$\beta$-transformation, natural extension and invariant measure
2000 ◽
Vol 20
(5)
◽
pp. 1271-1285
◽
Keyword(s):
For $\beta>1$, consider the $\beta$-transformation $T_\beta$. When $\beta$ is an integer, the natural extension of $T_\beta$ can be represented explicitly as a map on the unit square with an invariant measure: the corresponding two-dimensional Lebesgue measure. We show that, under certain conditions on $\beta$, the natural extension is defined on a simply connected region and an invariant measure is a constant multiple of the Lebesgue measure.We characterize those $\beta$ in terms of the $\beta$-expansion of one, and study the structure and size of the set of all such $\beta$.
1969 ◽
Vol 12
(5)
◽
pp. 673-674
◽
Keyword(s):
2009 ◽
Vol 29
(4)
◽
pp. 1119-1140
◽
Keyword(s):
1943 ◽
Vol 39
(1)
◽
pp. 51-53
Keyword(s):
1962 ◽
Vol 58
(2)
◽
pp. 417-419
◽
Keyword(s):
Keyword(s):