Complex-Domain Extension of Continued-Fraction IFS: A Study of Dynamical Topological Evolution Based on Complex Exponential Phases

Author: Zhang Suhang

Abstract

Traditional continued-fraction theory and the earlier framework of Fractal Continued-Fraction Geometry (FCFG) are mainly grounded in the real domain \mathbb{R}^+, exploring the static correspondence between arithmetic recursion and one-dimensional self-similar fractals (such as Cantor sets). However, the one-dimensional real domain lacks the geometric degrees of freedom needed to describe “rotation,” “phase,” and “dynamic evolution.” This paper proposes extending the Iterated Function System (IFS) of FCFG from the real domain to the complex plane \mathbb{C}. By introducing the Euler continued-fraction expansion of the natural exponential e^{i\theta} as a complex-parameter generation mechanism, this paper explores the topological features of complex-plane attractors (complex-domain Julia sets) generated by the complex continued-fraction mapping f_i(z)=\frac{1}{a_i+z}. This paper further proposes a conjecture: the periodicity or aperiodicity of the continued-fraction coefficient sequence \{a_i\} will directly determine the connectivity and Hausdorff dimension of the complex-plane limit set, thereby providing a discrete algebraic path from static geometry to complex dynamical systems.

Keywords: complex continued fractions; iterated function systems; complex dynamical systems; e^{i\theta} expansion; Julia sets; topological evolution

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1. Introduction

In the earlier theoretical construction of FCFG, we established the mapping f_i(x)=\frac{1}{a_i+x} on the real domain and proved the strict correspondence between finite continued fractions and finite IFS iteration. However, compressive mappings on the real domain can only generate disconnected, static one-dimensional fractal structures such as line segments and Cantor sets. They cannot naturally produce geometric objects with rotational symmetry, spiral structures, or dynamic phase features.

To compensate for this missing geometric dimension, this paper proposes extending the parameters and variables of continued fractions to the complex domain \mathbb{C}. In the complex plane, the fractional linear transformation f(z)=\frac{1}{a+z} (where a\in\mathbb{C}) is a Möbius transformation, which naturally includes translation, scaling, and rotation (phase change). This paper aims to explore how, when the complex exponential phase e^{i\theta} is introduced as a parameter input, the continued-fraction IFS can leap from a one-dimensional number-theoretic tool to a dynamical topological generator on the complex plane.

2. Complex Continued-Fraction Mappings and Phase Generation Mechanism

2.1 Construction of IFS Mappings on the Complex Plane

Let the sequence \{a_1,a_2,\dots,a_n\}\in\mathbb{C}. Define the family of compressive mappings on the complex plane:

f_i(z)=\frac{1}{a_i+z},\quad z\in\mathbb{C}.

The corresponding truncated value of the complex continued fraction is:

R_n=f_1\circ f_2\circ\cdots\circ f_n(z_0).

According to the Hutchinson theorem in complex dynamical systems, if the mapping family satisfies the contraction condition, then there exists a unique compact attractor K\subset\mathbb{C}.

2.2 Embedding of the Complex Exponential e^{i\theta} into Continued-Fraction Parameters

For the generated geometric figures to exhibit “rotation” and “spiral” features, the parameter a_i cannot be merely a real number; it must carry phase information.

This paper notes that the natural exponential e^z has a classical Euler continued-fraction expansion. Let z=i\theta; then the continuous phase e^{i\theta} can be expressed as an infinite complex continued fraction:

e^{i\theta}

=

1+\frac{i\theta}{1-\frac{i\theta}{2+i\theta-\frac{2i\theta}{3+i\theta-\dots}}}

Theoretical hypothesis: We take the truncations of this expansion at various levels as the parameter sequence a_i(\theta) in the complex mapping. This means that the rotational phase in continuous space can be strictly encoded as the coefficients of a discrete continued-fraction sequence, thereby driving the complex-plane IFS to generate fractal limit sets with specific spiral symmetry (such as variants of logarithmic spiral fractals).

3. Conjectures on the Topological Evolution of the Complex Plane under Coefficient Distributions

In the real domain, whether the coefficients a_i are constant or varying determines whether the fractal converges to a single point or breaks up into a Cantor set. In the complex domain, this evolution will manifest as drastic changes in topological connectivity.

Conjecture 3.1 (Periodic Sequences and Connectivity)

If the complex parameter sequence \{a_i\} exhibits a periodic distribution (corresponding to quadratic irrational approximation of complex numbers), then the corresponding IFS attractor K tends to be a highly symmetric connected set in the complex plane (a connected Julia set), with a definite topological boundary.

Conjecture 3.2 (Aperiodic Sequences and Complex Topology)

If \{a_i\} exhibits an aperiodic or ergodic distribution (for example, a_i=e^{i\theta_i} with irrational phases), the limit set K may evolve into a completely disconnected set (a complex-plane extension of Cantor dust) or exhibit an extremely complex fragmented topological structure. Its Hausdorff dimension will grow nonlinearly with the irregularity of the sequence.

(Note: This conjecture requires subsequent verification through specific numerical computations and connectivity determination theorems for Julia sets.)

4. Conclusion and Outlook

This paper takes a key step in FCFG’s transition from static real geometry to dynamic complex-domain geometry. By expanding e^{i\theta} as a continued fraction and substituting it into complex IFS mappings, this paper establishes a theoretical framework for describing the potential connection between “discrete algebraic sequences” and “continuous rotational fractals.”

The pure mathematical value of this paper lies in the following: it proposes a new parameter-generation method—rather than arbitrarily setting a_i by hand, it uses the phase expansion of the natural constant e to drive complex-plane fractals.

Subsequent research plan:

1. Write programs to plot complex Julia set images on the complex plane for truncated sequences with parameters a_i=e^{i\theta}.

2. Compute the attractor dimension for specific phases \theta (such as \theta=\pi/2).

3. Explore the potential theoretical interface between this framework and complex analytic dynamical systems (such as the Mandelbrot set).

References

[1] Falconer, K. Fractal Geometry: Mathematical Foundations and Applications. Wiley, 2014.

[2] Milnor, J. Dynamics in One Complex Variable. Princeton University Press, 2006.

[3] Euler, L. De Fractionibus Continuis. 1750.