The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers

Abstract

In this paper, we give a systematical study of the local structures and fractal indices of the limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers. For a given Pisot number in the interval ( 1 , 2 ) , we construct a finite family of non-negative matrices (maybe non-square), such that the corresponding fractal indices can be re-expressed as some limits in terms of products of these non-negative matrices. We are especially interested in the case that the associated Pisot number is a simple Pisot number, i.e., the unique positive root of the polynomial x k - x k - 1 - … - x - 1 ( k = 2 , 3 , … ). In this case, the corresponding products of matrices can be decomposed into the products of scalars, based on which the precise formulas of fractal indices, as well as the multifractal formalism, are obtained.