[Paper Review] $L^p$ spectrum and heat dynamics of locally symmetric spaces of higher rank

Lizhen Ji, Andreas Weber⋆|arXiv (Cornell University)|May 17, 2010

Advanced Algebra and Geometry33 references3 citations

TL;DR

This paper investigates the $L^p$ spectrum and dynamics of the heat semigroup on non-compact locally symmetric spaces of higher rank, generalizing results from rank one to higher rank settings. It shows that for $p \in (1,2)$, the $L^p$ heat semigroup exhibits chaotic behavior due to a continuous spectrum of eigenvalues, while for $p \geq 2$, such chaos does not occur, revealing a sharp phase transition at $p=2$. The analysis relies on Eisenstein series and spectral bounds in $L^p$ spaces.

ABSTRACT

The aim of this paper is to study the spectrum of the $L^p$ Laplacian and the dynamics of the $L^p$ heat semigroup on non-compact locally symmetric spaces of higher rank. Our work here generalizes previously obtained results in the setting of locally symmetric spaces of rank one to higher rank spaces. Similarly as in the rank one case, it turns out that the $L^p$ heat semigroup on $M$ has a certain chaotic behavior if $p\in(1,2)$ whereas for $p\geq 2$ such a chaotic behavior never occurs.

Motivation & Objective

• To extend $L^p$ spectral theory from rank one to higher rank locally symmetric spaces.
• To analyze the dynamics of the $L^p$ heat semigroup on non-compact, finite-volume locally symmetric spaces of higher rank.
• To determine under which $p$-values the $L^p$ heat semigroup exhibits chaotic behavior, particularly in relation to the spectrum of the $L^p$ Laplacian.
• To establish $L^p$-integrability of Eisenstein series for $p < 2$ and use them to identify continuous spectra of eigenvalues.
• To characterize the set of periods of the $L^p$ heat semigroup and relate them to the spectral structure on the imaginary axis.

Proposed method

• Derive an upper bound for Eisenstein series on $M = \Gamma \backslash X$ to establish their $L^p$-integrability for $p < 2$.
• Use the Helgason-Fourier transform and $L^p$ inversion formulas (Stanton-Tomas) to analyze the $L^p$ spectrum of the Laplacian.
• Define the parabolic region ${\cal P}_{M,p}(\rho_{\bf P})$ as the spectral set for $\Delta_{M,p}$, parameterized by $\rho_{\bf P}$ and $p$.
• Apply the theory of subspace chaos for $C_0$-semigroups, using the presence of eigenvalues on the imaginary axis to detect chaotic dynamics.
• Use the parametrization of the boundary of ${\cal P}_{M,p}(\rho_{\bf P})$ to compute the intersection with the imaginary axis and determine the set of periods.
• Leverage the fact that for $p \geq 2$, the point spectrum of $\Delta_{M,p}$ is discrete, while for $p \in (1,2)$, it contains a continuous open set in $\mathbb{C}$, leading to chaotic behavior.

Experimental results

Research questions

• RQ1 For which $p \in (1,\infty)$ does the $L^p$ heat semigroup on a higher rank locally symmetric space exhibit chaotic dynamics?
• RQ2 What is the structure of the $L^p$ spectrum of the Laplacian $\Delta_{M,p}$ on non-compact, finite-volume locally symmetric spaces of higher rank?
• RQ3 Under what conditions are Eisenstein series $E({\bf P}|\varphi,\Lambda)$ in $L^p(M)$ for $p < 2$?
• RQ4 How do the periods of the $L^p$ heat semigroup relate to the spectral properties of $\Delta_{M,p}$, particularly the intersection of the spectrum with the imaginary axis?
• RQ5 What is the critical threshold $c_p$ such that $e^{-t(\Delta_{M,p}-c)}$ becomes subspace chaotic for $c > c_p$ when $p \in (1,2)$?

Key findings

• For $p \in (1,2)$, the $L^p$ heat semigroup $e^{-t\Delta_{M,p}}$ exhibits subspace chaos, as the generator $\Delta_{M,p}$ has a continuous set of eigenvalues intersecting the imaginary axis for $c > c_p$.
• For $p \geq 2$, the $L^p$ heat semigroup is never subspace chaotic because the point spectrum of $\Delta_{M,p}$ is discrete and cannot intersect the imaginary axis in an infinite, non-discrete set.
• The critical threshold is $c_p = \frac{4||\rho_{\bf P}||^2}{p}\left(1 - \frac{1}{p}\right)$, above which $e^{-t(\Delta_{M,p}-c)}$ becomes chaotic for $p \in (1,2)$.
• The $L^p$ spectrum of $\Delta_{M,p}$ is contained in a parabolic region ${\cal P}_{M,p}'$, with the interior of the region containing eigenvalues for $p \in (1,2)$, but not for $p \geq 2$.
• For $p \in (1,2)$ and $c > c_p$, all but finitely many points in $[2\pi / r(c,p), \infty)$ are periods of the semigroup $e^{-t(\Delta_{M,p}-c)}$, where $r(c,p) = 2||\rho_{\bf P}||\left(1 - \frac{2}{p}\right)\sqrt{c - c_p}$.
• Eisenstein series $E({\bf P}|\varphi, \Lambda)$ are in $L^p(M)$ for $p < 2$ under suitable choices of $\Lambda$, enabling them to serve as eigenfunctions of $\Delta_{M,p}$.