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

Authors: Abraham Boyarsky and Sarah Cooper
Journal: Proc. Amer. Math. Soc. 80 (1980), 574-576
MSC: Primary 28D10; Secondary 39C05, 58F13
MathSciNet review: 587930
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Abstract: Let $ {\tau _a}:[0,1] \to [0,1],0 < a \leqslant \frac{1}{2}$, denote the class of transformations defined by

$\displaystyle {\tau _a}(x) = \left\{ {\begin{array}{*{20}{c}} {2x,} \hfill & {0... ...\hfill & {\frac{1}{2} \leqslant x \leqslant 1.} \hfill \\ \end{array} } \right.$

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.

References [Enhancements On Off] (What's this?)

  • [1] Z. S. Kowalski, Ergodic properties of piecewise monotonic transformations, Soc. Math. de France Astérisque 49 (1977), 145-149. MR 0499080 (58:17042)
  • [2] A. Lasota and J. A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481-488. MR 0335758 (49:538)
  • [3] T.-Y. Li and J. A. Yorke, Ergodic transformations from an interval into itself, Trans. Amer. Math. Soc. 235 (1978), 183-192. MR 0457679 (56:15883)

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