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A problem at the interface of differential geometry and dynamical systems gives rise to the question of what control of solutions of the Riccati equation 
+ x
2 = k(t) with positive right-hand side can be obtained from control of the forcing term k. We show that a known result about “relative” pinching is optimal and refine two known theorems. This gives improved regularity of horospheric foliations and may be of interest in control or filtering theory.
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