SOME ASPECTS OF THERMODYNAMIC FORMALISM OF PIECEWISE SMOOTH VECTOR FIELDS

The paper defines pressure for PSVFs via the time-one map on the space of trajectories and proves, for the petals constructions, that P(T1|Ω̂, −β log|det J_med T1|) equals log of the spectral radius of a finite matrix A built from transition weights of a Markov partition; see the definition of pressure and T1 on trajectories and Theorem 1’s statement asserting P = log(ρ(A)) with A irreducible, along with the explicit matrix representation in the proof sketch (A_ij = (med(P_j ∩ φ_Z^{-1}(P_i))/med(P_j))^β) and the conclusion via Perron–Frobenius/RPF formalism . The model solution follows the same scheme: remove a measure-zero singular set, code integer-time itineraries via a finite Markov partition, identify the dual RPF operator with a nonnegative matrix having entries p_{ji}^β, and conclude P = log ρ(A). It adds technical clarifications (e.g., conjugacy between T1 on Ω̂ and φ_Z on a full-measure domain, co-boundary normalization of the geometric potential, and use of Lebesgue invariance under divergence-zero, refractive PSVFs, as noted in the paper’s preliminaries) . Both arguments rely on the same finite-state coding and Perron–Frobenius/RPF mechanism, with the model providing more explicit justifications of steps the paper states succinctly.