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In this note, we establish an original result for the thermodynamic formalismin the context of expanding circle transformations with an indifferent fixedpoint. For an observable whose continuity modulus is linked to the dynamicsnear such a fixed point, by identifying an appropriate linear space to evaluatethe action of the transfer operator, we show that there is a strictly positiveeigenfunction associated with the maximal eigenvalue given as the exponentialof the topological pressure. Taking into account also the correspondingeigenmeasure, the invariant probability thus obtained is proved to be theunique Gibbs-equilibrium state of the system.

A Ruelle-Perron-Frobenius theorem for expanding circle maps with an indifferent fixed point