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Random Dynamics of a Family of Cubic Polynomials

Alexandre Miranda Alves, Gerardo Andrés Honorato Gutiérrez, Mostafa Salarinoghabi

Abstract

In this work, we study the non-autonomous dynamics generated by random iterations of the cubic family of the form $z^3 + cz$. The parameter sequence is chosen randomly from a bounded Borel subset of $\mathbb{C}$. We investigate topological properties of the corresponding Julia sets, with particular emphasis on conditions leading to total disconnectedness. We prove that the set of parameter sequences for which the Julia set is totally disconnected is dense in the parameter space. We also construct examples where the Julia set is totally disconnected but the associated non-autonomous system is not hyperbolic. Finally, under suitable probabilistic assumptions on the parameter distribution, we show that almost every sequence produces a totally disconnected Julia set.

Random Dynamics of a Family of Cubic Polynomials

Abstract

In this work, we study the non-autonomous dynamics generated by random iterations of the cubic family of the form . The parameter sequence is chosen randomly from a bounded Borel subset of . We investigate topological properties of the corresponding Julia sets, with particular emphasis on conditions leading to total disconnectedness. We prove that the set of parameter sequences for which the Julia set is totally disconnected is dense in the parameter space. We also construct examples where the Julia set is totally disconnected but the associated non-autonomous system is not hyperbolic. Finally, under suitable probabilistic assumptions on the parameter distribution, we show that almost every sequence produces a totally disconnected Julia set.
Paper Structure (7 sections, 24 theorems, 92 equations, 3 figures)

This paper contains 7 sections, 24 theorems, 92 equations, 3 figures.

Table of Contents

  1. Introduction
  2. An example totally disconnected and non hyperbolic
  3. Topological properties of $\mathcal{J}_{\omega}$ and main results
  4. The scenario in classical dynamics
  5. A family of cubic polynomials
  6. Green function (non-autonomous dynamic)
  7. Proofs of the Main Results

Key Result

Theorem 1

If all the critical points of a bounded hyperbolic polynomial sequence $\{F_n\}_{n=1}^{\infty}$, where $F_n(z) = f_{c_n}\circ\dots\circ f_{c_1}$, escape to infinity under iteration,

Figures (3)

  • Figure 1: Top: long hyperbolic (Markov) blocks (blue) enforce expansion, interspersed with isolated near-parabolic steps (red) where $|f'_n|\approx1$. Bottom: each block induces a finer clopen partition of the Cantor-like Julia set, showing that total disconnectedness persists arbitrarily deep in the tail.
  • Figure 2: The Mandlbrot set of the cubic family $z^3+cz$.
  • Figure 3:

Theorems & Definitions (55)

  • Definition 1.1
  • Theorem
  • Example 1.2
  • Definition 2.1: Bounded non-autonomous system
  • Remark 2.2
  • Example 2.3: Cubic family
  • Definition 2.4: Near-parabolic step in non-autonomous dynamics
  • Proposition 2.5: Near-parabolic steps destroy uniform hyperbolicity
  • proof
  • Proposition 2.6: Tail-imposed Cantor structure via Markov blocks
  • ...and 45 more