The transition between synchronization and chaos for random Blaschke products

Technical Summary: The Transition Between Synchronization and Chaos for Random Blaschke Products

Problem Statement
The paper addresses the fundamental question of how random dynamical systems transition between order (synchronization) and chaos. While routes to chaos in deterministic discrete-time systems are well-characterized (e.g., period-doubling, intermittency, Ruelle-Takens scenarios), the random setting remains less understood due to the added complexity of environmental stochasticity. The authors focus specifically on a class of random circle maps known as random Blaschke products. These are holomorphic self-maps of the unit disc $\mathbb{D}$ that preserve the unit circle $\mathbb{T}$. The central problem is to establish sufficient and necessary conditions under which such a cocycle exhibits either chaotic dynamics (with a unique absolutely continuous random invariant measure) or synchronization (convergence to a unique random fixed point, often on the boundary).

Methodology
The authors analyze the asymptotic behavior of admissible Blaschke product cocycles $(T, \sigma, P)$, where $T = (T_\omega)_{\omega \in \Omega}$ is a sequence of finite Blaschke products driven by an ergodic, measure-preserving transformation $\sigma$. The methodology relies on:

1. Hyperbolic Geometry: Utilizing the hyperbolic metric $d_\mathbb{D}$ on the unit disc and the Schwarz-Pick Lemma to analyze contraction properties.
2. Ergodic Theory: Applying Kingman's subadditive ergodic theorem and Birkhoff's ergodic theorem to study the asymptotic growth of derivatives and the convergence of pullback orbits $T^{(n)}_{\sigma^{-n}\omega}(z)$.
3. Boundary Convergence: Investigating the limit points of orbits starting in the interior of the disc. The authors distinguish between cases where orbits accumulate in the interior $\mathbb{D}$ versus the boundary $\mathbb{T}$.
4. Thermodynamic Formalism: Constructing random invariant measures (specifically absolutely continuous ones and Dirac measures) and calculating fiber-wise measure-theoretic entropy.
5. Specific 2-Map Cocycles: To illustrate the general theory, the authors construct a specific family of two-map cocycles ($T_0$ and $T_1$) and vary the probability $p$ of selecting $T_0$ to identify critical transitions.

Key Contributions and Results

1. Dichotomy of Dynamics (Theorem 1.1)
The paper establishes a sharp dichotomy for admissible Blaschke product cocycles. Exactly one of the following two scenarios holds:

• Case 1: Eventually Expanding on Average (EOA). If the cocycle is eventually expanding on average (defined by the limit of the average logarithmic derivative being positive), the system exhibits chaotic behavior. Specifically:
  • There exists a unique random attracting fixed point $x_\omega \in \mathbb{D}$.
  • There exists a unique absolutely continuous random invariant measure $\mu = \{\mu_\omega\}$ with marginal $P$, where the density is given by the Poisson kernel $P_{x_\omega}$.
  • This measure is the unique random physical measure, and its basin of attraction has full Lebesgue measure.
  • The system possesses positive measure-theoretic entropy.
• Case 2: Not Eventually Expanding on Average. If the system is not eventually EOA, it exhibits synchronization.
  • For almost every $\omega$, the pullback orbits $T^{(n)}_{\sigma^{-n}\omega}(z)$ accumulate on the unit circle $\mathbb{T}$.
  • Under specific integrability conditions (related to the decay of $1 - |T^{(n)}(0)|$), there exists a measurable random fixed point $x_\omega \in \mathbb{T}$ such that all forward trajectories in $\mathbb{D}$ and almost all trajectories on $\mathbb{T}$ converge to $x_\omega$.
  • The unique random physical measure is the Dirac measure $\delta_{x_\omega}$ supported on this boundary point.

2. Characterization of the Transition (Theorem 3.2)
For a specific 2-map cocycle family $T_{a,b}$, the authors identify a critical probability value $p_{a,b}$ that separates the two regimes:

• $p > p_{a,b}$: The system is EOA. Trajectories converge to a random fixed point inside the disc, and the physical measure is absolutely continuous.
• $p < p_{a,b}$: The system is not EOA. Trajectories synchronize to the boundary point $1$ (a common fixed point of the maps). The physical measure is the Dirac mass at $1$.
• $p = p_{a,b}$ (Critical Case): This case reveals a phenomenon distinct from deterministic dynamics.
  • Pullback: Pullback orbits still converge to the boundary point $1$.
  • Forward: In an i.i.d. setting, forward trajectories do not converge to $1$. Instead, they exhibit intermittency: trajectories escape from and return to arbitrarily small neighborhoods of the fixed point infinitely often. This contrasts with the deterministic parabolic case, where trajectories would typically converge to the fixed point.
  • The measure-theoretic entropy transitions from positive to zero at this critical value.

3. Entropy and Physical Measures
The paper proves that for eventually EOA cocycles, the fiber-wise measure-theoretic entropy is strictly positive. Conversely, in the synchronization regime (including the critical case), the entropy is zero. The authors also establish the equivalence between the convergence rates of pullback and pushforward trajectories in the synchronization regime.

Significance and Claims
The authors claim to provide a complete classification of the asymptotic behavior for this class of random holomorphic maps. The significance lies in:

• Bridging Deterministic and Random Dynamics: The paper highlights how random forcing alters the transition to chaos. Specifically, the critical case ($p = p_{a,b}$) demonstrates that while deterministic parabolic maps often lead to convergence (synchronization), the random i.i.d. composition can sustain intermittent behavior where trajectories fail to converge to the fixed point, despite the pullback converging.
• Physical Measures: The work rigorously identifies the random physical measures (both absolutely continuous and singular/Dirac) and their basins of attraction, providing a thermodynamic formalism for these systems.
• Generalizability: The results extend previous work on random Möbius transformations and iterated function systems to the broader class of Blaschke products, offering necessary and sufficient conditions for the existence of unique random fixed points and physical measures.

The paper concludes that the transition between order and chaos in random Blaschke products is governed by the average expansion rate, with the critical boundary exhibiting unique intermittent dynamics not present in the deterministic analogues.