Differentiability Properties of the Pre‐Image Pressure

We study the differentiability properties of the pre-image pressure. For a TDS ( X , T ) with finite topological pre-image entropy, we prove the pre-image pressure function P p r e ( T , • ) is Gateaux differentiable at f ∈ C ( X , R ) if and only if P p r e ( T , • ) has a unique tangent functional at f . Also, we obtain some equivalent conditions for P p r e ( T , • ) to be Fréchet differentiable at f .