Extreme value laws for fractal intensity functions in dynamical systems: Minkowski analysis

Typically, in the dynamical theory of extremal events, the function that gauges the intensity of a phenomenon is assumed to be convex and maximal, or singular, at a single, or at most a finite collection of points in phase–space. In this paper we generalize this situation to fractal landscapes, i.e. intensity functions characterized by an uncountable set of singularities, located on a Cantor set. This reveals the dynamical r^ole of classical quantities like the Minkowski dimension and content, whose definition we extend to account for singular continuous invariant measures. We also introduce the concept of extremely rare event, quantified by non–standard Minkowski constants and we study its consequences to extreme value statistics. Limit laws are derived from formal calculations and are verified by numerical experiments.

💡 Research Summary

This paper presents a significant extension of dynamical extreme value theory by analyzing intensity functions with fractal structures. Traditional approaches typically assume that the intensity function (which measures the severity of an event) attains its maximum or singularity at a single point or a finite set of points in phase space. In contrast, this work generalizes the theory to “fractal landscapes,” where the intensity function possesses an uncountable set of singularities located on a Cantor set.

The study reveals the dynamical role of classical fractal quantities like the Minkowski dimension and Minkowski content. Furthermore, it introduces the concept of “extremely rare events,” characterized by non-standard Minkowski constants, and explores their implications for extreme value statistics. The proposed limit laws are derived from formal calculations and validated through extensive numerical experiments.

The paper is structured as follows: After a brief introduction to dynamical extreme value theory and the numerical setup, the authors introduce a toy model called the “Cantor Ladder.” This is a hierarchical intensity function defined on the gaps of the ternary Cantor set, with values increasing as the gap size decreases. Using this model, they demonstrate that extreme value laws hold for both chaotic (asymmetric tent map) and non-chaotic, merely ergodic (irrational rotation) dynamical systems, even in the absence of correlation decay.

The Cantor Ladder model is then connected to a more general setting where the intensity function is defined as a function of the distance to a compact set K (the Cantor set). When the invariant measure of the dynamical system is Lebesgue measure, the extreme value statistics are shown to depend crucially on the Minkowski dimension D and the Minkowski content C of the set K. This provides a dynamical interpretation for these fundamental fractal quantities.

The framework is further generalized to systems with an arbitrary invariant measure μ. The authors define generalized Minkowski dimension and content relative to μ. When these quantities exist, they govern the scaling laws for extremes. A particularly novel scenario arises when considering singular continuous invariant measures, such as the Minkowski question-mark function. In this case, the standard scaling law breaks down, leading to “extremely rare events” characterized by non-standard Minkowski constants. This behavior is linked to the non-analytic properties of the question-mark function at rational points.

Finally, the paper investigates the case where the Hausdorff and Minkowski dimensions of the set K differ. A new scaling law for extreme values is derived and numerically verified, highlighting how the fine structure of a fractal set influences extreme event statistics.

In summary, this work bridges dynamical systems theory, fractal geometry, and extreme value statistics. It establishes that fractal properties of the intensity function and the invariant measure fundamentally shape the laws governing extreme events, opening new avenues for modeling complex natural phenomena with hierarchical or multi-scale structures. The methodological approach leans towards experimental mathematics, using heuristic arguments and numerical verification to propose new laws that await rigorous proof.