Baker domains and orbits disappearing to infinity

We study attracting orbits escaping to infinity in natural families of transcendental entire functions. We show that, if an attracting fixed point escapes to infinity while its multiplier tends to one, then the limiting function has a doubly parabolic Baker domain. Conversely, we show that any function with an invariant doubly parabolic Baker domain can be approximated locally uniformly by functions in its quasiconformal equivalence class having an attracting fixed point whose multiplier tends to one.

💡 Research Summary

This paper investigates a novel bifurcation phenomenon in the dynamics of transcendental entire functions, focusing on “orbits disappearing to infinity” within natural families (quasiconformal equivalence classes). The central results establish a precise two-way correspondence between such escaping attracting orbits and the existence of doubly parabolic Baker domains in the limiting function.

Core Theorems:

• Theorem A: Consider a path (f_t) in the quasiconformal equivalence class (M_f) of an entire function (f), with (f_t \to f) as (t \searrow 0). Suppose each (f_t) has an attracting fixed point (z_t) that escapes to infinity as (t \to 0). If the multiplier (\rho_t) of (z_t) tends to 1 horocyclically and there is a singular value (v_t) in its basin depending continuously on (t), then the limit function (f) possesses an invariant doubly parabolic Baker domain.
• Theorem B: Conversely, given any transcendental entire function (f) with an invariant doubly parabolic Baker domain (U), there exists a path (f_t) in (M_f) converging locally uniformly to (f), such that for (t>0), each (f_t) has an attracting fixed point (z_t) with multiplier (\rho_t) satisfying (|\rho_t| \nearrow 1) and (z_t \to \infty) as (t \to 0).

Methodology and Key Insights: The proofs employ a sophisticated blend of geometric and analytic techniques from holomorphic dynamics. For Theorem A, the analysis hinges on the geometric convergence of quotient surfaces. The attracting basin (U_t) of (z_t), when quotiented by the dynamics ((S_t = \widehat{U}_t / f_t)), is a complex torus determined by (\rho_t). The horocyclic convergence (\rho_t \nearrow 1) forces these tori to degenerate geometrically to a cylinder (Lemma 2.5), which is precisely the quotient structure of a doubly parabolic Baker domain (Lemma 2.7). The challenge is to transfer this convergence back to the original dynamical plane. The authors achieve this by constructing an (f)-invariant continuum (C) connecting the limit of the singular values (v_t) to infinity, showing a neighborhood of (C) is contained in the Fatou set for all small (t) (Lemma 3.1), and finally arguing that the containing Fatou component (U) of (f) must have the discrete grand orbit relation and cylindrical quotient characteristic of a doubly parabolic Baker domain.

Theorem B leverages the flexibility within the quasiconformal equivalence class (M_f) to construct an explicit deformation that “inverts” the internal dynamics of the given Baker domain, planting an attracting fixed point whose multiplier approaches 1 along the path.

Context and Significance: This work reveals a fundamental mechanism specific to infinite-type entire functions (those with infinitely many singular values). It contrasts sharply with known results for finite-type functions, where orbits disappearing to infinity are either non-existent or lead to “virtual cycles” involving asymptotic values and poles