Kannan Distortion Profiles and Intrinsic Threshold Geometry

Abstract

We introduce the finite-distortion Kannan profile, which measures the best Kannan coefficient achievable under a prescribed bi-Lipschitz change of metric. Two lower bounds distinguish geometric reweighting from the asymptotic rate of iteration. For threshold extensions, we obtain an exact decomposition into a switching contribution and the profile of the base dynamics. This yields an optimal piecewise affine remetrization and the sharp distortion threshold for genuine Kannan contractivity. We also compute the intrinsic chain geometry generated by a weighted directed excess, clarifying when a natural local-cost geometry is quantitatively close to the optimal one.