Maximising measure
In mathematics — specifically, in ergodic theory — a maximising measure is a particular kind of probability measure. Informally, a probability measure μ is a maximising measure for some function f if the integral of f with respect to μ is "as big as it can be". The theory of maximising measures is relatively young and quite little is known about their general structure and properties.
Definition
Let X be a topological space and let T : X → X be a continuous function. Let Inv(T) denote the set of all Borel probability measures on X that are invariant under T, i.e., for every Borel-measurable subset A of X, μ(T−1(A)) = μ(A). (Note that, by the Krylov-Bogolyubov theorem, if X is compact and metrizable, Inv(T) is non-empty.) Define, for continuous functions f : X → R, the maximum integral function β by
- [math]\displaystyle{ \beta(f) := \sup \left. \left\{ \int_{X} f \, \mathrm{d} \nu \right| \nu \in \mathrm{Inv}(T) \right\}. }[/math]
A probability measure μ in Inv(T) is said to be a maximising measure for f if
- [math]\displaystyle{ \int_{X} f \, \mathrm{d} \mu = \beta(f). }[/math]
Properties
- It can be shown that if X is a compact space, then Inv(T) is also compact with respect to the topology of weak convergence of measures. Hence, in this case, each continuous function f : X → R has at least one maximising measure.
- If T is a continuous map of a compact metric space X into itself and E is a topological vector space that is densely and continuously embedded in C(X; R), then the set of all f in E that have a unique maximising measure is equal to a countable intersection of open dense subsets of E.
References
- {{cite book
| last = Morris | first = Ian | title = Topics in Thermodynamic Formalism: Random Equilibrium States and Ergodic Optimisation | url = http://www.warwick.ac.uk/staff/Ian.Morris/thesis.ps | format = PostScript | year = 2006 | accessdate = 2008-07-05 | location = University of Manchester, UK | publisher = Ph.D. thesis
- Jenkinson, Oliver (2006). "Ergodic optimization". Discrete and Continuous Dynamical Systems 15 (1): 197–224. doi:10.3934/dcds.2006.15.197. ISSN 1078-0947. MR2191393
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