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Let W be an infinite Coxeter group. We initiate the study of the set E oflimit points of "normalized" positive roots (representing the directions of theroots) of W. We show that E is contained in the isotropic cone of the bilinearform B associated to a geometric representation, and illustrate this propertywith numerous examples and pictures in rank 3 and 4. We also define a naturalgeometric action of W on E, and then we exhibit a countable subset of E, formedby limit points for the dihedral reflection subgroups of W. We explain thatthis subset is built from the intersection with Q of the lines passing throughtwo positive roots, and finally we establish that it is dense in E.

Asymptotical behaviour of roots of infinite Coxeter groups