,

Resonances of a Transfer operator

This web page gives numerical illustrations (movies) related  to the paper [Fau09].

• Spectrum of resonances
• Evolution of densities by the transfer operator
• Fractal aspect of the trapped set
• Quantum effective evolution
  

Remark: all the movies are "gif animated" and you can download them and see them with xanim (or similar), which gives a better control for the view and the speed.
Credits: numerical computations are done in C++, using the library GSL and ROOT from cern.

Spectrum of resonances

For $\nu\in\mathbb{R}$, and $\varphi\in C^{\infty}\left(S^{1}\right)$, let

$$\left(\hat{F}_{\nu}\varphi\right)\left(x\right):=\varphi\left(2x\... ...\pi}\cos\left(2\pi x\right)\right),\qquad x\in S^{1}\equiv\mathbb{R}/\mathbb{Z}$$

• The operator $\hat{F}_{\nu}$ has a discrete spectrum in a Sobolev spaces $H^{-m}$, with $m\ll0$.
• The following movie shows the eigenvalues of $\hat{F}_{\nu}$, called Ruelle-Pollicott resonances as $\nu$ runs from $0\rightarrow80$. (Some explanations are given the paper [Fau09]).

To see the movie again, clik on "reload" on your web-browser.
(corresponds to figure 2 page 7  in [Fau09])

Frederic Faure, UJF Grenoble