2010/01/30 by David Ruelle · 7 citations
Mathematics · Physics and Astronomy · #Calabi–Yau manifold #Chaos control and synchronization #Diffeomorphism #Dimension (graph theory) #Function (biology) #Gravitational singularity #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematical physics #Mathematics #Measure (data warehouse) #Physics #Power series #Pure mathematics #Quantum chaos and dynamical systems #RADIUS #Radius of convergence #Schwarzian derivative #nlin.CD
paper · pdf · doi:10.1098/rsta.2010.0260
published in Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences 369(1935), 482-493 (Royal Society) · 12 pages
arxiv created 2010/01/30 · openalex publication_date 2010/12/13 · arxiv updated 2015/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let ρ be a Sinai-Ruelle-Bowen (SRB or 'physical') measure for the discrete time evolution given by a map f, and let ρ(A) denote the expectation value of a smooth function A. If f depends on a parameter, the derivative δρ(A) of ρ(A) with respect to the parameter is formally given by the value of the so-called susceptibility function Ψ(z) at z=1. When f is a uniformly hyperbolic diffeomorphism, it has been proved that the power series Ψ(z) has a radius of convergence r(Ψ)>1, and that δρ(A)=Ψ(1), but it is known that r(Ψ)<1 in some other cases. One reason why f may fail to be uniformly hyperbolic is if there are tangencies between the stable and unstable manifolds for (f,ρ). The present paper gives a crude, non-rigorous, analysis of this situation in terms of the Hausdorff dimension d of ρ in the stable direction. We find that the tangencies produce singularities of Ψ(z) for |z|<1 if d<1/2, but only for |z|>1 if d>1/2. In particular, if d>1/2, we may hope that Ψ(1) makes sense, and the derivative δρ(A)=Ψ(1) thus has a chance to be defined.