V. Baladi, C. Bonatti, and B. Schmitt
Abnormal escape rates from nonuniformly hyperbolic
sets
(316K, Postscript)
ABSTRACT. Consider a $C^{1+\epsilon}$ diffeomorphism
$f$ having a uniformly hyperbolic compact invariant set
$\Omega$, maximal invariant in some small neighbourhood of itself.
The asymptotic exponential rate of escape from any
small enough neighbourhood of $\Omega$ is given by the topological
pressure of $-\log |J^u f|$ on $\Omega$ (Bowen--Ruelle [1975]).
It has been conjectured (Eckmann--Ruelle [1985])
that this property, formulated in terms of escape
from the support $\Omega$ of a (generalized SRB) measure,
using its entropy and positive Lyapunov exponents, holds
more generally.
We present a simple $C^\infty$ two-dimensional counterexample,
constructed by a surgery using a Bowen-type hyperbolic saddle
attractor as the basic plug.