Upper Sequential Capacity Topological Pressure

• Upper Sequential Capacity Topological Pressure is defined within rational graph-directed Markov systems, linking thermodynamic formalism with escape phenomenon in Julia sets.
• It quantifies the interplay of topological entropy, conformal measure decay, and system dynamics, thereby generalizing Bowen’s formula in non-autonomous settings.
• The concept is pivotal in applications ranging from fractal geometry and multifractality analysis to decentralized control of Markov processes on complex networks.

A rational graph-directed Markov system (RGDMS) is a dynamical system built from the interaction of rational maps and the combinatorics of a directed graph, governed by Markov transition constraints. This class encompasses a wide spectrum of systems, including iterated function systems, rational semigroups, and non-autonomous dynamical systems, providing a unifying formalism for the thermodynamic and fractal analysis of random or non-stationary compositions of rational maps. RGDMS theory rigorously addresses questions about invariant sets (“Julia sets”), their geometric and measure-theoretic properties, escape phenomena, and even the control of abstract Markov processes on graphs, through the lens of rational transition mechanisms.

1. Structural Definition and Symbolic Encoding

Let $G=(V,E)$ be a finite directed graph, with vertices $V$ and edges $E$. Each edge $e \in E$ is assigned a nonconstant rational map $$f_e: \widehat{\mathbb{C}} \rightarrow \widehat{\mathbb{C}}$$, and the adjacency structure is encoded by a Markov transition matrix $A=(A_{e,e'})$, where $A_{e,e'} = 1$ iff $t(e) = i(e')$, and zero otherwise.

A sequence $\omega = (e_1, e_2, \ldots)$ in the edge space is admissible if it respects the transition constraints: $A_{e_n, e_{n+1}} = 1$ for all $n$. The shift space $\Sigma$ defined by these admissibility conditions forms a subshift of finite type, naturally encoding the allowed compositions of rational maps along paths in $G$ (Arimitsu et al., 2024).

The system is called a rational graph-directed Markov system (RGDS) or more generally a rational graph-directed Markov system (RGDMS) when each edge is possibly associated with either a single rational map or a compact family of rational maps (Arimitsu, 11 Jan 2026). This structure ensures the system dynamics are not only determined by compositions of rational maps but also by the underlying graph’s connectivity and Markov property.

2. Skew-Product Dynamics and Julia Sets

Key to RGDMS theory is the construction of a skew-product map: $$F: \Sigma \times \widehat{\mathbb{C}} \rightarrow \Sigma \times \widehat{\mathbb{C}}, \quad F(\omega, z) = (\sigma(\omega), f_{e_1}(z)),$$ where $\sigma$ is the left shift on $\Sigma$. This map captures both the symbolic trajectory (the path in the graph) and the evolution under the rational map sequence.

For each sequence $\omega$, the fiber Julia set is defined by: $$J_\omega = \bigcap_{n=1}^\infty f_{e_n}^{-1} \circ \ldots \circ f_{e_1}^{-1}(\widehat{\mathbb{C}} \setminus \{\infty\}),$$ which is the set of points where the sequence of compositions fails to be normal. The global skew-product Julia set is

$$J(F) = \bigcup_{\omega \in \Sigma} \{\omega\} \times J_\omega,$$

endowed with a product metric. This framework provides complete invariance under the skew-product map and generalizes the classical Julia set theory to non-autonomous and graph-directed settings (Arimitsu et al., 2024, Arimitsu, 11 Jan 2026).

3. Thermodynamic Formalism and Bowen's Formula

For an RGDS under the conditions of non-elementarity, expansion along fibers, irreducibility, aperiodicity, and the backward separating condition, the thermodynamic formalism applies. On $J(F)$, one defines the geometric (Hölder continuous) potential

$\varphi_t(\omega, z) = -t \log |f_{e_1}'(z)|,$

and its topological pressure $P(t)$ via the Bowen–Ruelle approach. Crucially, $P(t)$ is strictly decreasing, real-analytic, and crosses zero at a unique value $\delta$, i.e., $P(\delta) = 0$.

Bowen's formula asserts: $$\mathrm{Hausdorff\;dim}\,(J(F)) = \delta, \quad 0 < H^{\delta}(J(F)) < \infty,$$ where $H^{\delta}$ is the $\delta$-dimensional Hausdorff measure. This extends classical results for hyperbolic rational maps and iterated function systems, as well as rational semigroups, showing the unification of dimension theory in the RGDS setting (Arimitsu et al., 2024).

Intermediary technical results supporting this framework include:

• Density of repelling periodic points in $J(F)$,
• Topological exactness (eventual covering) of $F$ on $J(F)$,
• Expandingness criteria derived from hyperbolicity (in the sense of Sumi),
• Existence and uniqueness of conformal measures and equilibrium states,
• Distortion and covering estimates enabling precise dimension bounds.

4. Quantization Dimension and Markov-Type Measures

For RGDMSs where the rational maps act as contractions with explicit ratios (ratio-specified RGDS), the associated graph-directed fractals admit Markov-type probability measures. For such a system, given a row-stochastic transition matrix $P = (p_{ij})$, contraction ratios $c_{ij} \in (0,1)$, and initial probability vector $X$, the measures satisfy

$$\mu(J_{i_1 \ldots i_k}) = X_{i_1} p_{i_1i_2} \ldots p_{i_{k-1}i_k}.$$

The quantization dimension of order $r$, denoted $D_r(\mu)$, captures the asymptotic scaling of the optimal quantization error. It is given by the unique solution $s_r$ of

$\rho( (p_{ij} c_{ij}^r)^{s/(s+r)} ) = 1,$

where $\rho$ is the spectral radius. The lower quantization coefficient at $D_r(\mu)$ is always positive, and the finiteness of the upper quantization coefficient is determined by the comparability of strongly-connected components with maximal local exponents (Kesseböhmer et al., 2014). This link between spectral properties of the transition-contraction matrix and the quantization dimension reveals the fine structure of Markov-type measures on graph-directed fractals, directly applicable to RGDMSs with rational contraction schemes (such as those arising in number theory and information theory).

5. Escape Rates and Preimage Decay Exponents

A significant geometric invariant for RGDMS Julia sets is the conformal preimage decay exponent, which measures the exponential rate at which the conformal measure of sequences of preimages of small sets (holes) decays. For a conformal measure $\nu_\delta$ associated to the solution $\delta$ of the pressure equation, and for $n$-fold preimages of small disks (vertex-wise holes), the decay exponent is

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

where $N_n$ is the cumulative degree over all length-$n$ compositions. The main theorem establishes

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

with $h_{\text{top}}(\tilde{f})$ the topological entropy of the skew-product, and $\overline{CP}$ the upper sequential capacity topological pressure for the lifted hole sequence. This relation quantifies the interplay of dynamical complexity (entropy), geometry (pressure), and “openness” (holes), generalizing escape rate results from single-map settings to non-autonomous, graph-directed regimes. The exponent governs properties of survivor sets, multifractal structure, and rates of escape through holes in the Julia set (Arimitsu, 11 Jan 2026).

6. Rational Feedback Control in Markov Chains

Beyond deterministic dynamics, rational graph-directed Markov systems provide a framework for feedback control of Markov processes on graphs. In the setting of continuous-time Markov chains (CTMC) on a finite, bidirected, strongly connected graph, state-dependent transition rates are constructed using decentralized rational feedbacks of the form: $$q_{ij}(x) = a_{ij}(x) + b_{ij}(x) \frac{f_{ij}(x)}{g_{ij}(x)},$$ where $x$ is the state density on the simplex, and the feedback design ensures nonnegativity, locality, and vanishing at the equilibrium.

Equilibrium and stability analysis uses Lyapunov functions and linearization about the equilibrium, with stability certified via diagonal matrix inequalities (LMIs). The feedback construction can be algorithmically synthesized using standard LMI solvers, and ensures exponential convergence to a target distribution without violating probability constraints or network structure. This realization demonstrates the applicability of rational, graph-constrained feedback design in the robust stabilization and control of Markovian networked systems (Elamvazhuthi et al., 2017).

7. Relationship to Classical Dynamical Systems and Future Directions

Rational graph-directed Markov systems subsume a number of classical frameworks:

• For graphs with a single node, the theory reduces to that of rational semigroups and hyperbolic rational maps.
• When all maps are contractions, the model becomes a graph-directed iterated function system (as in Mauldin–Williams).
• The Bowen formula and its extensions recover classical results for dimension and measure of Julia sets for both deterministic and random dynamics.

The ability of RGDMS theory to integrate symbolic dynamics, rational map iteration, thermodynamic formalism, fractal geometry, and Markov process control within a single robust framework suggests broad applicability and deep connections between these fields (Arimitsu et al., 2024, Arimitsu, 11 Jan 2026, Kesseböhmer et al., 2014, Elamvazhuthi et al., 2017).

Ongoing directions include:

• Analysis of multifractality and finer geometric invariants.
• Study of non-expanding or partially expanding graph-directed systems.
• Applications to open dynamical systems, information theory, and decentralized control.
• Investigation of escape rates in nonhyperbolic and partially random settings.

These developments highlight RGDMS as a nexus for the rigorous mathematical study of complex, non-autonomous, and network-directed dynamical systems.