Luchezar Stoyanov

Spectra of Ruelle transfer operators for Axiom A flows

For Axiom A flows on basic sets satisfying certain additional conditions we prove strong spectral estimates for Ruelle transfer operators similar to these of Dolgopyat [D2] for geodesic flows on compact surfaces (for general potentials) and transitive Anosov flows on compact manifolds with C jointly non-integrable horocycle foliations (for the Sinai-Bowen-Ruelle potential). Here we deal with general potentials. As is now well known, such results have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions and partial differential operators, closed orbit counting functions, decay of correlations for Holder continuous potentials. The situation considered here is much more difficult than that in [D2] since in general the local geometry of the basic set, especially in higher dimensions, can be rather complicated. In fact, in higher dimensions even the Anosov case for general potentials presents substantial difficulties.

Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function

Let s0 < 0 be the abscissa of absolute convergence of the dynamical zeta function Z(s) for several disjoint strictly convex compact obstacles Ki ⊂ R N , i = 1, . . . , κ0, κ0 ≥ 3, and let Rχ(z) = χ(−∆D − z )χ, χ ∈ C 0 (R N ), be the cut-off resolvent of the Dirichlet Laplacian −∆D in Ω = RN \ ∪0 i=1Ki. We prove that there exists σ1 < s0 such that the cut-off resolvent Rχ(z) has an analytic continuation for Im(z) < −σ1, |Re(z)| ≥ J1 > 0.

Non-integrability of open billiard flows and Dolgopyat-type estimates

We consider open billiard flows in IR and show that the standard symplectic form dα in IR satisfies a specific non-integrability condition over their non-wandering sets Λ. This allows to use the main result in [St3] and obtain Dolgopyat type estimates for spectra of Ruelle transfer operators under simpler conditions. We also describe a class of open billiard flows in IR (n ≥ 3) satisfying a certain pinching condition, which in turn implies that the (un)stable laminations over the non-wandering set are C1.

Correlations for pairs of periodic trajectories for open billiards

In this paper we prove two asymptotic estimates for pairs of closed trajectories for open billiards similar to those established by Pollicott and Sharp (2006 Invent. Math. 163 10–24) for closed geodesics on negatively curved compact surfaces. The first of these estimates holds for general open billiards in any dimension. The more intricate second estimate is established for open billiards satisfying the so-called Dolgopyat type estimates. This class of billiards includes all open billiards in the plane and open billiards in (N ≥ 3) satisfying some additional conditions.

Hyperbolic Dynamics, Fluctuations and Large Deviations

We present, in the simplest possible form, the so called {\em martingale problem} strategy to establish limit theorems. The presentation is specially adapted to problems arising in partially hyperbolic dynamical systems. We will discuss a simple partially hyperbolic example with fast-slow variables and use the martingale method to prove an averaging theorem and study fluctuations from the average. The emphasis is on ideas rather than on results. Also, no effort whatsoever is done to review the vast literature of the field.

Sharp large deviations for some hyperbolic systems

We prove a sharp large deviation principle concerning intervals shrinking with sub-exponential speed for certain models involving the Poincare map related to a Markov family for an Axiom A flow restricted to a basic set

Regular decay of ball diameters and spectra of Ruelle operators for contact Anosov flows

\Lambda

Singularities of the Scattering Kernel Related to Trapping Rays

satisfying some additional regularity assumptions.

Tangent bundles to regular basic sets in hyperbolic dynamics

For Anosov flows on compact Riemann manifolds we study the rate of decay along the flow of diameters of balls

Lens rigidity in scattering by non-trapping obstacles

B^s(x,\ep)