# American Institute of Mathematical Sciences

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

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

 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.

Citation: Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198
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##### References:
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 $\frac12 .$
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.
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.
$\beta=1/t=1.84$ 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 $\nu .$
Right eigenvectors are polynomials for $\lambda>\frac12 .$ First eigenvector with $|\lambda|<\frac12$ shown for comparison. Parameters as in Figure 3.
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