FIXED-ROTATION MINKOWSKI AVERAGING: CLASSIFICATION AND SHARP RATES

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Abstract We study a discrete process on planar convex bodies in which, at each step, a body is replaced by a weighted Minkowski average of itself and its rotation by a fixed angle. Up to translation and uniform scaling, this produces a rigid averaging dynamical system. We give a complete classification of the limit shapes. If the angle is an irrational multiple of $2\pi $ , the iterates converge to a disk. If the angle is rational, they converge to the average of finitely many rotated copies of the initial body. We also obtain sharp convergence rates. In the rational case, the decay is uniform and exponential with an explicit constant depending only on the weight and the denominator of the angle. For irrational angles, we prove quantitative rates under a mild number-theoretic condition that holds for almost every angle: low regularity inputs have polynomial decay up to a logarithmic factor, while real analytic inputs have stretched exponential decay. For angles with bounded continued fraction coefficients, we give matching lower bounds along subsequences. These results describe the global attractors of the dynamics and indicate the absence of chaotic behaviour.

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