A functional limit theorem for a dynamical system with an observable maximised on a Cantor set

TL;DR

The paper tackles extreme-value behavior for heavy-tailed observables maximized on a dynamically defined Cantor set within a uniformly expanding interval map. It develops decorated point processes and enriched functional limit theorems (FLT) in the FFT25 framework, proving convergence of Rare Events Point Processes (REPP) to a decorated Poisson process and convergence of normalized sums $S_n(t)$ to a decorated $\alpha$-stable Lévy process $V$ in the enriched space $F'([0,1])$. The extremal index $\theta$ is computed from the Cantor geometry (with $\theta=1-\lambda$ and, for the middle-$\frac{1}{3}$ Cantor set, $\theta=\tfrac{1}{3}$), and explicit decorations $e_V^{s}$ describe cluster excursions around jumps. The results hold for $0<\alpha<1$ and subject to small-jumps and moment conditions for $1<\alpha<2$, providing a refined probabilistic description of extreme clustering on fractal attractors with potential climate-risk applications. Overall, the work extends the enriched limit theory to fractal maximal sets, delivering precise asymptotics for both point-process decorations and sums of heavy-tailed observables in dynamical systems.

Abstract

We consider heavy-tailed observables maximised on a dynamically defined Cantor set and prove convergence of the associated point processes as well as functional limit theorems. The Cantor structure, and its connection to the dynamics, causes clustering of large observations: this is captured in the `decorations' on our point processes and functional limits, an application of the theory developed in a paper by the latter three authors.

Figures (2)

• Figure 1: Graph of the function $\psi$ for the ternary Cantor set.
• Figure 2: Graph of the the distribution function of $\psi$ for the ternary Cantor set.

Theorems & Definitions (24)

• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Lemma 2.4
• proof
• Definition 3.1
• Remark 3.2
• Theorem 4.1
• Theorem 4.2