Spectrum of an operator arising in a dynamical problem - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T04:19:10Z http://mathoverflow.net/feeds/question/43050 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43050/spectrum-of-an-operator-arising-in-a-dynamical-problem Spectrum of an operator arising in a dynamical problem Benoît Kloeckner 2010-10-21T13:38:33Z 2010-10-21T16:11:17Z <p>(Question edited according to Denis Serre comment).</p> <p>While studying the action of dilating map of the circle on probability measures, I ran across the following operator: $$\mathcal{K}^* : L^2_0(\mu)\to L^2_0(\mu)\quad u\mapsto \varphi' u\circ \varphi$$ where $\varphi$ is a $C^2$ dilating map of $S^1$, of degree $d$ say, $\mu$ is the unique absolutely continuous measure fixed by $\varphi$ (it has $C^1$ density) and $L^2_0$ means the subset of $L^2$ of functions having zero mean <em>with respect to Lebesgue measure</em>. </p> <p>In the model case $\varphi(x)=dx \mod 1$, Fourier analysis shows easily that the spectrum is the closed disc of radius $d$. In fact every $\lambda$ such that <code>$|\lambda| &lt; d$</code> is an eigenvalue of the adjoint operator $\mathcal{K}$. In the general case it seems easier to work with $\mathcal{K}^*$ since the expression of $\mathcal{K}$ involves $\mu$.</p> <p>My question is the following: what can be said in general on the spectrum of $\mathcal{K}^*$? references are welcome if they exist (note that this question led me far away from my usual domain of mathematics, so that this might be a dumb or very easy question -- or might be tough).</p>