Speaker: Tuomas Sahlsten
Title: Fractal Uncertainty Principle without Markov structure
Abstract:
Fractal Uncertainty Principle (FUP) is a tool in harmonic analysis developed by
Dyatlov and Zahl in 2016 that roughly says an L^2 function cannot be simultaneously
localised near a fractal set in both position and frequency. When applied to trapped
sets arising from chaotic dynamics, using semiclassical methods, FUP implies novel
applications in mathematical physics and PDEs such as distribution of scattering
resonances, advances towards Quantum Unique Ergodicity conjecture by Rudnick and
Sarnak, control- and observability of the Schrödinger equation and exponential decay
for the solutions of the damped wave equation on Anosov surfaces.
FUP was proved for porous sets in R by Bourgain and Dyatlov in 2018 by adapting a
special case of the Beurling-Malliavin multiplier theorem, and recently (May 2023)
FUP was generalised to R^d in a breakthrough work by A. Cohen for line porous sets
by introducing a version of Beurling-Malliavin multiplier theorem in R^d. Other ways
to prove FUP, that have stronger or more quantitative bounds in some cases, involve
Fourier dimension and discretised sum-product techniques (Bourgain-Dyatlov ’17),
Dolgopyat’s method in hyperbolic dynamics (Dyatlov-Jin ’17) or additive energy
estimates (Dyatlov-Zahl ’16, Cladek-Tao ’20).
Past works to prove FUP require Markov structure from the underlying dynamics (to
establish porosity), but with many natural chaotic systems such as fractals arising
from random matrix products or higher dimensional horseshoes/solenoids, there may
be complicated overlaps in the system, so it has been unclear if FUP can be
established here and what assumptions are needed. In this talk I will introduce a
method we developed with Simon Baker (Loughborough) based on disintegrating complex
transfer operators using random models of Dolgopyat operators, which implies FUP for
a wide range of hyperbolic dynamical systems without any geometric Markov structure.