Strongly asymptotically stable Frobenius-Perron operators

Abstract:

Let $(X,\Sigma ,\mu )$ be a $\sigma$-finite measure space and let $T : L^1(X,\Sigma ,\mu ) \to L^1(X,\Sigma ,\mu )$ be a Frobenius-Perron operator. In 1997 Bartoszek and Brown proved that if $T$ overlaps supports and if there exists $h \in L^1(X,\Sigma ,\mu )$, $h > 0$ on $X$, such that $Th = h$, then $T$ is (strongly) asymptotically stable. In the note we prove that instead of assuming that $h > 0$ on $X$, it is enough to assume that $h\geq 0$ and $h\neq 0$. More precisely, we prove that $T$ is asymptotically stable if and only if $T$ overlaps supports and there exists $h\in L^1(X,\Sigma ,\mu )$, $h\geq 0$, $h\neq 0$, such that $Th=h$.