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

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 $J_{\gamma}$ to satisfy that $J_{\gamma}$ is a Jordan curve but not a quasicircle, the unbounded component of the complement of $J_{\gamma}$ is a John domain and the bounded component of the complement of $J_{\gamma}$ 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

Semi-hyperbolic fibered rational maps and rational semigroups

We consider fiber-preserving complex dynamics on fiber bundles whose fibers are Riemann spheres and whose base spaces are compact metric spaces. In this context, without any assumption on (semi-)hyperbolicity, we show that the fiberwise Julia sets are uniformly perfect. From this result, we show that, for any semigroup $G$ generated by a compact family of rational maps on the Riemann sphere of degree two or greater, the Julia set of any subsemigroup of $G$ is uniformly perfect. We define the semi-hyperbolicity of dynamics on fiber bundles and show that, if the dynamics on a fiber bundle is semi-hyperbolic, then the fiberwise Julia sets are porous, and the dynamics is weakly rigid. Moreover, we show that if the dynamics is semi-hyperbolic and the fiberwise maps are polynomials, then under some conditions, the fiberwise basins of infinity are John domains. We also show that the Julia set of a rational semigroup (a semigroup generated by rational maps on the Riemann sphere) that is semi-hyperbolic, except at perhaps finitely many points in the Julia set, and which satisfies the open set condition, is either porous or equal to the closure of the open set. Furthermore, we derive an upper estimate of the Hausdorff dimension of the Julia set.Comment: June 10, 2006 version. 31 pages, published in Ergodic Theory and Dynamical Systems. A typo in Theorem 2.6 is fixed. See also http://www.math.sci.osaka-u.ac.jp/~sumi/welcomeou-e.htm

Random complex dynamics and devil's coliseums

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 $G$ of polynomials whose planar postcritical set is bounded and the associated random dynamics are studied. In general, the Julia set of such a $G$ may be disconnected. We show that if $G$ is such a semigroup, then regarding the associated random dynamics, the chaos of the averaged system disappears in the $C^{0}$ sense, and the function $T_{\infty}$ of probability of tending to $\infty$ is H\"{o}lder continuous on the Riemann sphere and varies only on the Julia set of $G$. Moreover, the function $T_{\infty}$ has a kind of monotonicity. It turns out that $T_{\infty}$ is a complex analogue of the devil's staircase, and we call $T_{\infty}$ a "devil's coliseum." We investigate the details of $T_{\infty}$ when $G$ is generated by two polynomials. In this case, $T_{\infty}$ varies precisely on the Julia set of $G$, which is a thin fractal set. Moreover, under this condition, we investigate the pointwise H\"{o}lder exponents of $T_{\infty}$ by using some geometric observations, ergodic theory, potential theory and function theory. In particular, we show that for almost every point $z$ in the Julia set of $G$ with respect to an invariant measure, $T_{\infty}$ is not differentiable at $z.$ 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

Random complex dynamics and semigroups of holomorphic maps

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 $T$ which 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 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