Alternative proofs of linear response for piecewise expanding unimodal maps

TL;DR

This paper presents two new, more conceptual proofs for the differentiability of SRB measures in unimodal maps, including higher-order differentiability and a formula for the derivative, with potential for higher-dimensional extensions.

Contribution

The paper introduces two novel proofs of linear response for unimodal maps, improving conceptual understanding and extending applicability to higher dimensions.

Findings

Differentiability of SRB measure at tangent maps proved

Higher-order differentiability established under smoothness conditions

Linear response formula recovered and extended

Abstract

We give two new proofs that the SRB measure of a C^2 path f_t of unimodal piecewise expanding C^3 maps is differentiable at 0 if f_t is tangent to the topological class of f_0. The arguments are more conceptual than the one in our previous paper, but require proving Holder continuity of the infinitesimal conjugacy (a new result, of independent interest) and using spaces of bounded p-variation. The first new proof gives differentiability of higher order if f_t is smooth enough and stays in the topological class of f_0 and if the observable smooth enough (a new result). In addition, this proof does not require any information on the decomposition of the SRB measure into regular and singular terms, making it potentially amenable to extensions to higher dimensions. The second new proof allows us to recover the linear response formula (i.e., the formula for the derivative at 0) obtained in our previous paper and gives additional information on this formula.