Bowen parameter and Hausdorff dimension for expanding rational semigroups

Proceedings of the London Mathematical Society

2010

Self Cite

229

We investigate the random dynamics of rational maps on the Riemann sphereĈ and the dynamics of semigroups of rational maps onĈ. We show that regarding random complex dynamics of polynomials, in most cases, the chaos of the averaged system disappears, due to the cooperation of the generators. We investigate the iteration and spectral properties of transition operators. We show that under certain conditions, in the limit stage, 'singular functions on the complex plane' appear. In particular, we consider the functions T that represent the probability of tending to infinity with respect to the random dynamics of polynomials. Under certain conditions these functions T are complex analogues of the devil's staircase and Lebesgue's singular functions. More precisely, we show that these functions T are continuous onĈ and vary only on the Julia sets of associated semigroups. Furthermore, by using ergodic theory and potential theory, we investigate the non-differentiability and regularity of these functions. We find many phenomena that can hold in the random complex dynamics and the dynamics of semigroups of rational maps, but cannot hold in the usual iteration dynamics of a single holomorphic map. We carry out a systematic study of these phenomena and their mechanisms.

“…For recent work on the dynamics of rational semigroups, see the author's papers [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40] and [25,[41][42][43][44].…”

Section: Introductionmentioning

confidence: 99%

Random complex dynamics and semigroups of holomorphic maps

Proceedings of the London Mathematical Society

2010

Self Cite

229

“…Another crucial property of the Bowen parameter is the following fact proved as one of the main results of [40]. Remark 2.17 ( [28,41]). Let f = (f 1 , .…”

Section: We Definementioning

confidence: 95%

“…However, if we do not assume the open set condition, then there are a lot of examples for which the Bowen parameter is strictly larger than the Hausdorff dimension of the Julia set. In fact, the Bowen parameter can be strictly larger than two ( [28,41]). Thus, it is very natural to ask when we have this situation and what happens if we have such a case.…”

Section: Introductionmentioning

confidence: 99%

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Transversality family of expanding rational semigroups

Advances in Mathematics

Urbański

2013

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We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a d-parameter family of such semigroups satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the Julia set is the minimum of 2 and the zero of the pressure function. Moreover, the Hausdorff dimension of the exceptional set of parameters is estimated. We also show that if the zero of the pressure function is greater than 2, then typically the 2-dimensional Lebesgue measure of the Julia set is positive. Some sufficient conditions for a family to satisfy the transversality conditions are given. We give non-trivial examples of families of semigroups of non-linear polynomials with the transversality condition for which the Hausdorff dimension of the Julia set is typically equal to the zero of the pressure function and is less than 2. We also show that a family of small perturbations of the Sierpinski gasket system satisfies that for a typical parameter value, the Hausdorff dimension of the Julia set (limit set) is equal to the zero of the pressure function, which is equal to the similarity dimension. Combining the arguments on the transversality condition, thermodynamical formalisms and potential theory, we show that for each a ∈ C with |a| = 0, 1, the family of small perturbations of the semigroup generated by {z 2 , az 2 } satisfies that for a typical parameter value, the 2-dimensional Lebesgue measure of the Julia set is positive.

“…dim H (J(h 1 , h 2 )). Then[31, Theorem 3.15] implies that there exist a transformation ϕ(z) = αz + β with α, β ∈ C, α = 0, two complex numbers a 1 , a 2 and a positive integer d such that we have d = d 1 = d 2 and for each j = 1, 2, ϕ • h j • ϕ −1 (z) = a j z d . By Lemma 2.14, we obtain that d ≥ 3.…”

mentioning

confidence: 99%

The space of 2-generator postcritically bounded polynomial semigroups and random complex dynamics

Advances in Mathematics

2016

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We investigate the dynamics of 2-generator semigroups of polynomials with bounded planar postcritical set and associated random dynamics on the Riemann sphere. Also, we investigate the space B of such semigroups. We show that for a parameter h in the intersection of B, the hyperbolicity locus H and the closure of the disconnectedness locus (the space of parameters for which the Julia set is disconnected), the corresponding semigroup satisfies either the open set condition (and Bowen's formula) or that the Julia sets of the two generators coincide. Also, we show that for such a parameter h, if the Julia sets of the two generators do not coincide, then there exists a neighborhood U of h such that for each parameter in U , the Hausdorff dimension of the Julia set of the corresponding semigroup is strictly less than 2. Moreover, we show that the intersection of the connectedness locus and B ∩ H has dense interior. By using the results on the semigroups corresponding to these parameters, we investigate the associated functions which give the probability of tending to ∞ (complex analogues of the devil's staircase or Lebesgue's singular functions) and complex analogues of the Takagi function.