Abnormal escape rates for . . .

Consider a C 1+ffl diffeomorphism f having a uniformly hyperbolic compact invariant set \Omega\Gamma 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 \Gamma log jJ u f j 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 1 two-dimensional counterexample, constructed by a surgery using a Bowen-type hyperbolic saddle attractor as the basic plug