Negativity of Lyapunov Exponents and Convergence of Generic Random Polynomial Dynamical Systems and Random Relaxed Newton’s Methods

Sumi, Hiroki

doi:10.1007/s00220-021-04070-6

Cited by 8 publications

(5 citation statements)

References 38 publications

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“…More recently, Sumi obtained a number of very interesting results concerning action of semigroups of rational maps in both topological and probabilistic context (e.g., [18], [17]).…”

Section: Introduction the Resultsmentioning

confidence: 99%

Total disconnectedness of Julia sets of random quadratic polynomials

Lech¹,

Zdunik²

2021

Ergod. Th. Dynam. Sys.

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For a sequence of complex parameters $(c_n)$ we consider the composition of functions $f_{c_n} (z) = z^2 + c_n$ , the non-autonomous version of the classical quadratic dynamical system. The definitions of Julia and Fatou sets are naturally generalized to this setting. We answer a question posed by Brück, Büger and Reitz, whether the Julia set for such a sequence is almost always totally disconnected, if the values $c_n$ are chosen randomly from a large disc. Our proof is easily generalized to answer a lot of other related questions regarding typical connectivity of the random Julia set. In fact we prove the statement for a much larger family of sets than just discs; in particular if one picks $c_n$ randomly from the main cardioid of the Mandelbrot set, then the Julia set is still almost always totally disconnected.

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“…More recently, Sumi obtained a number of very interesting results concerning action of semigroups of rational maps in both topological and probabilistic context (e.g., [18], [17]).…”

Section: Introduction the Resultsmentioning

confidence: 99%

Total disconnectedness of Julia sets of random quadratic polynomials

Lech¹,

Zdunik²

2021

Ergod. Th. Dynam. Sys.

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“…The latter phenomena are called noise-induced order. For details on randomness-induced phenomena, the reader is referred to the authors' previous paper [1] and references therein, say [6,8,9,13,[17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Non-i.i.d. random holomorphic dynamical systems and the generic dichotomy

Sumi

Watanabe²

2022

Nonlinearity

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We consider non-i.i.d. random holomorphic dynamical systems whose choice of maps depends on Markovian rules. We show that generically, such a system is mean stable or chaotic with full Julia set. If a system is mean stable, then the Lyapunov exponent is uniformly negative for every initial value and almost every random orbit. Moreover, we consider families of random holomorphic dynamical systems and show that the set of mean stable systems has full measure under certain conditions. The latter is a new result even for i.i.d. random dynamical systems.

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“…We will quickly review this theory in Subsection 2.2. For details, see [27,28]. See also [29,30] for the author generalizing the setting from i.i.d.…”

Section: Introduction 1backgroundmentioning

confidence: 99%

On the stochastic bifurcations regarding random iterations of polynomials of the form $z^{2} + c_{n}$

Watanabe¹

2022

Preprint

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In this paper, we consider random iterations of the form z 2 + c n where c n are complex-valued independent random variables following the uniform distribution on the closed disk with center c and radius r. The aim of this paper is twofold. First, we investigate the bifurcation of our proposed random iterations and give quantitative estimates of bifurcations parameters. Second, we apply the estimates to the (dis)connectedness of random Julia sets. Here, we reveal the relationships between the bifurcation radius and connectedness of random Julia sets. In particular, we prove that almost every random Julia set is totally disconnected if the central parameter is c = −1 and the radial parameter r is much smaller than expected.

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Negativity of Lyapunov Exponents and Convergence of Generic Random Polynomial Dynamical Systems and Random Relaxed Newton’s Methods

Cited by 8 publications

References 38 publications

Total disconnectedness of Julia sets of random quadratic polynomials

Total disconnectedness of Julia sets of random quadratic polynomials

Non-i.i.d. random holomorphic dynamical systems and the generic dichotomy

On the stochastic bifurcations regarding random iterations of polynomials of the form $z^{2} + c_{n}$

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