Random complex dynamics and semigroups of holomorphic maps

Abstract

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.

Published in Proc. London. Math. Soc. (2011), 102 (1), 50--112. 56 pages, 5 figures

References in corpus (4)

• Dynamics of postcritically bounded polynomial semigroups II: fiberwise dynamics and the Julia sets
• Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups
• Dynamics of postcritically bounded polynomial semigroups I: connected components of the Julia sets
• Bowen Parameter and Hausdorff Dimension for Expanding Rational Semigroups

Cited by in corpus (24)

• Dynamics of postcritically bounded polynomial semigroups II: fiberwise dynamics and the Julia sets
• Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups
• Dynamics of postcritically bounded polynomial semigroups I: connected components of the Julia sets
• Real analyticity of Hausdorff dimension for expanding rational semigroups
• Cooperation principle, stability and bifurcation in random complex dynamics
• Random complex dynamics and devil's coliseums
• Pointwise Hölder Exponents of the Complex Analogues of the Takagi Function in Random Complex Dynamics
• Negativity of Lyapunov Exponents and Convergence of Generic Random Polynomial Dynamical Systems and Random Relaxed Newton's Methods
• The space of $2$-generator postcritically bounded polynomial semigroups and random complex dynamics
• Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups
• Non-i.i.d. random holomorphic dynamical systems and the probability of tending to infinity
• Non-i.i.d. random holomorphic dynamical systems and the generic dichotomy
• Bowen Parameter and Hausdorff Dimension for Expanding Rational Semigroups
• Transversality Family of Expanding Rational Semigroups
• Interaction cohomology of forward or backward self-similar systems
• On the stochastic bifurcations regarding random iterations of polynomials of the form $z^2+c_n$
• Random local complex dynamics
• Escaping set and Julia set of transcendental semigroups
• Dynamics of infinitely generated nicely expanding rational semigroups and the inducing method
• Spectral gap property for random dynamics on the real line and multifractal analysis of generalised Takagi functions
• Dynamics of postcritically bounded polynomial semigroups III: classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles
• Networks of coupled quadratic nodes
• Regularity and irregularity of fiber dimension of non-autonomous dynamical systems
• Entropy of random chaotic interval map with noise which causes coarse-graining