Thermodynamic formalism for correspondences

Abstract: In this work, we investigate the Variational Principle and develop a thermodynamic formalism for correspondences. We define the measure-theoretic entropy for transition probability kernels and topological pressure for correspondences. Based on these two notions, we establish the following results: The Variational Principle holds and equilibrium states exist for continuous potential functions, provided that the correspondence satisfies some expansion property called forward expansiveness. If, in addition, the correspondence satisfies the specification property and the potential function is Bowen summable, then the equilibrium state is unique. On the other hand, for a distance-expanding, open, strongly transitive correspondence and a Hölder continuous potential function, there exists a unique equilibrium state, and the backward orbits are equidistributed. Furthermore, we investigate the Variational Principle for general correspondences. In conformal dynamics, we establish the Variational Principle for the Lee--Lyubich--Markorov--Mazor--Mukherjee anti-holomorphic correspondences, which are matings of some anti-holomorphic rational maps with anti-Hecke groups and not forward expansive. We also show a Ruelle--Perron--Frobenius Theorem for a family of hyperbolic holomorphic correspondences of the form $\boldsymbol{f}_c (z)= z^{q/p}+c$.

• The paper introduces a thermodynamic formalism for correspondences, extending classical entropy and topological pressure to multi-valued settings.
• It rigorously proves a variational principle showing the link between topological pressure and measure-theoretic entropy, ensuring existence and uniqueness of equilibrium states in expansive cases.
• Applications to holomorphic and anti-holomorphic dynamics illustrate advanced analytical methods for characterizing equilibrium and equidistribution in complex, multi-valued systems.

Thermodynamic Formalism for Set-Valued Dynamical Correspondences

Introduction and Motivations

This work introduces and systematically studies the thermodynamic formalism for correspondences—set-valued (multi-valued) dynamical systems generalizing classical single-valued maps. The framework unifies concepts of topological pressure, measure-theoretic entropy, and variational principles for this broader class, inspired by longstanding developments in the ergodic theory of deterministic maps and by emerging needs in complex dynamics, particularly in the analysis of holomorphic and anti-holomorphic dynamical correspondences.

Set-valued dynamical systems (or correspondences) naturally arise when a state may evolve to multiple possible future states, and can be seen in control theory, game theory, mathematical economics, viability theory, and especially in several branches of complex dynamics. While much of ergodic theory for single-valued maps has been successfully extended to non-invertible and even random processes, the measure-theoretic, variational, and statistical structure for true multi-valued dynamics has been comparatively underdeveloped. This paper fills that gap by developing formalism and demonstrating its strength on previously open analytic examples.

Main Definitions and Formalism

Entropy and Pressure for Correspondences

The authors introduce a measure-theoretic entropy $h_\mu(Q)$ for transition probability kernels (i.e., Markov processes) $Q$ supported by a correspondence $T:X \to 2^X$, and a notion of topological pressure $P(T, \phi)$ for a correspondence $T$ and continuous potential $\phi$ defined on the graph of $T$:

• The entropy is defined in terms of the shift-invariant Markov process on orbit spaces, generalizing the Kolmogorov–Sinai entropy for maps.
• The topological pressure naturally extends the classical pressure and entropy by considering growth rates of weighted $(n, \epsilon)$-separated sets of multi-valued orbits, following the tradition of Bowen and Ruelle.

Variational Principle: Markovian and General Cases

A central focus is the Variational Principle for correspondences: a maximizing formula linking topological pressure and measure-theoretic entropy. The paper conjectures, and establishes under substantial generality, that for suitable $T$ and $\phi$,

$$P(T, \phi) = \sup_{Q, \mu} \left\{ h_\mu(Q) + \int_X \int_{T(x)} \phi(x, y)\,dQ_x(y)\,d\mu(x) \right\}$$

where $Q$ runs over transition probability kernels supported on $T$ and $\mu$ over $Q$-invariant measures.

This extends the classical Variational Principle for single-valued maps and offers a new paradigm for studying statistical mechanics and equilibrium in deterministic but non-uniquely evolving systems.

Key technical advances:

• Proof of the principle for forward expansive correspondences (multi-valued maps where distinct orbits diverge at a uniform rate), including existence and description of equilibrium states.
• Uniqueness of equilibrium states under specification and Bowen summability conditions.
• For distance-expanding, open, and strongly transitive correspondences with Hölder potentials, a unique equilibrium state is shown to exist, and equidistribution of backward orbits is established—mirroring the powerful machinery available for expansive single-valued systems.

For general correspondences (not necessarily expansive), the paper establishes at least one direction—a lower bound resembling the variational upper bound, matching the classical inequality with possible non-attainability.

Applications to Complex Dynamics

The motivation and culmination of this theory lies in its application to new objects in complex dynamics:

Anti-Holomorphic and Holomorphic Correspondences

The authors apply their formalism to two prominent families:

• The Lee–Lyubich–Makarov–Mazor–Mukherjee anti-holomorphic correspondences, arising in the theory of matings of rational maps with anti-Hecke groups, which had lacked a variational framework because they are not forward expansive. The implementation of the variational principle here is nontrivial and extends the analytic toolkit for such fractal Julia sets.
• The unicritical hyperbolic holomorphic correspondences of the form $f_c(z) = z^{q/p} + c$, where for certain $c$ and with respect to an appropriate metric, the authors show these correspondences satisfy openness, distance-expansion, and strong transitivity, thus inheriting the existence and equidistribution of equilibrium states as a direct result of their general theory.

Analytical Advances

Strong numerical results include explicit characterizations:

• For anti-holomorphic correspondences, the topological pressure matches the supremum over Markov measure-process pairs, even in settings without uniform expansion.
• For holomorphic correspondences $z^{q/p} + c$, existence and uniqueness of equilibrium states, and explicit weak* convergence/equidistribution of backward orbits are shown for large open regions in parameter space.

These applications demonstrate not just conceptual reach but also real analytic applications—answering open questions in the field and providing methods for direct computation and further research.

Implications and Theoretical Advances

This paper marks a significant synthesis and extension of dynamical, ergodic, and thermodynamic formalism to the context of correspondences, with several major consequences:

• Broader class of dynamical systems: The techniques support the study of systems where uniqueness of evolution fails globally—important for modeling in random, controlled, and even partially specified physical systems.
• Unified perspective: The use of orbit spaces and shift maps shows that much of the classical single-valued ergodic theory remains structurally intact when suitably reframed.
• Connections to Markov processes: The structure of the variational principle through transition kernels provides conceptual and quantitative bridges between deterministic, random, and set-valued dynamics.

Future theoretical developments are envisioned in several directions:

• Further relaxation of expansivity and regularity conditions for the variational principle.
• Construction and analysis of equilibrium states and Gibbs measures in less regular or more degenerate (non-expansive, non-open) settings.
• Application to the study of random dynamical systems, qualitative physics, and higher-dimensional complex dynamical correspondences.

The results here set a foundation for systematic ergodic and thermodynamic analysis in the complex, multi-valued, and random settings that naturally arise in many contemporary mathematical and applied fields.

Conclusion

The paper provides a rigorous and general thermodynamic formalism for dynamical correspondences, including:

• New, robust definitions of topological pressure and measure-theoretic entropy for set-valued dynamics,
• Strong proofs of the Variational Principle in expansive and regular settings, with explicit existence and uniqueness results for equilibrium states,
• Applications to analytically challenging examples in complex dynamics, notably holomorphic and anti-holomorphic matings, and
• A conceptual framework that unites deterministic and random dynamics.

This work facilitates both deeper theoretical exploration and practical analysis of a wide class of dynamical systems previously out of reach for the full machinery of thermodynamic formalism.