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1. CJM 2008 (vol 60 pp. 658)
Inverse Pressure Estimates and the Independence of Stable Dimension for Non-Invertible Maps We study the case of an Axiom A holomorphic non-degenerate
(hence non-invertible) map $f\from\mathbb P^2
\mathbb C \to \mathbb P^2 \mathbb C$, where $\mathbb P^2 \mathbb C$
stands for the complex
projective space of dimension 2. Let $\Lambda$ denote a basic set for
$f$ of unstable index 1, and $x$ an arbitrary point of $\Lambda$; we
denote by $\delta^s(x)$ the Hausdorff dimension of $W^s_r(x) \cap
\Lambda$, where $r$ is some fixed positive number and $W^s_r(x)$ is
the local stable manifold at $x$ of size $r$; $\delta^s(x)$ is called
\emph{the stable dimension at} $x$. Mihailescu and
Urba\'nski introduced a notion of inverse topological pressure,
denoted by $P^-$, which takes into consideration preimages of points.
Manning and McCluskey study the case of hyperbolic diffeomorphisms on
real surfaces and give formulas for Hausdorff dimension. Our
non-invertible situation is different here since the local unstable
manifolds are not uniquely determined by their base point, instead
they depend in general on whole prehistories of the base points. Hence
our methods are different and are based on using a sequence of inverse
pressures for the iterates of $f$, in order to give upper and lower
estimates of the stable dimension. We obtain an estimate of the
oscillation of the stable dimension on $\Lambda$. When each point $x$
from $\Lambda$ has the same number $d'$ of preimages in $\Lambda$,
then we show that $\delta^s(x)$ is independent
of $x$; in fact $\delta^s(x)$ is shown to be equal in this case with
the unique zero of the map $t \to P(t\phi^s - \log d')$. We also
prove the Lipschitz continuity of the stable vector spaces over
$\Lambda$; this proof is again different than the one for
diffeomorphisms (however, the unstable distribution is not always
Lipschitz for conformal non-invertible maps). In the end we include
the corresponding results for a real conformal setting.
Keywords:Hausdorff dimension, stable manifolds, basic sets, inverse topological pressure Categories:37D20, 37A35, 37F35 |