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arXiv · 1805.05910

Linear response theory for diffeomorphisms with tangencies of stable and unstable manifolds. [A contribution to the Gallavotti-Cohen chaotic hypothesis.]

David Ruelle
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Abstract

This note presents a non-rigorous study of the linear response for an SRB (or `natural physical') measure $\rho$ of a diffeomorphism $f$ in the presence of tangencies of the stable and unstable manifolds of $\rho$. We propose that generically, if $\rho$ has no zero Lyapunov exponent, if its stable dimension is sufficiently large (greater than 1/2 or perhaps 3/2) and if it is exponentially mixing in a suitable sense, then the following formal expression for the first derivative of $\rho(\phi)$ with respect to $f$ along $X$ is convergent: $$ \Psi(z)=\sum_{n=0}^\infty z^n\int\rho(dx)\,X(x)\cdot\nabla_x(\phi\circ f^n)\qquad{\rm for}\qquad z=1 $$ This suggests that an SRB measure may exist for small perturbations of $f$, with weak differentiability.

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David Ruelle. 2018-05-15. Linear response theory for diffeomorphisms with tangencies of stable and unstable manifolds. [A contribution to the Gallavotti-Cohen chaotic hypothesis.]. https://doi.org/10.1088/1361-6544/aae740

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