97-87 Nicolai Haydn

Convergence of the natural approximations of piecewise monotone interval map (41K, LaTeX) Feb 21, 97

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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.

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