Weak continuity of invariant measures for a class of piecewise monotonic transformations

Abstract:

Let ${\tau _a}:[0,1] \to [0,1],0 < a \leqslant \frac {1}{2}$, denote the class of transformations defined by \[ {\tau _a}(x) = \left \{ {\begin {array}{*{20}{c}} {2x,} \hfill & {0 \leqslant x \leqslant \frac {1}{2},} \hfill \\ {(2 - a) - 2(1 - a)x,} \hfill & {\frac {1}{2} \leqslant x \leqslant 1.} \hfill \\ \end {array} } \right .\] For ${a^ \ast } \leqslant a < \frac {1}{2}$, the transformation ${\tau _a}$ admits unique absolutely continuous invariant measures ${\mu _a}$ whose density function ${f_a}(x)$ assumes the value 0 on an interval which expands as $a \uparrow \frac {1}{2}$. Furthermore, the measures ${\mu _a}$ converge weakly to the measure ${\mu _{1/2}} = \frac {1}{2}{\delta _{1/2}} + \frac {1}{2}{\delta _1}$ as $a \uparrow \frac {1}{2}$, where ${\mu _{1/2}}$ is an invariant measure under ${\tau _a}$, and ${\delta _y}$ denotes the Dirac measure at the point y.