scholarly journals Random complex dynamics and semigroups of holomorphic maps

2010 ◽  
Vol 102 (1) ◽  
pp. 50-112 ◽  
Author(s):  
Hiroki Sumi
Keyword(s):  
Complex Dynamics ◽  
Holomorphic Maps ◽  
Random Complex

scholarly journals Random local complex dynamics

2018 ◽  
Vol 40 (8) ◽  
pp. 2156-2182
Author(s):  
LORENZO GUERINI ◽  
HAN PETERS
Keyword(s):  
Fixed Point ◽  
Probability Measure ◽  
Complex Dynamics ◽  
Linear Part ◽  
Interesting Case ◽  
Holomorphic Maps ◽  
Deterministic Case ◽  
Local Complex ◽  
The Family ◽  
The Stability

The study of the dynamics of an holomorphic map near a fixed point is a central topic in complex dynamical systems. In this paper, we will consider the corresponding random setting: given a probability measure $\unicode[STIX]{x1D708}$ with compact support on the space of germs of holomorphic maps fixing the origin, we study the compositions $f_{n}\circ \cdots \circ f_{1}$, where each $f_{i}$ is chosen independently with probability $\unicode[STIX]{x1D708}$. As in the deterministic case, the stability of the family of the random iterates is mostly determined by the linear part of the germs in the support of the measure. A particularly interesting case occurs when all Lyapunov exponents vanish, in which case stability implies simultaneous linearizability of all germs in $\text{supp}(\unicode[STIX]{x1D708})$.

scholarly journals Random complex dynamics and devil's coliseums

Nonlinearity ◽  
2015 ◽  
Vol 28 (4) ◽  
pp. 1135-1161 ◽  
Author(s):  
Hiroki Sumi
Keyword(s):  
Complex Dynamics ◽  
Random Complex

scholarly journals Periodic points of post-critically algebraic holomorphic endomorphisms

2021 ◽  
pp. 1-33
Author(s):  
VAN TU LE
Keyword(s):  
Complex Analysis ◽  
Complex Dynamics ◽  
Rational Maps ◽  
Higher Dimension ◽  
Holomorphic Maps ◽  
Rational Map ◽  
Critical Set ◽  
Princeton University ◽  
Dimension One ◽  
Hyperbolicity Assumption

Abstract A holomorphic endomorphism of ${{\mathbb {CP}}}^n$ is post-critically algebraic if its critical hypersurfaces are periodic or preperiodic. This notion generalizes the notion of post-critically finite rational maps in dimension one. We will study the eigenvalues of the differential of such a map along a periodic cycle. When $n=1$ , a well-known fact is that the eigenvalue along a periodic cycle of a post-critically finite rational map is either superattracting or repelling. We prove that, when $n=2$ , the eigenvalues are still either superattracting or repelling. This is an improvement of a result by Mattias Jonsson [Some properties of 2-critically finite holomorphic maps of P2. Ergod. Th. & Dynam. Sys.18(1) (1998), 171–187]. When $n\geq 2$ and the cycle is outside the post-critical set, we prove that the eigenvalues are repelling. This result improves one obtained by Fornæss and Sibony [Complex dynamics in higher dimension. II. Modern Methods in Complex Analysis (Princeton, NJ, 1992) (Annals of Mathematics Studies, 137). Ed. T. Bloom, D. W. Catlin, J. P. D’Angelo and Y.-T. Siu, Princeton University Press, 1995, pp. 135–182] under a hyperbolicity assumption on the complement of the post-critical set.

Frontiers in Complex Dynamics

2014 ◽  
Author(s):  
Araceli Bonifant ◽  
Misha Lyubich ◽  
Scott Sutherland
Keyword(s):  
Historical Perspective ◽  
Complex Dynamics ◽  
Short Paper ◽  
Combinatorial Group Theory ◽  
Entropy Theory ◽  
Holomorphic Dynamics ◽  
Eightieth Birthday ◽  
Several Variables ◽  
Fields Medal ◽  
Combinatorial Group

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing.

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