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

Abstract

The paper investigates 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, the geometric structure of the set of maximum points is described; in particular, it is shown that this set can be fractal.