arXiv:2607.28689v1
Uniqueness of the equilibrium state in the dynamics of holomorphic correspondences
math.DS37A3537A6037F80
Abstract
This paper concerns the study of the existence of a unique equilibrium state for a Hölder continuous function under the dynamics of a holomorphic correspondence defined on the Riemann sphere. We mainly work with the correspondence restricted on the support of the Dinh-Sibony measure and identify topologically interesting correspondences, namely distance expanding ones. Further, we consider Hölder continuous potentials defined on the support of the Dinh-Sibony measure, for which we prove the uniqueness of equilibrium state. Along the way, we also prove some interesting topological results related to holomorphic correspondences. Finally, we establish a result connecting the Ruelle operator for holomorphic correspondences and the unique equilibrium state under a suitable hypothesis. The concluding part of the paper is devoted to some discussion related to the hypothesis involved and providing some examples.
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01Statements3 reported findingsCorrect
The two main equilibrium-state statements are supported. Theorem 1.1 obtains a unique equilibrium state for each Hölder potential by lifting the distance-expanding correspondence to its compact forward-orbit shift. Theorem 1.2 identifies that state with the projection of a Ruelle eigenmeasure when the additional eigenfunction hypothesis printed there holds.
Theorem 1.1Correct
Unique equilibrium state for a Hölder potential
Page 3 · Theorem 1.1 · arXiv:2607.28689v1
For a distance-expanding holomorphic correspondence on the declared compact dynamically stable set, Theorem 1.1 gives existence and uniqueness of the equilibrium state for every Hölder potential, under the stated topological-transitivity condition. The pressure is defined through the forward-orbit space used in the manuscript, and the resulting measure projects to the correspondence with the variational equality claimed. Multiplicity of branches is retained in the orbit space, so the conclusion does not improperly replace the correspondence by its underlying set-valued relation.
Theorem 1.2Correct
Projection of the normalized Ruelle eigenmeasure
Pages 3–4 and 24–25 · Theorem 1.2 · arXiv:2607.28689v1
Theorem 1.2 assumes, in addition to the hypotheses of Theorem 1.1, that the positive eigenfunction for the lifted Ruelle operator is constant. Under this condition the eigenmeasure itself, after normalization, is the lifted equilibrium measure and its zeroth-coordinate projection has the stated invariance and variational properties. The proof uses constancy exactly in converting the conformal eigenmeasure into an invariant probability; it does not assert the identification for a general nonconstant eigenfunction.
Pressure formulationCorrect
Compatibility of orbit-space and correspondence pressure
Pages 18–23 · Sections 4–5 · arXiv:2607.28689v1
The potential on the correspondence is pulled back through the zeroth-coordinate map. Legal orbit words, including their branch multiplicities, correspond to inverse branches of the orbit shift, so the partition sums and invariant-measure entropy used in the variational principle are the same quantities defined for the correspondence. This establishes that uniqueness upstairs yields uniqueness for the projected equilibrium object in the manuscript's formulation.
02Proofs3 reported findingsCorrect
Proofs of Theorems 1.1 and 1.2. The forward-orbit space records a legal branch at every time and turns the correspondence into a single-valued left shift. Proposition 4.6 proves compactness and openness and transfers the distance-expansion constant to this shift. Topological transitivity is also lifted, so Theorem 5.1 applies the standard expanding-map thermodynamic formalism to obtain a unique Gibbs/equilibrium measure. The projection intertwines orbit evaluation with the correspondence action, and the entropy/pressure definition in the paper is precisely the lifted one; hence projecting the unique lifted measure gives the equilibrium state asserted in Theorem 1.1 without losing branch multiplicities.
Orbit-space and Ruelle argumentsCorrect and complete
Proofs of Theorems 1.1 and 1.2
Pages 18–25 · Sections 5.1–5.3 · arXiv:2607.28689v1
The forward-orbit space records a legal branch at every time and turns the correspondence into a single-valued left shift. Proposition 4.6 proves compactness and openness and transfers the distance-expansion constant to this shift. Topological transitivity is also lifted, so Theorem 5.1 applies the standard expanding-map thermodynamic formalism to obtain a unique Gibbs/equilibrium measure. The projection intertwines orbit evaluation with the correspondence action, and the entropy/pressure definition in the paper is precisely the lifted one; hence projecting the unique lifted measure gives the equilibrium state asserted in Theorem 1.1 without losing branch multiplicities.
Section 4Correct and complete
Expansion and openness of the orbit shift
Pages 18–21 · Propositions 4.3–4.6 · arXiv:2607.28689v1
The metric weights later coordinates geometrically, so a discrepancy at the first differing branch is expanded by the shift until it reaches the base-coordinate scale. Local inverse branches of the correspondence give openness, while compactness follows from closedness of the incidence relations in the finite product. The proof verifies these properties on the dynamically stable orbit set and not on an unjustified larger space.
Theorem 1.2 proofCorrect and complete
Ruelle normalization and projection
Pages 23–25 · Theorems 5.2–5.3 and proof of Theorem 1.2 · arXiv:2607.28689v1
The Ruelle–Perron–Frobenius theorem supplies a positive eigenfunction, eigenmeasure, and pressure eigenvalue for the expanding shift. Multiplying the eigenmeasure by the eigenfunction gives the invariant Gibbs probability in general. Under Theorem 1.2's constancy assumption this multiplication changes only normalization, so the projected eigenmeasure equals the already unique equilibrium state. The intertwining identity for test functions proves the stated invariance on the correspondence.
03Novelty0 reported findingsNo non-novelty findings
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