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DCDS

Let $f$ be a real-valued function defined on the phase space of a
dynamical system. Ergodic optimization is the study of those orbits,
or invariant probability measures, whose ergodic $f$-average is as
large as possible.

In these notes we establish some basic aspects of the theory: equivalent definitions of the maximum ergodic average, existence and generic uniqueness of maximizing measures, and the fact that every ergodic measure is the unique maximizing measure for some continuous function. Generic properties of the support of maximizing measures are described in the case where the dynamics is hyperbolic. A number of problems are formulated.

In these notes we establish some basic aspects of the theory: equivalent definitions of the maximum ergodic average, existence and generic uniqueness of maximizing measures, and the fact that every ergodic measure is the unique maximizing measure for some continuous function. Generic properties of the support of maximizing measures are described in the case where the dynamics is hyperbolic. A number of problems are formulated.

keywords:
Maximizing measure.

DCDS

Let $M_{\phi}$ denote the set of Borel probability measures invariant
under a topological action $\phi$ on a compact metrizable space $X$.
For a continuous function $f:X\to\R$, a measure $\mu\in\M_{\phi}$ is called
$f$-maximizing if $\int f\, d\mu = s u p\{\int f dm:m\in\M_{\phi}\}$. It
is shown that if $\mu$ is any ergodic measure in $\M_{\phi}$, then there
exists a continuous function whose unique maximizing measure is
$\mu$. More generally, if $\mathcal E$ is a non-empty collection of
ergodic measures which is weak$^*$ closed as a subset of $\M_{\phi}$, then
there exists a continuous function whose set of maximizing measures
is precisely the closed convex hull of $\mathcal E$. If moreover
$\phi$ has the property that its entropy map is upper
semi-continuous, then there exists a continuous function whose set
of equilibrium states is precisely the closed convex hull of
$\mathcal E$.

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