Phil Howard: Continuity properties of transport coefficients in simple maps
Abstract:
We consider families of dynamics that can be described in terms of Perron-Frobenius operators with exponential mixing properties. For piecewise $C^2$ expanding interval maps we rigorously prove continuity properties of the drift $J(\lambda)$ and of the diffusion coefficient $D(\lambda)$ under parameter variation. Our main result is that $D(\lambda)$ has a modulus of continuity of order $\O(|\delta\lambda|\cdot|\log|\delta\lambda|)^{2})$, i.e. $D(\lambda)$ is Lipschitz continuous up to quadratic logarithmic corrections. For a special class of piecewise linear maps we provide more precise estimates at specific parameter values. Our analytical findings are verified numerically for the latter class of maps by using exact formulas for the transport coefficients. We numerically observe strong local variations of all continuity properties.