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A Julia-Fatou Theory via Random Systems with Complete Connections

Yoshiyuki Endo

Abstract

We develop a Julia-Fatou theory for random dynamical systems of continuous self-maps on a compact metric space, driven by random systems with complete connections (RSCCs). This framework allows the selection rule to depend on the evolving state and, in general, on the entire past, going beyond the Markovian graph directed Markov system setting. For each state we define Julia, Fatou, and kernel Julia sets via equicontinuity of admissible composition families, and we introduce a pathwise and skew product viewpoint. Under natural compactness and continuity assumptions on the RSCC, we study the associated averaged dynamics on the product space and prove Cooperation Principle I: if the kernel Julia set is empty at every state and the admissible maps are open, then the iterates of the adjoint transition operator are equicontinuous on the whole space of probability measures, and along almost every admissible path the fiberwise Julia set has zero mass for any given finite measure. We further identify a new phenomenon specific to RSCCs, namely emptiness jumps of kernel Julia sets along admissible state trajectories, and provide criteria excluding such jumps, including discreteness of the state space and a propagation mechanism under phi-irreducibility. Several examples, motivated by reinforcement and feedback mechanisms, illustrate both the jump phenomenon and the applicability of the Cooperation Principle I in non-Markovian settings.

A Julia-Fatou Theory via Random Systems with Complete Connections

Abstract

We develop a Julia-Fatou theory for random dynamical systems of continuous self-maps on a compact metric space, driven by random systems with complete connections (RSCCs). This framework allows the selection rule to depend on the evolving state and, in general, on the entire past, going beyond the Markovian graph directed Markov system setting. For each state we define Julia, Fatou, and kernel Julia sets via equicontinuity of admissible composition families, and we introduce a pathwise and skew product viewpoint. Under natural compactness and continuity assumptions on the RSCC, we study the associated averaged dynamics on the product space and prove Cooperation Principle I: if the kernel Julia set is empty at every state and the admissible maps are open, then the iterates of the adjoint transition operator are equicontinuous on the whole space of probability measures, and along almost every admissible path the fiberwise Julia set has zero mass for any given finite measure. We further identify a new phenomenon specific to RSCCs, namely emptiness jumps of kernel Julia sets along admissible state trajectories, and provide criteria excluding such jumps, including discreteness of the state space and a propagation mechanism under phi-irreducibility. Several examples, motivated by reinforcement and feedback mechanisms, illustrate both the jump phenomenon and the applicability of the Cooperation Principle I in non-Markovian settings.
Paper Structure (15 sections, 21 theorems, 96 equations)

This paper contains 15 sections, 21 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Setting and Main Results
  4. Organization of the Paper
  5. Formulation of the Julia--Fatou Theory for Random Systems with Complete Connections
  6. Julia Set, Fatou Set and Kernel Julia Set
  7. Propagation of Kernel Julia Set Emptiness and the Jump Phenomenon
  8. Admissible Infinite Sequences and Skew Product Dynamics
  9. Families of Probability Measures Associated with Random Systems with Complete Connections
  10. Generalization of Cooperation Principle I
  11. Settings for Cooperation Principle I
  12. Product Space and Markov Operators
  13. Transition Operators and Their Iterations
  14. Cooperation Principle I
  15. Examples

Key Result

Theorem 2.3

Let $\{(W,\mathcal{W}), (X,\mathcal{X}), u, P\}$ be a RSCC, and fix an arbitrary state $w_{0}\in W$. Then there exists a unique probability measure $\mathbf{P}_{w_{0}}$ on $(X^{\mathbb{N}}, \mathcal{X}^{\mathbb{N}})$ and a sequence of $X$-valued random variables $(\xi_{n})_{n\in\mathbb{N}}$ defined where $\xi^{(n)}=(\xi_{1},\ldots,\xi_{n})$, $\zeta_{n}=w_{0}\xi^{(n)}$, and $\zeta^{(n)}=(\zeta_{1}

Theorems & Definitions (53)

  • Definition 2.1: MR1070097
  • Theorem 2.3: MR1070097
  • Definition 2.4
  • Definition 2.6: Admissible words and reachable states
  • Definition 2.8
  • Definition 2.9: Admissible composition families
  • Definition 2.10: Julia and Fatou sets at a state
  • Lemma 2.11
  • Definition 2.12: Forward and backward invariance
  • Lemma 2.13: Invariance of Julia--Fatou sets
  • ...and 43 more