# Geometry of Bicharacteristics

## Abstract

If *p* is of spectral type 1 on *Σ* there is no bicharacteristic with a limit point in *Σ*. When *p* is of spectral type 2 the spectral property of *F*_{ p } itself is not enough to determine completely the behavior of bicharacteristics and we need to look at the third order term of the Taylor expansion of *p* around the reference characteristic to obtain a complete picture of the behavior of bicharacteristics. We prove that there is no bicharacteristic with a limit point in *Σ* near *ρ* if and only if the cube of the vector field *H*_{ S }, introduced in Chap. 2, annihilates *p* on *Σ* near *ρ*. This suggests that the behavior of bicharacteristics near *Σ* closely relates to the possibility of microlocal factorization discussed in Chap. 2. In the last section, we prove that at every effectively hyperbolic characteristic there exists exactly two bicharacteristics that are transversal to *Σ* having which as a limit point. This makes the difference of the geometry of bicharacteristics clearer.

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