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Let $P^1$ denote one-dimensional real projective space, and for each $A \in GL(2,\mathbb{R})$ let $\overline{A}$ denote the homeomorphism of $P^1$ induced by $A$. I am currently reading a paper which makes use of transfer operators of the form $$(Lf)(x):=\int f\left(\overline{A}x\right)d\mathbb{P}(A)$$ where $\mathbb{P}$ is a probability measure on $GL(2,\mathbb{R})$ such that the support of $\mathbb{P}$ is not contained in a compact subgroup and does not simultaneously preserve a finite union of one-dimensional subspaces. Here the operator $L$ acts on suitable Banach spaces of continuous functions $f \colon P^1 \to \mathbb{R}$, and the $N^{\mathrm{th}}$ iterate of $L$ may be viewed as describing the "average effect" of applying $N$ independent matrices selected according to the distribution $\mathbb{P}$. The authors of the paper which I am reading note that $L$ is quasicompact (that is, has spectral radius strictly larger than its essential spectral radius) on $C^\varepsilon(P^1)$ when $\varepsilon>0$ is sufficiently small, citing the book "Products of Random Matrices with Applications to Schrödinger Operators" by Bougerol and Lacroix (Theorem V.2.5 on page 106). I am not currently very familiar with the literature on this type of operator and am curious to know whether these operators are known to have good properties on Banach spaces of more regular functions:

Are there circumstances in which operators of the above form are known to be quasicompact on $C^1(P^1)$, or on other spaces of everywhere-differentiable functions $f$?

The proof given by Bougerol and Lacroix (and the earlier proof by Le Page on which it is based) for the case of the action of $L$ on $C^\varepsilon(P^1)$ does not seem to me to adapt well to the case of $C^\alpha(P^1)$ with arbitrary $\alpha \in (0,1)$, but I am wondering if this question has been studied elsewhere. In the cases of interest to me the measure $\mathbb{P}$ has the following properties: $\mathbb{P}$ is a convex combination of countably many atoms, the support of $\mathbb{P}$ generates a noncompact subgroup of $SL(2,\mathbb{R})$ which does not simultaneously preserve a finite union of one-dimensional subspaces of $\mathbb{R}^2$, and $\mathbb{P}(\{A \colon \|A\| \geq t\})$ decreases exponentially as $t \to \infty$. In particular the system of random matrices has two distinct Lyapunov exponents (see e.g. Bougerol & Lacroix p.27-30).

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Interesting question. Do you know if in your case the Lyapunov exponents of the action are distinct? – Anthony Quas May 29 '13 at 18:56
In the cases I am considering the support of $\mathbb{P}$ lies in $SL(2,\mathbb{R})$ and does not simultaneously preserve a finite union of linear subspaces of $\mathbb{R}^2$, so the Lyapunov exponents are distinct (Bougerol & Lacroix p.27-30). If they did preserve such a set then this would seem to significantly obstruct uniform convergence of $L^nf$, which makes me doubt that $L$ has nice spectral properties in such a context. – Ian Morris May 29 '13 at 21:30
Am I missing something here? If the matrices are all rotations, there's no finite union that's preserved, but still both exponents are 0? – Anthony Quas May 29 '13 at 23:52
I've edited the question to make my standing hypotheses a little clearer. In the cases of interest to me the support of $\mathbb{P}$ is not contained in a compact group, which eliminates the case of irrational rotations. – Ian Morris May 30 '13 at 9:42

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