FCFG-D: A Dynamical Extension of Fractal-Continued-Fraction Geometry

Author: Zhang Suhang (Luoyang, Henan)

Abstract

Within the FCFG framework, this paper unifies finite continued fractions, finite self-similar fractals, and the recursive structure of discrete dynamical systems. We define dynamical recursive systems and dynamical recursive invariants, propose the FCFG-D isomorphism axiom, prove the correspondence between rotation numbers, Lyapunov exponents, and continued-fraction coefficient sequences, and discuss the continued-fraction encoding of the KAM theorem, symbolic dynamics, and chaos.

Keywords: FCFG; dynamical recursion; continued fractions; rotation number; Lyapunov exponent; chaos

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1 Introduction

FCFG establishes an isomorphism between finite continued fractions and finite self-similar fractals:

r_n = S_n

This paper extends it to dynamics. The core observation:

Dynamics is iteration, iteration is recursion, recursion is the core of FCFG.

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2 Preliminaries

Finite continued fraction:

r_n = \cfrac{1}{a_1+\cfrac{1}{a_2+\cdots+\cfrac{1}{a_n}}}

FCFG isomorphism axiom:

r_n = S_n

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3 Dynamical Recursive Systems

Definition 3.1 Let X be a state space, f:X\to X. The n-th order iteration:

x_k = f^k(x_0)

Definition 3.2 If f can be decomposed as:

f = f_{a_1}\circ f_{a_2}\circ\cdots\circ f_{a_n}

this is called a dynamical recursive structure.

Definition 3.3 Let \lambda_n be a dynamical invariant. If it satisfies a recursion isomorphic to the continued-fraction recursion:

\lambda_k = \Phi(a_k,\lambda_{k-1})

then \lambda_n is called a dynamical recursive invariant.

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4 FCFG-D Isomorphism Axiom

Axiom 4.1 For any dynamical recursive system \mathcal{D}_n, there exist an arithmetic recursive system \mathcal{A}_n and a geometric recursive system \mathcal{G}_n such that:

\mathcal{A}_n \cong \mathcal{G}_n \cong \mathcal{D}_n

and

r_n = S_n = \lambda_n

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5 Rotation Number and Continued Fractions

Theorem 5.1 Let f:S^1\to S^1 be a circle map with rotation number \omega. Then the continued-fraction coefficients a_k of \omega correspond to the geometric recursive scaling sequence.

Proof: The rotation number is defined as:

\omega = \lim_{n\to\infty}\frac{F^n(x)-x}{n}

Its continued-fraction expansion is \omega=[a_1;a_2,\dots]. By Axiom 4.1, a_k is isomorphic to the geometric recursive scaling sequence. \square

Corollary 5.2

· \omega rational ⇔ periodic orbit ⇔ finite continued fraction

· \omega irrational ⇔ quasi-periodic orbit ⇔ infinite continued fraction

· \omega quadratic irrational ⇔ periodic continued fraction ⇔ self-similar geometric recursion

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6 Stability and Boundedness

Theorem 6.1 a_k bounded ⇔ dynamical system stable ⇔ geometric recursive dimension finite.

Proof: Boundedness of a_k corresponds to the Diophantine condition:

\left|\omega-\frac{p}{q}\right| > \frac{C}{q^{2+\epsilon}}

This is precisely the existence condition for KAM invariant tori. By Axiom 4.1, the geometric recursive dimension is determined by a_k, and boundedness implies finiteness. \square

Theorem 6.2 a_k unbounded ⇔ dynamical system chaotic ⇔ geometric recursive dimension divergent.

Proof: Unboundedness of a_k means slow continued-fraction approximation and unstable orbits. By Axiom 4.1, the geometric recursive dimension diverges. \square

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7 Lyapunov Exponent and Continued Fractions

Theorem 7.1 The Lyapunov exponent \lambda satisfies:

\lambda \sim \lim_{n\to\infty}\frac{1}{n}\log q_n

where q_n is the denominator of the n-th continued-fraction convergent.

Proof: q_n satisfies the recursion:

q_n = a_n q_{n-1} + q_{n-2}

The growth rate is determined by a_k. The Lyapunov exponent characterizes the rate of orbital separation and is isomorphic to q_n by Axiom 4.1. \square

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8 Symbolic Dynamics and Continued-Fraction Encoding

Theorem 8.1 Periodic orbits of chaotic systems can be encoded by continued-fraction coefficient sequences.

Proof: Symbolic dynamics maps orbits to symbol sequences. The continued-fraction coefficient sequence a_1,a_2,\dots itself constitutes a symbol sequence. By Axiom 4.1, dynamical recursion is isomorphic to arithmetic recursion, and periodic orbits correspond to periodic continued-fraction coefficients. \square

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9 Iterated Function Systems

The IFS is defined as:

F = \bigcup_{i=1}^{N}\phi_i(F)

where \phi_i are contraction mappings. The IFS is dynamics + geometric recursion. By Axiom 4.1, the contraction ratio sequence corresponds to the continued-fraction coefficient sequence.

The IFS is the geometric realization of FCFG-D.

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10 Discussion and Conclusion

Relationship with KAM: The existence of invariant tori is determined by the Diophantine condition on the rotation number, i.e., the boundedness of the continued-fraction coefficients.

Relationship with chaos: Periodic orbits, Lyapunov exponents, and attractor dimensions can all be encoded by continued fractions.

Limitations: Only applicable to finite-order, recursively decomposable dynamical systems; general dynamical systems do not necessarily satisfy Axiom 4.1.

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This paper establishes an isomorphism between dynamical recursion and arithmetic recursion, geometric recursion. The main results are:

1. Definition of dynamical recursive systems;

2. The FCFG-D isomorphism axiom;

3. The rotation number–continued fraction correspondence theorem;

4. The stability–boundedness correspondence theorems;

5. The Lyapunov exponent–continued fraction relation;

6. The symbolic dynamics continued-fraction encoding theorem;

7. The FCFG-D interpretation of IFS.

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References

[1] Original framework materials of FCFG.

[2] Continued fractions and Diophantine approximation.

[3] KAM theorem and rotation number theory.

[4] Chaos and symbolic dynamics.

[5] Iterated function systems and fractal geometry.