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### Open Access Journals

JMD

We establish stable ergodicity for diffeomorphisms with partially hyperbolic attractors whose
Lyapunov exponents along the center direction are all positive with respect to SRB measures.

DCDS

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.

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