Bounds on Eigenfunctions of Quantum Cat Maps

Abstract

We study $\ell^\infty$ norms of $\ell^2$-normalized eigenfunctions of quantum cat maps. For maps with short quantum periods (constructed by Bonechi and de Bièvre), we show that there exists a sequence of eigenfunctions $u$ with $\|u\|_{\infty}\gtrsim (\log N)^{-1/2}$. For general eigenfunctions we show the upper bound $\|u\|_\infty\lesssim (\log N)^{-1/2}$. Here the semiclassical parameter is $h=(2πN)^{-1}$. Our upper bound is analogous to the one proved by Bérard for compact Riemannian manifolds without conjugate points.

Figures (5)

• Figure 1: The plot of the maximal $\ell^\infty$-norm of an $\ell^2$-normalized eigenfunction of $M_{N,0}$ where $M_{N,0}$ is associated with $A=2312$. The lower and upper bounds from Theorems \ref{['thm:lowerbound']} and \ref{['thm:upperbound']}, $(2\log_\lambda N)^{-1/2}$ and $(\log_\lambda N)^{-1/2}$, respectively, give the solid red lines. The dotted blue line is the trivial lower bound $N^{-1/2}$. Finally, the large red dots correspond to the sequence $N_k$ from Theorem \ref{['thm:lowerbound']}.
• Figure 2: The plots of a maximal $\ell^\infty$-norm, $\ell^2$-normalized eigenfunction of $M_{N,0}$, where $M_{N,0}$ corresponds to $A=2312$. Specifically, each plot point corresponds to the absolute value of the $i$th coordinate of the eigenfunction for $0 \leq i \leq N-1$. Note that $N=2911$ is an element of the sequence $N_k$ in Corollary \ref{['lem:quantumperiod']}, while $N=991$ is not.
• Figure 3: In each of the two columns, the left image shows the Wigner function for $M^j g$. This particular Gaussian is centered at $(1/2, 1/2)$, a fixed point for $A$. The right images in each of the columns shows the Wigner function for $M^j e^0_{N/2}$. Halfway through the period, $W_{M^je^0_{N/2}}(x)$ becomes less chaotic, something that does not happen for $W_{M^j g}(x)$.
• Figure 4: The left image shows the Wigner function for the eigenfunction \ref{['eq:gaussian']}, where the Gaussian is centered at $(1/2, 1/2)$, a fixed point. The right image shows the Wigner function for the eigenfunction \ref{['eq:delta']}. Notice how the left image illustrates localization at $(1/2, 1/2)$, while the right image does not. We conjecture that the limit of eigenfunctions \ref{['eq:delta']} equidistributes in the sense of semiclassical measures.
• Figure 5: The plot of $\|M_{N,0}^j\|_{\ell^1 \rightarrow \ell^\infty}$ for $0 \leq j \leq 50$ and several values of $N$. $M_{N,0}$ corresponds to $A=2312$. We also plot the upper bound $\sqrt{|b|/N}$ for $N=855$.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Lemma 3: dyatlov2021semiclassical, Lemma 2.5
• Lemma 4
• Theorem 5: Bonechi-DeBievre2000_Article_ExponentialMixingAndTimeScales, Prop. 11
• proof
• Corollary 6
• proof
• Proposition 7
• proof