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Fractal Continued-Fraction Geometry (FCFG): A Framework for the Correspondence Between Continued-Fraction Recursion and IFS Fractal Iteration

Author: Suhang Zhang,

                Luoyang, Henan

Abstract

Within the framework of Fractal Continued-Fraction Geometry (FCFG), this paper systematically establishes a rigorous correspondence between the arithmetic recursion of continued fractions and the self-similar fractal structures of Iterated Function Systems (IFS). Based on classical continued-fraction theory, Lagrange's classification theorem, and Hutchinson's theorem on the existence of fractals, we define standard mapping rules between finite continued-fraction truncations and finite geometric iterations. We rigorously construct the transformation from layer-by-layer continued-fraction nesting to compressive iterative mappings, and we distinguish the topological characteristics of the limit sets corresponding to periodic and non-periodic continued fractions.

The entire paper is restricted to structural explorations in pure number theory, dynamical systems, and fractal geometry. It does not involve physical field theories or macroscopic paradigm extrapolations. The aim is to establish a rigorous, computable, and verifiable foundational mathematical basis for the FCFG theory.

Keywords: Fractal Continued-Fraction Geometry (FCFG); Continued fractions; Iterated Function System (IFS); Self-similar fractals; Hausdorff dimension; Quadratic irrationals

1. Introduction

Continued-fraction theory is a classical branch of number theory and Diophantine approximation, used to characterize the rate of rational approximation of real numbers, the approximation order of irrationals, and numerical recursive structures. Fractal geometry, on the other hand, uses self-similar iteration, compressive mappings, and limit-set topology as its core tools to characterize complex geometric structures generated by nonlinear recursion. Both fundamentally rely on the underlying logic of "layer-by-layer recursion, iterative convergence, and self-similar generation," yet they have long belonged to two independent research languages, lacking a unified corresponding system.

The core motivation of Fractal Continued-Fraction Geometry (FCFG) is to bridge the homologous structures of arithmetic recursion and geometric self-similarity, establishing a rigorously definable and step-by-step computable transformation channel between "the recursion of numbers" and "the iteration of shapes."

Focusing on the one-dimensional real domain and using Iterated Function Systems (IFS) as the standard geometric carrier, this paper constructs a family of compressive mappings adapted to finite continued-fraction structures. We prove that a finite continued-fraction truncation is equivalent to a finite-step IFS composite iteration. Furthermore, we classify the topological characteristics of the limit sets based on the periodicity of the continued fractions, providing a foundational framework for subsequent dimensional calculations, dynamical system measure analysis, and higher-dimensional generalizations within FCFG.

2. Basic Definitions and Core FCFG Mapping Construction

2.1 Standard Definition of Finite Continued Fractions

Definition 2.1 (Finite Simple Continued Fraction)

Let a_1, a_2, \dots, a_n be positive integers (the classical simple continued fraction convention; this framework can be naturally extended to positive real parameters). A finite n-th order continued fraction is defined as:

r_n = \frac{1}{a_1 + \dfrac{1}{a_2 + \dfrac{1}{\ddots + \dfrac{1}{a_n}}}} 

Each sequence \{a_i\}_{i=1}^n uniquely determines a finite continued fraction value, corresponding to a finite-level arithmetic nested recursion.

2.2 IFS Mapping Construction Adapted to Continued Fractions (Core FCFG Correspondence)

Definition 2.2 (FCFG Continued-Fraction–Geometric Mapping)

For any positive integer parameter a_i, define a family of single-variable functions:

f_i(x) = \frac{1}{a_i + x}, \quad x \in \mathbb{R}^+ 

For a finite sequence, take the set of functions:

\mathcal{F} = \{f_1, f_2, \dots, f_n\} 

We call \mathcal{F} the FCFG Iterated Function System adapted to the n-th order continued fraction.

According to the fundamental theory of fractal geometry, if the mappings satisfy the contraction condition, Hutchinson's theorem guarantees that this IFS possesses a unique compact attractor K \subset \mathbb{R}^+, which is the geometric limit set corresponding to this continued fraction sequence.

2.3 Rigorous Correspondence Proposition for Finite Recursion

Proposition 2.1

Within the FCFG–IFS mapping system constructed in this paper, the finite continued-fraction truncation value satisfies:

r_n = f_1 \circ f_2 \circ \dots \circ f_n (0) 

That is, the finite-level continued-fraction arithmetic nesting is strictly equivalent to the composite iteration of finite-step IFS compressive mappings.

This proposition establishes the foundational correspondence of the FCFG framework: the numerical recursion process of continued fractions completely corresponds to the geometric iterative generation process of fractals, achieving step-by-step homology between arithmetic structures and geometric structures.

3. Periodicity Classification and Topological Differences of Limit Sets

3.1 Topological Characteristics of Periodic Continued Fractions

Proposition 3.2

According to Lagrange's theorem: a real number can be expanded into an eventually periodic continued fraction if and only if it is a quadratic irrational.

Within the FCFG–IFS framework, if the partial quotient sequence \{a_i\} is an eventually periodic sequence, the corresponding family of iterative mappings possesses a cyclic iterative structure. Its limit attractor K exhibits closed, discrete, and strictly self-similar topological characteristics, and the orbital iteration possesses regressiveness and periodic closure.

This property characterizes the discrete fractal geometric attributes of quadratic irrationals within the FCFG system.

3.2 Geometric Conjecture for Non-Periodic Continued Fractions

Conjecture 3.1 (FCFG Non-Periodic Limit Set Complexity Conjecture)

For non-periodic continued fractions—including higher-degree algebraic numbers and transcendental numbers (such as \pi, e) with non-cyclic partial quotient sequences—the corresponding IFS limit set no longer possesses a simple closed self-similar structure.

We conjecture within the FCFG framework that the limit sets corresponding to such sequences possess higher geometric complexity. Their Hausdorff dimension, ergodicity, and asymptotic iterative behavior can serve as core topological invariants to distinguish the geometric characteristics of transcendental numbers from those of lower-degree algebraic numbers.

This paper only proposes this structural conjecture; rigorous proofs and numerical verification are left for future work.

4. Conclusion and Outlook

4.1 Conclusion

Based on the FCFG framework, this paper completes the rigorous correspondence construction between finite continued-fraction recursive structures and IFS self-similar iteration:

1. We established a step-by-step matching arithmetic–geometric mapping, proving that finite continued-fraction truncation is equivalent to finite-step geometric iteration, thereby solidifying the core foundation of FCFG's "homology of numerical and geometric recursion."

2. Relying on classical number theory theorems, we rigorously distinguished the topological differences of limit sets corresponding to periodic and non-periodic continued fractions, avoiding the boundary oversights of traditional statements.

3. The entire process is restricted to pure mathematical structural analysis, forming a self-consistent, rigorous, and extensible foundational pre-research framework for FCFG.

4.2 Outlook for Future Work

Based on the foundational framework of this paper, subsequent actionable research can be carried out as follows:

1. For classical partial quotient sets (such as a_i \in \{1, 2\} or Fibonacci sequences), precisely calculate or numerically approximate the Hausdorff dimension of the FCFG–IFS attractors using the box-counting method and Moran's equation.

2. Establish measure-theoretic connections and ergodicity analysis between the FCFG iterative structure and the Gauss map dynamical system.

3. Generalize the one-dimensional FCFG mapping framework to the complex domain and higher-dimensional spaces, constructing higher-dimensional fractal topological systems corresponding to continued-fraction recursion.

Acknowledgements

This work is a foundational study of the Fractal Continued-Fraction Geometry (FCFG) system, completing structural construction based on classical number theory and fractal geometry theory.