Rational Graph-Directed Markov Systems

• Rational Graph-Directed Markov Systems are frameworks integrating directed graphs, rational maps, and symbolic dynamics to analyze complex fractal Julia sets.
• They unify thermodynamic formalism, Bowen’s formula, and quantization measures to provide precise dimension estimates and invariant properties in fractal geometry.
• Decentralized rational feedbacks for Markov chains showcase practical applications in process stabilization and control within dynamic systems.

A rational graph-directed Markov system (RGDMS) formalizes the dynamics of rational maps assigned to a directed graph, synthesizing graph-theoretic structure, symbolic dynamics, and complex analytic iteration on the Riemann sphere. Central research objectives include classification and fractal analysis of the corresponding Julia sets, characterization of conformal and thermodynamic invariants, and control-theoretic implications for process stabilization. Recent work has unified various strands: Bowen’s formula for Hausdorff dimension, quantization theory for Markov measures, entropy/pressure analysis, and rational decentralized feedback for Markov chains.

1. Algebraic and Dynamical Structure of RGDMS

Let $G=(V,E)$ be a finite directed graph, $i(e), t(e)$ denoting the initial and terminal vertices of an edge $e\in E$. Each $e$ corresponds to a non-constant rational map $f_e:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ of degree $\deg f_e\geq1$. The Markov adjacency matrix $A=(A_{e,e'})$ satisfies $A_{e,e'}=1$ iff $t(e)=i(e')$ and $0$ otherwise. The tuple $S=(V,E,(f_e)_{e\in E})$ defines the rational graph-directed Markov system.

For control-theoretic applications (e.g., (Elamvazhuthi et al., 2017)), let $(V,E)$ be bidirected and strongly connected, $x(t)$ a simplex-constrained vector of densities, and $q_{ij}(x)$ the state-dependent transition rates connected by Kolmogorov forward equations. Decentralized rational feedbacks of form $q_{ij}(x)=a_{ij}(x)+b_{ij}(x)f_{ij}(x)/g_{ij}(x)$ are constructed to stabilize equilibria with structure constraints matching $G$.

2. Symbolic Space, Skew Product, and Julia Sets

Associate with $S$ a subshift of finite type: $$\Sigma = \{ \omega=(e_1,e_2,...) \in E^{\mathbb{N}} : A_{e_n,e_{n+1}}=1 \ \forall n \}$$ and left shift $\sigma$. The skew-product map is

$$F : \Sigma \times \widehat{\mathbb{C}} \to \Sigma \times \widehat{\mathbb{C}},\quad F((e_n)_n, z) = ( (e_{n+1})_n, f_{e_1}(z) )$$

endowed with product metric.

The fiber Julia set for $\omega\in\Sigma$ is

$$J_\omega = \bigcap_{n=1}^\infty f_{e_n}^{-1}\circ\cdots\circ f_{e_1}^{-1}(\widehat{\mathbb{C}}\setminus\{\infty\})$$

defining $$J(F) = \bigcup_{\omega\in\Sigma}\{\omega\}\times J_\omega$$. Hypotheses including non-elementarity ($|J_i|\geq3$), expandingness ($|(F^n)'|\geq C\lambda^n$), irreducibility, and aperiodicity ensure topological exactness and density of repelling periodic points (Arimitsu et al., 2024).

3. Thermodynamic Formalism and Bowen’s Formula

The thermodynamic approach introduces a geometric potential $\phi_t(\omega,z) = -t\log|f_{e_1}'(z)|$. The topological pressure

$$P(t) = \lim_{n\to\infty}\frac{1}{n}\log\sum_{\omega_1\cdots\omega_n\in\Sigma^n}\sup_{z\in J_{\omega_1\cdots\omega_n}} \exp{S_n\phi_t(\omega,z)}$$

where $S_n\phi_t$ is the $n$-th Birkhoff sum. The variational principle yields

$$P(t) = \sup\left\{h_\mu(F) + \int \phi_t d\mu : \mu\ \text{F-invariant on}\ J(F)\right\}$$

There exists a unique $\delta\in\mathbb{R}$ with $P(\delta)=0$ (Walters). Provided the backward separating condition is met, Bowen’s formula asserts

$$\dim_H J(F) = \delta,\quad 0 < \mathcal{H}^\delta(J(F)) < \infty$$

and enables equilibrium/conformal measure constructions ($$m_\delta\circ F^{-1}=e^{P(\delta)-\phi_\delta}m_\delta$$) (Arimitsu et al., 2024).

4. Quantization Dimensions and Markov-Type Measures

Markov-type measures $\mu$ on ratio-specified graph-directed fractals arise from a stochastic matrix $P=(p_{ij})$, contraction ratios $c_{ij}$, and initial probability vector $X$. For $r>0$, the quantization error $e_{n,r}(\mu)$, dimensions $D_r^+(\mu), D_r^-(\mu)$, and quantization coefficients $Q_r^+(\mu,s), Q_r^-(\mu,s)$ are central (Kesseböhmer et al., 2014). The main result provides the quantization dimension $s_r$ by solving the Perron–Frobenius equation for $\psi(s)=\rho(A(s))=1$, with $A(s)$ defined by $a_{ij}(s)=(p_{ij}c_{ij}^r)^{s/(s+r)}$.

The lower quantization coefficient $Q_r^-(\mu,s_r)>0$ always holds, while $Q_r^+(\mu,s_r)<\infty$ iff maximal strongly connected components with $s_r(H)=s_r$ are pairwise incomparable. This criterion links fractal geometry on graph-directed structures to Markov quantization analysis.

5. Conformal Preimage-Decay Exponent and Entropy Relations

In non-autonomous, graph-directed dynamics, backward invariance is quantified via the conformal preimage-decay exponent $e_\delta(R)$ (Arimitsu, 11 Jan 2026):

• Given equilibrium measure $\nu_\delta$, holes $H_i(R)$ in Julia sets, and $n$-fold preimages $HP_n(R)$ under composition, the decay exponent is

$$e_\delta(R) = -\limsup_{n\to\infty}\frac{1}{n}\log\left[\nu_\delta(HP_n(R)) / N_n\right]$$

where $N_n$ is the total degree sum.

• The main theorem links $e_\delta(R)$ to dynamical invariants:

$$e_\delta(R) = h_{\mathrm{top}}(\tilde{f}) - \overline{CP}(\tilde{f}, -\delta\log|\tilde{f}'|, \widetilde{HP}(R)) > 0$$

with $h_{\mathrm{top}}$ the topological entropy and $\overline{CP}$ the sequential capacity topological pressure for lifted holes.

This exponent governs mass decay in survivor sets, connects escape rates and Bowen-type dimension formulas, and bridges geometric and thermodynamic quantities in graph-directed rational dynamics.

6. Rational Feedback Control for Markov Chains

Decentralized stabilization of continuous-time Markov chains evolving on finite bidirected graphs employs rational feedbacks as transition rates (Elamvazhuthi et al., 2017). For each edge $(i,j)\in E$,

$$q_{ij}(x) = m^+_{ij}(x) + m^-_{ji}(x)\frac{x_i}{x_j}$$

with $m^+_{ij}(x):=\max\{\ell_{ij}(x),0\}$, $m^-_{ji}(x):=\max\{-\ell_{ji}(x),0\}$, and $\ell_{ij}(x)=x_j^*x_i-x_i^*x_j$. This law preserves graph sparsity and equilibrium properties ($q_{ij}(x^*)=0$). Local exponential stability is certified via Lyapunov diagonalization and Linear Matrix Inequality (LMI) methods for gain synthesis.

Algorithmic controller design leverages basis matrices, artificial drift, and convex LMI constraints to compute decentralized feedbacks implementable via local densities and respecting the Markov graph structure.

7. Connections to Classical and Multifractal Theory

Special cases of RGDMS encompass rational semigroups (single-vertex graphs), Markov iterated function systems (IFS), and classical hyperbolic Julia sets (Ruelle’s thermodynamic formalism). Mauldin–Williams graphs with contraction ratios and the separation condition recover classic graph-directed attractor theory (Kesseböhmer et al., 2014, Arimitsu et al., 2024).

The general RGDMS framework offers a unified platform for applying pressure/entropy theory, dimension formulas, conformal measures, and escape-rate analysis to the study of fractal and probabilistic properties of dynamics generated by non-i.i.d., non-autonomous rational compositions. This suggests broad utility in fractal geometry, thermodynamic formalism, Markov process control, and multifractal analysis.