2026/06/10 by Yin Cai, Xiang Fang, Hongdou Qu · 1 citation
#math.PR
We prove exact Fourier-dimension formulas for scalar dyadic Mandelbrot cascades pushed forward to fixed nondegenerate C2 embedded arcs and fixed nondegenerate C2 Jordan curves in \mathbb R2. Let W be in the minimal Kahane--Peyriere regime. For each fixed nondegenerate C2 embedded arc γ:[0,1]→\mathbb R2, the pushforward μγ of the interval cascade satisfies, almost surely on non-extinction, dim\mathrm F(μγ)=Aloc(W), where Aloc(W) = supq>1 max\ 0, (q-1-log2\mathbb E[Wq])/(q) \, with the q-term interpreted as 0 when \mathbb E[Wq]=∞. The analogous formula holds for scalar circle cascades pushed forward by fixed nondegenerate C2 Jordan curves γ:\mathbb T→\mathbb R2, with the pushforward denoted by μγ\mathbb T. This extends the scalar circle endpoint formula from the canonical circle to fixed parametrized arcs and Jordan curves. The main new issue beyond the canonical circle is the loss of the explicit trigonometric phase and, for arcs, the presence of endpoint stationary regimes. We prove the arc lower bound by a finite-r annular Fourier theorem based on an endpoint-safe phase decomposition, phase-bin coefficient estimates, predictable capping, complex Freedman concentration, and an r-tail compensator. The Jordan lower bound follows by first-generation dyadic cutting into two fixed arcs. The matching upper bounds use deterministic curved-support obstructions together with the scalar-circle minimum lower local-dimension theorem.