Given a closed Riemannian manifold, we show how to close an orbit of the geodesic flow by a small perturbation of the metric in the C1 topology.

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We define an analogue of the Casimir element for a graded affine Hecke algebra $$ \mathbb{H} $$ , and then introduce an approximate square-root called the Dirac element. Using it, we define the Dirac cohomology$H$^{$D$}($X$) of an $$ \mathbb{H} $$ -module$X$, and show that$H$^{$D$}($X$) carries a representation of a canonical double cover of the Weyl group $$ \widetilde{W} $$ . Our main result shows that the $$ \widetilde{W} $$ -structure on the Dirac cohomology of an irreducible $$ \mathbb{H} $$ -module$X$determines the central character of$X$in a precise way. This can be interpreted as$p$-adic analogue of a conjecture of Vogan for Harish-Chandra modules. We also apply our results to the study of unitary representations of $$ \mathbb{H} $$ .

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Let$F$be families of meromorphic functions in a domain$D$, and let$R$be a rational function whose degree is at least 3. If, for any$f∈$$F$, the composite function$R(f)$has no fixed-point in$D$, then$F$is normal in$D$. The number 3 is best possible. A new and much simplified proof of a result of Pang and Zalcman concerning normality and, shared values is also given.