Instytut Matematyczny Polskiej Akademii Nauk / pl / Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

Streszczenie

Let $M=I$ or $M=\mathbb {S}^1$ and let $k\geq 1$. We exhibit a new infinite class of Polish groups by showing that each group $\mathop {\rm Diff}\nolimits _+^{k+{\rm AC}}(M)$, consisting of those $C^k$ diffeomorphisms whose $k$th derivative is absolutely continuous, admits a natural Polish group topology which refines the subspace topology inherited from $\mathop {\rm Diff}\nolimits _+^k(M)$. By contrast, the group $\mathop {\rm Diff}\nolimits _+^{1+{\rm BV}}(M)$, consisting of $C^1$ diffeomorphisms whose derivative has bounded variation, admits no Polish group topology whatsoever.