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Abstract
In this work we exhibit a one-parameter
family of $C^1$-diffeomorphisms $F_\alpha$ of the 2-sphere,
where $\alpha>1$, such that the equator $\S^1$ is an attracting set
for every $F_\alpha$ and $F_\alpha|_{\S^1}$ is the identity. For $\alpha>2$
the Lebesgue measure on the equator is a non ergodic physical
measure having uncountably many ergodic components. On the other
hand, for $1<\alpha\leq 2$ there is no physical measure for $F_\alpha$. If
$\alpha<2$ this follows directly from the fact that the $\omega$-limit
of almost every point is a single point on the equator (and the
basin of each of these points has zero Lebesgue measure). This is no
longer true for $\alpha=2$, and the non existence of physical measure
in this critical case is a more subtle issue.
Mathematics Subject Classification: Primary: 37C40. Secondary: 37A10.
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