%0 Journal Article %1 MR534172 %A Ruelle, D. %D 1979 %J Adv. in Math. %K Lyapunov_exponent linear_SDE random_matrices %N 1 %P 68--80 %T Analycity properties of the characteristic exponents of random matrix products %U http://dx.doi.org/10.1016/0001-8708(79)90029-X %V 32 %X The main result of this interesting article is the following theorem. Let $Ømega$ be a compact space and $\tau$ a homeomorphism of $Ømega$. Let $E$ be a continuous vector bundle over $Ømega$ with fibers $E(x)$ isomorphic to $R^m$, $xınØmega$, where $m$ is a fixed positive integer. Denote by $O_x$ the origin of the fiber $E(x)$. A continuous vector bundle map $T$ is a continuous map $TEE$ such that $T(E(x))E(\tau(x))$ for each $xınØmega$ and such that the restriction of $T$ to $E(x)$ is linear. An important example of this situation arises when $Ømega$ is a compact differentiable manifold, $\tau$ is a diffeomorphism of $Ømega$, $E$ is the tangent bundle of $Ømega$ and $T$ is the differential of $\tau$. Denote by $T$ the Banach space of continuous vector bundle maps $T$ on $E$ with the sup-norm (where we have fixed norms on each $E(x)$ varying continuously with $xınØmega$). Now define an open subset $P$ of $T$ consisting of those $TınT$ for which the following condition holds. For each $xınØmega$ there is a proper closed convex cone $C(x)E(x)$ such that for each $xınØmega$, (1) $C(x)\cup(-C(x))$ depends continuously on $x$; (2) $TC(x)\subseteqØ_\tau(x)\\cupint\,C(\tau(x))\cupint(-C(\tau(x)))$; and (3) the apex of $C(x)$ is $O_x$. Theorem: For each $\tau$-invariant probability measure $\rho$ on $Ømega$, the function (highest characteristic exponent) $\chi(T,\rho)=łim_n\rightarrowınftyn^-1ınt\rho(dx)·log\|T(\tau^n-1x)T(x)\|=n^-1ınt\rho(dx)log\|T(\tau^n-1x)T(x)\|$ is real analytic for $TınP$. The author also considers exterior products and obtains analogous information on the sums of the highest $p$ characteristic exponents of $T$, as well as the case of complex bundles.

@article{MR534172, abstract = {The main result of this interesting article is the following theorem. Let $\Omega$ be a compact space and $\tau$ a homeomorphism of $\Omega$. Let $E$ be a continuous vector bundle over $\Omega$ with fibers $E(x)$ isomorphic to ${\bf R}^m$, $x\in\Omega$, where $m$ is a fixed positive integer. Denote by $O_x$ the origin of the fiber $E(x)$. A continuous vector bundle map $T$ is a continuous map $T\colon E\rightarrow E$ such that $T(E(x))\subseteq E(\tau(x))$ for each $x\in\Omega$ and such that the restriction of $T$ to $E(x)$ is linear. An important example of this situation arises when $\Omega$ is a compact differentiable manifold, $\tau$ is a diffeomorphism of $\Omega$, $E$ is the tangent bundle of $\Omega$ and $T$ is the differential of $\tau$. Denote by $\scr T$ the Banach space of continuous vector bundle maps $T$ on $E$ with the sup-norm (where we have fixed norms on each $E(x)$ varying continuously with $x\in\Omega$). Now define an open subset $\scr P$ of $\scr T$ consisting of those $T\in\scr T$ for which the following condition holds. For each $x\in\Omega$ there is a proper closed convex cone $C(x)\subseteq E(x)$ such that for each $x\in\Omega$, (1) $C(x)\cup(-C(x))$ depends continuously on $x$; (2) $TC(x)\subseteq\{O_{\tau(x)}\}\cup\text{int}\,C(\tau(x))\cup\text{int}(-C(\tau(x)))$; and (3) the apex of $C(x)$ is $O_x$. Theorem: For each $\tau$-invariant probability measure $\rho$ on $\Omega$, the function (highest characteristic exponent) $\chi(T,\rho)=\lim_{n\rightarrow\infty}n^{-1}\int\rho(dx)·log\|T(\tau^{n-1}x)\cdots T(x)\|=\inf n^{-1}\int\rho(dx)log\|T(\tau^{n-1}x)\cdots T(x)\|$ is real analytic for $T\in\scr P$. The author also considers exterior products and obtains analogous information on the sums of the highest $p$ characteristic exponents of $T$, as well as the case of complex bundles. }, added-at = {2009-10-29T21:41:52.000+0100}, author = {Ruelle, D.}, biburl = {https://www.bibsonomy.org/bibtex/2a1cb313e2cb3269020f136778dffcbf2/peter.ralph}, coden = {ADMTA4}, description = {Analycity properties of the characteristic exponents of random matrix products}, fjournal = {Advances in Mathematics}, interhash = {6a0ea6dd86331e0817ba36989143b556}, intrahash = {a1cb313e2cb3269020f136778dffcbf2}, issn = {0001-8708}, journal = {Adv. in Math.}, keywords = {Lyapunov_exponent linear_SDE random_matrices}, mrclass = {58F15 (28D99 82A15)}, mrnumber = {MR534172 (80e:58035)}, mrreviewer = {Michael Keane}, number = 1, pages = {68--80}, timestamp = {2009-10-29T21:41:52.000+0100}, title = {Analycity properties of the characteristic exponents of random matrix products}, url = {http://dx.doi.org/10.1016/0001-8708(79)90029-X}, volume = 32, year = 1979 }