37 research outputs found

    Dynamics of postcritically bounded polynomial semigroups II: fiberwise dynamics and the Julia sets

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    We investigate the dynamics of semigroups generated by polynomial maps on the Riemann sphere such that the postcritical set in the complex plane is bounded. Moreover, we investigate the associated random dynamics of polynomials. Furthermore, we investigate the fiberwise dynamics of skew products related to polynomial semigroups with bounded planar postcritical set. Using uniform fiberwise quasiconformal surgery on a fiber bundle, we show that if the Julia set of such a semigroup is disconnected, then there exist families of uncountably many mutually disjoint quasicircles with uniform dilatation which are parameterized by the Cantor set, densely inside the Julia set of the semigroup. Moreover, we give a sufficient condition for a fiberwise Julia set JΞ³J_{\gamma} to satisfy that JΞ³J_{\gamma} is a Jordan curve but not a quasicircle, the unbounded component of the complement of JΞ³J_{\gamma} is a John domain and the bounded component of the complement of JΞ³J_{\gamma} is not a John domain. We show that under certain conditions, a random Julia set is almost surely a Jordan curve, but not a quasicircle. Many new phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are found and systematically investigated.Comment: 24 pages, 1 figure. Published in J. London Math. Soc. (2) 88 (2013) 294--318. See also http://www.math.sci.osaka-u.ac.jp/~sumi/welcomeou-e.htm

    Random complex dynamics and devil's coliseums

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    We investigate the random dynamics of polynomial maps on the Riemann sphere and the dynamics of semigroups of polynomial maps on the Riemann sphere. In particular, the dynamics of a semigroup GG of polynomials whose planar postcritical set is bounded and the associated random dynamics are studied. In general, the Julia set of such a GG may be disconnected. We show that if GG is such a semigroup, then regarding the associated random dynamics, the chaos of the averaged system disappears in the C0C^{0} sense, and the function T∞T_{\infty} of probability of tending to ∞\infty is H\"{o}lder continuous on the Riemann sphere and varies only on the Julia set of GG. Moreover, the function T∞T_{\infty} has a kind of monotonicity. It turns out that T∞T_{\infty} is a complex analogue of the devil's staircase, and we call T∞T_{\infty} a "devil's coliseum." We investigate the details of T∞T_{\infty} when GG is generated by two polynomials. In this case, T∞T_{\infty} varies precisely on the Julia set of GG, which is a thin fractal set. Moreover, under this condition, we investigate the pointwise H\"{o}lder exponents of T∞T_{\infty} by using some geometric observations, ergodic theory, potential theory and function theory. In particular, we show that for almost every point zz in the Julia set of GG with respect to an invariant measure, T∞T_{\infty} is not differentiable at z.z. We find many new phenomena of random complex dynamics which cannot hold in the usual iteration dynamics of a single polynomial, and we systematically investigate them.Comment: Published in Nonlinearity 28 (2015) 1135-1161. See also http://www.math.sci.osaka-u.ac.jp/~sumi

    Preservation of External Rays in non-Autonomous Iteration

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    We consider the dynamics arising from the iteration of an arbitrary sequence of polynomials with uniformly bounded degrees and coefficients and show that, as parameters vary within a single hyperbolic component in parameter space, certain properties of the corresponding Julia sets are preserved. In particular, we show that if the sequence is hyperbolic and all the Julia sets are connected, then the whole basin at infinity moves holomorphically. This extends also to the landing points of external rays and the resultant holomorphic motion of the Julia sets coincides with that obtained earlier using grand orbits. In addition, if a finite set of external rays separate the Julia set for a particular parameter value, then the rays with the same external angles separate the Julia set for every parameter in the same hyperbolic component

    Random complex dynamics and semigroups of holomorphic maps

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    We investigate the random dynamics of rational maps on the Riemann sphere and the dynamics of semigroups of rational maps on the Riemann sphere. 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 TT which represent the probability of tending to infinity with respect to the random dynamics of polynomials. Under certain conditions these functions TT are complex analogues of the devil's staircase and Lebesgue's singular functions. More precisely, we show that these functions TT are continuous on the Riemann sphere 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 which 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.Comment: Published in Proc. London. Math. Soc. (2011), 102 (1), 50--112. 56 pages, 5 figure
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