Published online by Cambridge University Press: 20 November 2009
We consider iterations of smooth non-invertible maps on manifolds of real dimension 4, which are hyperbolic, conformal on stable manifolds and finite-to-one on basic sets. The dynamics of non-invertible maps can be very different than the one of diffeomorphisms, as was shown for example in [4, 7, 12, 17, 19], etc. In  we introduced a notion of inverse topological pressure P− which can be used for estimates of the stable dimension δs(x) (i.e the Hausdorff dimension of the intersection between the local stable manifold Wsr(x) and the basic set Λ, x ∈ Λ). In  it is shown that the usual Bowen equation is not always true in the case of non-invertible maps. By using the notion of inverse pressure P−, we showed in  that δs(x) ≤ ts(ϵ), where ts(ϵ) is the unique zero of the function t → P−(tφs, ϵ), for φs(y):= log|Dfs(y)|, y ∈ Λ and ϵ > 0 small. In this paper we prove that if Λ is not a repellor, then ts(ϵ) < 2 for any ϵ > 0 small enough. In  we showed that a holomorphic s-hyperbolic map on
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