Boundary cases for β and intermittency

Investigate the behavior of multiplicative cascades satisfying the hierarchical symmetry axiom A1 in the boundary cases β → 0 (maximal intermittency) and β → 1 (vanishing intermittency), beyond the current assumptions β ∈ (0,1) and C > 0.

Background

The main results assume nontrivial intermittency (C > 0) and contraction parameter β strictly between 0 and 1. Under these conditions, A1 determines a unique log-Poisson law and yields stability.

The authors highlight that the extreme limits of β—approaching 0 or 1—are not covered by the current analysis and merit separate study to understand how characterization, classification, or stability may degenerate or change in these regimes.

References

Several directions remain open. The present results assume nontrivial intermittency ($C > 0$) and $\beta \in (0,1)$. The boundary cases $\beta \to 0$ (maximal intermittency) and $\beta \to 1$ (vanishing intermittency) deserve separate analysis.

— Hierarchical symmetry selects log-Poisson cascades: classification, uniqueness, and stability  (2604.01632 - Freeburg, 2 Apr 2026) in Section 6, Concluding remarks