Hölder exponents and the fractal structure of level sets of self-affine functions associated with the $Q_s$-representation of numbers

Abstract

We investigate a class of locally complicated self-affine functions defined in terms of the $Q_s$-representation of numbers. Local Hölder exponents are computed at points with given asymptotic frequencies of digits in their $Q_s$-representation. Conditions under which these functions have continuum level sets are also found. For self-affine functions satisfying additional conditions, we describe the geometric structure of the set of maximum points; in particular, we show that this set can be fractal.