The X-ray transform on a general family of curves on Finsler surfaces
Abstract
We consider a general family of curves
Γ
on a compact oriented Finsler surface
(M,F)
with boundary
∂M
. Let
φ∈
C
∞
(M)
and
ω
a smooth 1-form on
M
. We show that
∫
γ(t)
{φ(γ(t))+
ω
γ(t)
(
γ
˙
(t))}dt=0
holds for every
γ∈Γ
whose endpoints belong to
∂M
,
γ(a)∈∂M
,
γ(b)∈∂M
if and only if
φ=0
and
ω
is exact.
Similar results were proved when
M
is closed and some additional conditions on Gaussian curvature are imposed.
We also study the cohomological equations of Anosov generelized thermostats on a closed Finsler surface. Finally, we gave conditions when thermostat is of Anosov type.