Pointwise convergence over fractals for dispersive equations with homogeneous symbol

Abstract

We study the fractal pointwise convergence for the equation $i\hbar\partial_tu + P(D)u = 0$, where the symbol $P$ is real, homogeneous and non-singular. We prove that for initial data $f\in H^s(\mathbb{R}^n)$ with $s>(n-α+1)/2$ the solution $u$ converges to $f$ $\mathcal{H}^α$-a.e, where $\mathcal{H}^α$ is the $α$-dimensional Hausdorff measure. We improve upon this result depending on the dispersive strength of $P$. On the other hand, for a family of polynomials $P$ and given $α$, we exploit a Talbot-like effect to construct initial data whose solutions $u$ diverge in sets of Hausdorff dimension $α$. To compute the dimension of the sets of divergence, we adopt the Mass Transference Principle from Diophantine approximation. We also construct counterexamples for quadratic symbols like the saddle to show that our positive results are sometimes best possible.