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Say I have a map ${\bf X}_n = {\bf F}({\bf X}_{n-1})$, ${\bf X_n} \in \mathbb{R}^N$. The Frobenius-Perron operator $L$ transfers the probability measure in phase space according to the dynamics defined by the map, ${\bf F}$. That is, $L \mu_n = \mu_{n+1}$, where $\mu_n$ is the probability distribution in phase space at time $n$. $\mu_0$ is the initial distribution.

We know that the map has an invariant measure $\mu$. In other words, $L$ has an eigenvalue 1 with eigenfunction $\mu$. My question is: under what conditions, (on ${\bf F}$ and $L$) will $L$ have an at most countably infinite set of eigenvalues? I know that compact operators on separable Hilbert spaces can be shown to have at most countably infinite set of eigenvalues but I have never studied the space of measures formally and am not sure how to think about this.

Thank you very much for your time!

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Let's play on a compact manifold, e.g. $M = \mathbb T^d$, the $d$-dimensional torus.

Let's also play with slightly nicer measures, as in some sense the class of Borel measures is quite large. In applications, one usually works with a subset of suitable measures, e.g. absolutely continuous measures with some regularity assumption on the density.

Let's assume that $F$ is a local diffeomorphism, so that $\mu_1, \mu_2, \cdots$ are absolutely continuous if $\mu_0$ is. Let $\rho_n$ be the density of $\mu_n$. Then, $\rho_{n+1}$ is given as $\mathcal L \rho_n = \rho_{n+1}$, where $\mathcal L$ is the (Ruelle) transfer operator: $$ \mathcal L \rho(x) = \sum_{y \in F^{-1} x} \frac{\rho(y)}{|\det dF_y|} \, . $$ (you should check this using the change of variables formula!)

When $F$ is expanding ($\| dF^n_x v\| \geq C \lambda^n$ for some $C > 0, \lambda > 1$ and at all points $x \in M$), $F$ spreads mass out and so the transfer operator $\mathcal L$ has a smoothing effect on densities. Formally, one often works with the space $\operatorname{Lip}$ of Lipschitz-continuous functions on $M$. One can show, then, that regarded as an operator on $\operatorname{Lip}$ that $\mathcal L$ is quasicompact, i.e., there exists some $\gamma \in [0,1)$ such that the spectrum of $\mathcal L$ outside of $\{ z \in \mathbb C : |z| \leq \gamma \}$ consists of finitely many eigenvalues (sometimes called Ruelle-Pollicot resonances).

I should note: quasicompactness of the transfer operator can be used to obtain exponential decay of correlations: this is perhaps the most significant application of these ideas.

For compactness, see the paper of Grundlach and Latushkin, who obtain an explicit formula for the radius of the essential spectrum of $\mathcal L$ in the above setting with respect to successively finer spaces of densities.

Lastly: the expanding case is the simplest to describe. Roughly speaking, $\mathcal L$ has a tendency to 'pile up' mass along directions where $dF$ is contracting (to see this in action: suppose $F$ has an attracting fixed point somewhere and watch mass accumulate to a Dirac delta measure at that fixed point!). As a result, for more general $F$ one usually works with specialized Banach spaces of measures. See, for example, the book Decay of Correlations by Baladi.

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    $\begingroup$ If you're interested, Vaughn Climenhaga has a few blog posts on spectral gaps for transfer operators: vaughnclimenhaga.wordpress.com/2013/01/30/… $\endgroup$ – A Blumenthal Mar 11 '17 at 17:55
  • $\begingroup$ Thank you very, very much! I asked the question with decay of correlations in mind - so your answer was more than what I was hoping for. Thanks also for the references - the book by Baladi looks very illuminating! I have a fundamental question: is the joint density defined when we have an SRB measure on the attractor? Since the SRB measure has a density only along unstable directions, can we still use the equation for the transfer operator that you have written (I understand how it is derived when we have a density). $\endgroup$ – rivendell Mar 11 '17 at 18:36
  • $\begingroup$ ^I am not specifically looking at diffeomorphisms. Let's say I am interested in any system that has an invariant measure that can be classified as an SRB measure. $\endgroup$ – rivendell Mar 11 '17 at 20:11

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