Alternative proofs of linear response for piecewise expanding unimodal maps

作者：Baladi Viviane^*; Smania Daniel

来源：Ergodic Theory and Dynamical Systems, 2010, 30(01): 1-20.

DOI：10.1017/S0143385708001077

We give two new proofs that the Sinai-Ruelle-Bowen (SRB) measure t bar right arrow mu(t) 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 original proof of Baladi and Smania [Linear response formula for piecewise expanding unimodal maps. Nonlinearity 21 (2008). 677-711], but require proving Holder continuity of the infinitesimal conjugacy alpha (a new result, of independent interest) and using spaces of bounded p-variation. The first new proof gives differentiability of higher order of integral psi d mu(t) if f(t) is smooth enough and stays in the topological class of f(0) and if psi is 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 by Baladi and Smania, by an argument more conceptual than the 'brute force' cancellation mechanism used by Baladi and Smania.

• 出版日期2010-2