Are there really phase transitions in 1-d heat conduction models?

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Recently, it has been claimed (O. V. Gendelman and A. V. Savin, Phys. Rev. Lett. 84, 2381 (2000); A.V.Savin and O.V.Gendelman, arXiv: cond-mat/0204631 (2002)) that two nonlinear classical 1-d lattice models show transitions, at finite temperatures, where the heat conduction changes from being finite to being infinite. These are the well known Frenkel-Kontorova (FK) model and a model for coupled rotators. For the FK model we give strong theoretical arguments why such a phase transition is not to be expected. For both models we show numerically that the effects observed by Gendelman et al. are not true phase transitions but are rather the expected cross-overs associated to the conductivity divergence as T→ 0 and (for the FK model) T→∞.