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Extreme instability of the horocycle flow

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Abstract

Let g be aC 3 negatively curved Riemannian metric on a compact connected orientable surfaceS. LetB be the collection of all metrics resulting from sufficiently small conformal changes of the metricg. (1) Then there is a constantA > 0 such that ifB then the\(\bar d\) distance between the horocycle flowĥ t (Margulis parametrization) of (S, ĝ) and the rescaled horocycle flowh ct of (S, g) is at leastA (∀c > 0). No other dynamical system is known to have such extreme instability. (2) Fix ε > 0. Then there is anN > 0 so that if we are given samples {ξ} N0 {η} N0 which arose from the horocycle flows corresponding to two of the metricsĝ, gB, then either the two samples are\(\bar d\) farther thanA/2 apart or the two surfaces are closer than ε. This holds even if these samples are slightly inaccurate.

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Troubetzkoy, S.E. Extreme instability of the horocycle flow. J. Anal. Math. 57, 37–63 (1991). https://doi.org/10.1007/BF03041065

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  • DOI: https://doi.org/10.1007/BF03041065

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