Good measures on Cantor space |
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Authors: | Ethan Akin |
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Affiliation: | Department of Mathematics, The City College (CUNY), 137 Street and Convent Avenue, New York City, New York 10031 |
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Abstract: | While there is, up to homeomorphism, only one Cantor space, i.e. one zero-dimensional, perfect, compact, nonempty metric space, there are many measures on Cantor space which are not topologically equivalent. The clopen values set for a full, nonatomic measure is the countable dense subset is clopen of the unit interval. It is a topological invariant for the measure. For the class of good measures it is a complete invariant. A full, nonatomic measure is good if whenever are clopen sets with , there exists a clopen subset of such that . These measures have interesting dynamical properties. They are exactly the measures which arise from uniquely ergodic minimal systems on Cantor space. For some of them there is a unique generic measure-preserving homeomorphism. That is, within the Polish group of such homeomorphisms there is a dense, conjugacy class. |
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Keywords: | Cantor set measure on Cantor space ordered measure spaces unique ergodicity generic conjugacy class Rohlin property |
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