Difference between revisions of "Weakly-wandering set"
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| − | ''for an invertible measurable transformation | + | {{TEX|done}} |
| + | ''for an invertible measurable transformation $T$ of a [[Measurable space|measurable space]] $(X,\mathfrak B)$'' | ||
| − | A measurable subset | + | A measurable subset $A\subset X$ for which there is an infinite sequence of integers $n_i$ such that the sets $T^{n_i}$ are mutually disjoint (here, invertibility of $T$ is understood to mean existence and measurability of $T^{-1}$). If $T$ has a $\sigma$-finite [[Quasi-invariant measure|quasi-invariant measure]] $\mu$ (defined on $\mathfrak B$), then a necessary and sufficient condition for $T$ to have a finite [[Invariant measure|invariant measure]] equivalent to $\mu$ is that $\mu A=0$ for any weakly-wandering set $A$. |
A word of warning. In [[Topological dynamics|topological dynamics]] there is a notion of weakly non-wandering point (see [[#References|[3]]]) which has nothing to do with the notion defined above. | A word of warning. In [[Topological dynamics|topological dynamics]] there is a notion of weakly non-wandering point (see [[#References|[3]]]) which has nothing to do with the notion defined above. | ||
Latest revision as of 12:04, 23 August 2014
for an invertible measurable transformation $T$ of a measurable space $(X,\mathfrak B)$
A measurable subset $A\subset X$ for which there is an infinite sequence of integers $n_i$ such that the sets $T^{n_i}$ are mutually disjoint (here, invertibility of $T$ is understood to mean existence and measurability of $T^{-1}$). If $T$ has a $\sigma$-finite quasi-invariant measure $\mu$ (defined on $\mathfrak B$), then a necessary and sufficient condition for $T$ to have a finite invariant measure equivalent to $\mu$ is that $\mu A=0$ for any weakly-wandering set $A$.
A word of warning. In topological dynamics there is a notion of weakly non-wandering point (see [3]) which has nothing to do with the notion defined above.
References
| [1] | A.B. Hajian, S. Kakutani, "Weakly wandering sets and invariant measures" Trans. Amer. Math. Soc. , 110 : 1 (1964) pp. 136–151 |
| [2] | A. Hajian, Y. Itô, "Weakly wandering sets and invariant measures for a group of transformations" J. Math. Mech. , 18 : 12 (1969) pp. 1203–1216 |
| [3] | D.V. Anosov, I.V. Bronshtein, "Topological dynamics" , Dynamical Systems I , Encycl. Math. Sci. , I , Springer (1988) pp. 197–219 (Translated from Russian) |
How to Cite This Entry:
Weakly-wandering set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weakly-wandering_set&oldid=33108
Weakly-wandering set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weakly-wandering_set&oldid=33108
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article