The Minkowski measure and multifractal anomalies in invariant measures of parabolic dynamic systems

Abstract

The author has recently shown that an important singular non-random measure defined in 1900 by Hermann Minkowski is multifractal and has the characteristic that a is almost surely infinite. Hence α_max = ∞ and its f(α) distribution has no descending right-side corresponding to a decreasing f(α). Its being left-sided creates many very interesting complications. Denjoy observed in 1932 that this Minkowski measure is the restriction to [0, 1] of the attractor measure for the dynamical system on the line based on the maps (Math) and (Math). This paper points out that it follows from Denjoy’s observation that the new “multifractal anomalies” due to the left-sidedness of f(α) extend to the invariant measures of certain dynamical systems.

The author’s original approach to multifractals, based on the distribution of the coarse Hölder α, also injects approximate measures µ_ε(dt) that have been “coarsened” by replacing the continuous t by multiples of ε > 0. This theory therefore involves a sequence of observable approximant functions f _ε(α); their graphs are not left-sided.