Sharp stability results for almost conformal maps in even dimensions (original) (raw)
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Transactions of the American Mathematical Society, 1998
We prove that for every rational map on the Riemann sphere f : C ¯ → C ¯ f:\overline {\mathbb {C}} \to \overline {\mathbb {C}} , if for every f f -critical point c ∈ J c\in J whose forward trajectory does not contain any other critical point, the growth of | ( f n ) ′ ( f ( c ) ) | |(f^{n})’(f(c))| is at least of order exp Q n \exp Q \sqrt n for an appropriate constant Q Q as n → ∞ n\to \infty , then HD ess ( J ) = α 0 = HD ( J ) \operatorname {HD}_{\operatorname {ess}} (J)=\alpha _{0}=\operatorname {HD} (J) . Here HD ess ( J ) \operatorname {HD}_{\operatorname {ess}} (J) is the so-called essential, dynamical or hyperbolic dimension, HD ( J ) \operatorname {HD} (J) is Hausdorff dimension of J J and α 0 \alpha _{0} is the minimal exponent for conformal measures on J J . If it is assumed additionally that there are no periodic parabolic points then the Minkowski dimension (other names: box dimension, limit capacity) of J J also coincides with HD ( J ) \operatorname {HD}(J)...