C. McMullen. Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps. Preprint, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....(b) The topological pressure function t 7 P(t) is not differentiable at the point t = h. We would like to conclude this section with the remark that recently parabolic bifurcations have been extensively investigated and lots of interesting dimension results have been obtained (see for ex. BZ] [Mc1], Sh] DSZ] HS] Zi1] and [Zi2] x3. Rational functions whose all critical points contained in the Julia set are non recurrent. We recall that if T : X X is a continuous map of a topological space X, then for every point x 2 X, the limit set of x denoted by (x) is defined to be the set ....

....was explored in [DU7] from the point of view of the fractal properties of the Julia set. We have dealt with NCP maps in [U3] and [U4] exploring their fractal and ergodic properties. Their geometric properties have been studied in [CJY] and the rigidity type results have been obtained in [PU2] In [MC1] the reader may find some properties of geometrically finite maps proven earlier in [U3] for NCP maps. The paper [Ha] contains some ergodic properties of geometrically finite maps. We start our description of fractal and ergodic properties of NCP maps with the following characterization of conical ....

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C. McMullen, Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps, Preprint 1997.

....it makes sense) replacing only in Theorem 1.9 the property not critically finite with parabolic orbifold for which J(f) J(g) CI by non linear and removing condition (1) Clearly T (f) is a subset of the set of conical points of f (for the definition of the latter see [6] comp. 7] and [8]) and therefore as an immediate consequence of Theorem 1.2 in [7] we get the following. Theorem 1.5. If f : CI CI is tame, then there exists at most one value t for which a t conformal measure exists and is supported on T (f ) Additionally, for such a t there exists exactly one t conformal ....

C. McMullen, Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps, Preprint Berkeley 1997.

....studied H: dimJ(z 2 c) using a Monte Carlo algorithm [3] See also [28] and [23] for calculations for quadratic polynomials. This paper belongs to a three part series. Parts I and II study the continuity of Hausdorff dimension in families of Kleinian groups and rational maps [18] [19]. The bibliographies to parts I and II provide further references. Notation. A i B means A=C B CB for some implicit constant C; A B means A=B 1. 2 Markov partitions and the eigenvalue algorithm In this section we define Markov partitions for conformal dynamical systems equipped with ....

....but not expanding. As c decreases from 1=4 along the real axis, the map f c undergoes a sequence of period doubling bifurcations at parabolic points c n converging to the Feigenbaum point c Feig Gamma1:401155. The map f c is expanding for all c 2 (c Feig ; 1=4] outside this sequence c n . In [19] we show: Theorem 5.1 The function H: dimJ(f c ) is continuous on the interval (c Feig ; 1=4] The graph of dimJ(f c ) for c 2 [ Gamma1; 1=2] as determined by the eigenvalue algorithm, is plotted in Figure 8. One striking feature is the discontinuity at the parabolic point c = 1=4, studied in ....

C. McMullen. Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps. Preprint, 1997.

....counterexamples to continuity of dimension come from geometrically finite handlebodies with cusps (x8) A version of Theorem 1. 1 (without continuity of ) was proved by Taylor [33] See Anderson and Canary for related results on cores and limits [2] 3] This paper belongs to a three part series [25], 24] Part II gives parallel results in the setting of iterated rational maps. The theory of conformal densities is available for both rational maps and Kleinian groups; it is in anticipation of the applications in Part II that we work with conformal densities here, rather than with ....

C. McMullen. Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps. Preprint, 1997.

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