97-380 V. Baladi, C. Bonatti, and B. Schmitt
Abnormal escape rates from nonuniformly hyperbolic sets (316K, Postscript) Jun 27, 97
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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.

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