Linear Response For Equilibrium Measures For Piecewise Expanding Unimodal Maps

Equilibrium measures,including SRB measures and measures of maximal entropy,are one of the most important classes of invariant measures in dynamical systems.The pioneering works of Ruelle,in proving the linear response formula of SRB measures in uniformly hyperbolic systems,boost a lot of research results on the response theory for dynamical systems beyond uniform hyperbolicity.and therefore the linear response theory becomes one of the most popular topics in smooth ergodic theory in recent years.In this paper,we shall continue the works by Baladi and Smania on a family {f_t} of the piecewise expanding unimodal maps,and we shall extend the linear response formula of SRB measures to the general equilibrium measures.Moreover,we shall obtain explicit formulae for the linear response in the case of equilibrium measures.Here are our key techniques in the proof:（a）Under the topological conjugate tangency condition.we establish the smooth dependence of the conjugate map on the parameter.Moreover,using the conjugate map.we shall convert a two-parameter Ruelle transfer operator on the system f_t to a Ruelle transfer operator over fixed base dynamical system f₀ and associated with twoparameter potential function,such that the corresponding topological pressure is unchanged;（b）For the two-parameter Ruelle transfer operator over fixed base dynamical system f₀,we apply the classical analytic perturbation theory to calculate the second order partial derivatives of the topological pressure.