Nicolai Haydn
Convergence of the natural approximations of piecewise monotone
interval map
(41K, LaTeX)
ABSTRACT. We consider piecewise monotone interval mappings which are topologically mixing
and satisfy the Markov property. It has previously been shown that the invariant
densities of the natural approximations converge exponentially fast in
uniform pointwise topology to the
invariant density of the given map provided it's derivative is piecewise
Lipshitz continuous. In this paper we show that in general one does not
obtain exponential convergence in the bounded variation norm.
Here we prove that if the derivative of the interval map is
H\"{o}lder continuous
and its variation is well approximable ($\gamma$-uniform variation for
$\gamma>0$), then the densities converge exponentially fast in the
norm.