite Hankel matrices, Numerische Mathematik, 85 (2000), 553-577.  doi: 10.1007/PL00005392.

[5]

M.-D. Choi, Tricks or treats with the Hilbert matrix, Amer. Math. Monthly, 90 (1983), 301-312.  doi: 10.2307/2975779.

[6]

D. J. Driebe, The bernoulli map, in Fully Chaotic Maps and Broken Time Symmetry, vol. 4 of Nonlinear Phenomena and Complex Systems, Springer Netherlands, 1999, 19-43.

[7]

K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications 3rd edition, J. Wiley & sons, 2014.

[8]

D. Hilbert, Ein Beitrag zur Theorie des Legendre'schen Polynoms, Acta Mathematica, 18 (1894), 155-159.  doi: 10.1007/BF02418278.

[9]

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

[10]

T. Kato, Perturbation Theory for Linear Operators Classics in Mathematics, Springer Berlin Heidelberg, 1995.

[11]

P. Shmerkin, On the exceptional set for absolute continuity of Bernoulli convolutions, Geometric and Functional Analysis, 24 (2014), 946-958.  doi: 10.1007/s00039-014-0285-4.

[12]

J. Shohat and J. Tamarkin, The Problem of Moments Mathematical surveys and monographs, American Mathematical Society, 1943.

[13]

B. Solomyak, Notes on Bernoulli convolutions, in Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 1, vol. 72 of Proceedings of Symposia in Pure Mathematics, AMS, 2004,207-230.

[14]

B. Solomyak, Y. Peres and W. Schlag, Sixty years of Bernoulli convolutions, Progress in Probability, 46 (2000), 39-65. 

[15]

T. Tao, An Epsilon of Room I Graduate Studies in Mathematics, American Mathematical Society, 2010. doi: 10.1090/gsm/117.

Figure 1.  Eigenvalues of two matrix approximations of the Hutchinson operator for the Bernoulli convolution with $t=0.8.$ The matrix size, $N=499$ and 500, does not influence the leading eigenvalues $t^k, k=0,...,3$ while virtually all remaining eigenvalues are different. The circle has radius $\frac12 .$
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Figure 2.  Polynomial approximations $v_{t,n}$ of the Bernoulli convolution measure $\nu_t$ in the smooth case $t=0.8$ (left) and the more fractal case $t=0.6$ (right). In the left picture, one could not distinguish at this scale between the approximations of degree $n\geq 8$. The gray line shows a histogram with 2000 bins based on $2^{20}$ points generated by the 'chaos game' algorithm.
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Figure 3.  Above: leading left eigenvectors $e_0,e_1,e_2$ of $T_N$ for the Bernoulli convolution with $t=0.8$ and $N=500.$ Below: scaled eigenvector $e_3,$ integral of $e_2$ and iterated integral of $e_3.$ The latter two coincide with $e_1,$ up to a constant. Thus $e_1,e_2,e_3$ are the first 3 derivatives of the density $e_0$ of the self-similar measure.
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Figure 4.  $\beta=1/t=1.84$ was chosen near the Pisot parameter 1.8393... Below: the self-similar measure. Above: cumulative sum of second eigenvector. Although $\nu$ is not a differentiable function, the second eigenvector looks like a derivative of $\nu .$
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Figure 5.  Right eigenvectors are polynomials for $\lambda>\frac12 .$ First eigenvector with $|\lambda|<\frac12$ shown for comparison. Parameters as in Figure 3.
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