Differentiability properties of the pre-image pressure (Q444322)

Summary: 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 \(P_{\text{pre}}(T, \bullet)\) is Gateaux differentiable at \(f \in C(X, \mathbb R)\) if and only if \(P_{\text{pre}}(T, \bullet)\) has a unique tangent functional at \(f\). Also, we obtain some equivalent conditions for \(P_{\text{pre}}(T, \bullet)\) to be Fréchet differentiable at \(f\).