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### Open Access Journals

DCDS

Given a compact manifold $X,$ a continuous
function $g:X\to \R{},$ and a map $T:X\to X,$ we study properties of
the $T$-invariant Borel probability measures that maximize the
integral of $g$.

We show that if $X$ is a $n$-dimensional connected Riemaniann manifold, with $n \geq 2$, then the set of homeomorphisms for which there is a maximizing measure supported on a periodic orbit is meager.

We also show that, if $X$ is the circle, then the "topological size'' of the set of endomorphisms for which there are $g$ maximizing measures with support on a periodic orbit depends on properties of the function $g.$ In particular, if $g$ is $\mathcal{C}^1$, it has interior points.

We show that if $X$ is a $n$-dimensional connected Riemaniann manifold, with $n \geq 2$, then the set of homeomorphisms for which there is a maximizing measure supported on a periodic orbit is meager.

We also show that, if $X$ is the circle, then the "topological size'' of the set of endomorphisms for which there are $g$ maximizing measures with support on a periodic orbit depends on properties of the function $g.$ In particular, if $g$ is $\mathcal{C}^1$, it has interior points.

DCDS

Let $f$ be a homeomorphism of the closed annulus $A$ that preserves
the orientation, the boundary components and that has a lift $\tilde f$ to the
infinite
strip
$\tilde A$ which is transitive.
We show that, if the rotation number of $\tilde f$ restricted to both
boundary components of $A$ is
strictly positive, then there exists a closed nonempty connected set
$\Gamma\subset\tilde A$ such that $\Gamma\subset]-\infty,0]\times[0,1]$,
$\Gamma$ is unbounded,
the projection of $\Gamma$ to $A$ is dense, $\Gamma-(1,0)\subset\Gamma$ and
$\tilde{f}(\Gamma)\subset \Gamma.$ Also, if $p_1$ is the projection on the
first coordinate of $\tilde A$, then there exists $d>0$ such that, for any
$\tilde z\in\Gamma,$
$$\limsup_{n\to\infty}\frac{p_1(\tilde f^n(\tilde z))-p_1(\tilde z)}{n}<-d.$$

DCDS

Let $M$ be a compact $n$-dimensional Riemanian manifold, End($M$) the set of the endomorphisms of $M$ with the usual $\mathcal{C}^0$ topology and $\phi:M\to\mathbb{R}$ continuous.
We prove, extending the main result of [2], that there exists a dense subset of $\mathcal{A}$ of End($M$) such that, if $f\in\mathcal{A}$, there exists a $f$ invariant measure $\mu_{\max}$ supported on a periodic orbit that maximizes the integral of $\phi$ among all $f$ invariant Borel probability measures.

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