An analogue of the Stefan-Sussmann Theorem on manifolds with boundary is proven for normal distributions. These distributions contain vectors transverse to the boundary along its entirety. Plain integral manifolds are not enough to integrate a normal
distribution; the next best integrals are so-called neat integral manifolds with boundary. The conditions on the distribution for this integrability is expressed in terms of adapted collars and integrability of a pulled-back distribution on the interior and on the boundary.
A construction is given for the recovery of a disjoint union of strictly convex smooth planar obstacles from travelling-time information. The obstacles are required to be such that no Euclidean line meets more than two of them.
We consider travelling times of billiard trajectories in the exterior of an obstacle K on a two-dimensional Riemannian manifold M. We prove that given two obstacles with almost the same travelling times, the generalised geodesic flows on the non-trap
ping parts of their respective phase-spaces will have a time-preserving conjugacy. Moreover, if M has non-positive sectional curvature we prove that if K and L are two obstacles with strictly convex boundaries and almost the same travelling times then K and L are identical.
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 simila
r 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 $t geq n.$
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].
It was proved in cite{NS1} that obstacles $K$ in $R^d$ that are finite disjoint unions of strictly convex domains with $C^3$ boundaries are uniquely determined by the travelling times of billiard trajectories in their exteriors and also by their so c
alled scattering length spectra. However the case $d = 2$ is not properly covered in cite{NS1}. In the present paper we give a separate different proof of the same result in the case $d = 2$.
In this work we study strong spectral properties of Ruelle transfer operators related to a large family of Gibbs measures for contact Anosov flows. The ultimate aim is to establish exponential decay of correlations for Holder observables with respect
to a very general class of Gibbs measures. The approach invented in 1997 by Dolgopyat cite{D1} and further developed in cite{St2} is substantially refined here, allowing to deal with much more general situations than before, although we still restrict ourselves to the uniformly hyperbolic case. A rather general procedure is established which produces the desired estimates whenever the Gibbs measure admits a Pesin set with exponentially small tails, that is a Pesin set whose preimages along the flow have measures decaying exponentially fast. We call such Gibbs measures regular. Recent results in cite{GSt} prove existence of such Pesin sets for hyperbolic diffeomorphisms and flows for a large variety of Gibbs measures determined by Holder continuous potentials. The strong spectral estimates for Ruelle operators and well-established techniques lead to exponential decay of correlations for Holder continuous observables, as well as to some other consequences such as: (a) existence of a non-zero analytic continuation of the Ruelle zeta function with a pole at the entropy in a vertical strip containing the entropy in its interior; (b) a Prime Orbit Theorem with an exponentially small error.
We prove that if two non-trapping obstacles in $mathbb{R}^n$ satisfy some rather weak non-degeneracy conditions and the scattering rays in their exteriors have (almost) the same travelling times or (almost) the same scattering length spectrum, then they coincide.
We prove a comprehensive version of the Ruelle-Perron-Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is t
hat the Holder constant of the function generating the operator appears only polynomially, not exponentially as in previous known estimates.
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.
