  Our paper is devoted to the study of the holonomy groups of Finsler surfaces
using the methods of infinite dimensional Lie theory. The notion of
infinitesimal holonomy algebra will be introduced, by the smallest Lie algebra
of vector fields on an indicatrix, containing the curvature vector fields and
their horizontal covariant derivatives with respect to the Berwald connection.
We obtain that the topological closure of the holonomy group contains the
exponential image of any tangent Lie algebra of the holonomy group. A class of
Randers surfaces is determined, for which the infinitesimal holonomy algebra
coincides with the curvature algebra. We prove that for all projectively flat
Randers surfaces of non-zero constant flag curvature the infinitesimal holonomy
algebra has infinite dimension and hence the holonomy group cannot be a Lie
group of finite dimension. Finally, in the case of the Funk metric we prove
that the infinitesimal holonomy algebra is a dense subalgebra of the Lie
algebra of the full diffeomorphism group and hence the topological closure of
the holonomy group is the orientation preserving diffeomorphism group of the
circle.
