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Register a new userSharp large deviations for hyperbolic flows
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For hyperbolic flows $varphi_t$ we examine the Gibbs measure of points $w$ for which $$int_0^T G(varphi_t w) dt - a T in (- e^{-epsilon n}, e^{- epsilon n})$$ as $n to infty$ and $T geq n$, provided $epsilon > 0$ is sufficiently small. This is similar to local central limit theorems. The fact that the interval $(- e^{-epsilon n}, e^{- epsilon n})$ is exponentially shrinking as $n to infty$ leads to several difficulties. Under some geometric assumptions we establish a sharp large deviation result with leading term $C(a) epsilon_n e^{gamma(a) T}$ and rate function $gamma(a) leq 0.$ The proof is based on the spectral estimates for the iterations of the Ruelle operators with two complex parameters and on a new Tauberian theorem for sequence of functions $g_n(t)$ having an asymptotic as $ n to infty$ and $t geq n.$
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We obtain spectral estimates for the iterations of Ruelle operator $L_{f + (a + i b)tau + (c + i d) g}$ with two complex parameters and H{o}lder functions $f,: g$ generalizing the case $Pr(f) =0$ studied in [PeS2]. As an application we prove a sharp large deviation theorem concerning exponentially shrinking intervals which improves the result in [PeS1].
We give an exponential upper bound on the probabilitywith which the denominator of the $n$th convergent in the regular continued fraction expansion stays away from the mean $frac{npi^2}{12log2}$. The exponential rate is best possible, given by an analytic function related to the dimension spectrum of Lyapunov exponents for the Gauss transformation.
Given Holder continuous functions $f$ and $psi$ on a sub-shift of finite type $Sigma_A^{+}$ such that $psi$ is not cohomologous to a constant, the classical large deviation principle holds (cite{OP}, cite{Kif}, cite{Y}) with a rate function $I_psigeq 0$ such that $I_psi (p) = 0$ iff $p = int psi , d mu$, where $mu = mu_f$ is the equilibrium state of $f$. In this paper we derive a uniform estimate from below for $I_psi$ for $p$ outside an interval containing $tilde{psi} = int psi , dmu$, which depends only on the sub-shift, the function $f$, the norm $|psi|_infty$, the Holder constant of $psi$ and the integral $tilde{psi}$. Similar results can be derived in the same way e.g. for Axiom A diffeomorphisms on basic sets.
Let (X,T) be a dynamical system, where X is a compact metric space and T a continuous onto map. For weak Gibbs measures we prove large deviations estimates.
In this paper two independent and unitarily invariant projection matrices P(N) and Q(N) are considered and the large deviation is proven for the eigenvalue density of all polynomials of them as the matrix size $N$ converges to infinity. The result is formulated on the tracial state space $TS({cal A})$ of the universal $C^*$-algebra ${cal A}$ generated by two selfadjoint projections. The random pair $(P(N),Q(N))$ determines a random tracial state $tau_N in TS({cal A})$ and $tau_N$ satisfies the large deviation. The rate function is in close connection with Voiculescus free entropy defined for pairs of projections.
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