# Classifying immersions into IR4 over stable maps of by Harold Levine

By Harold Levine

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Extra info for Classifying immersions into IR4 over stable maps of 3-manifolds into IR2

Example text

La mesure µf est m´elangeante, et donc ergodique. Nous rappellerons les notions d’entropie topologique et d’entropie m´etrique plus loin. Depuis 1977, on sait que l’entropie topologique d’un endomorphisme de Pk (C) de degr´e alg´ebrique d 2 est k log d. La minoration est due `a Misiurewicz et Przytycki [23] et la majoration est due ` a Gromov [17]. Comme la mesure d’´equilibre µf est de jacobien constant dk , la formule de Rohlin-Parry [24] dit que l’entropie m´etrique de µf vaut k log d. Enfin, le principe variationnel affirme que l’entropie topologique est le supremum des entropies m´etriques.

On pose Γn = x, f (x), . . , f ◦n−1 (x) , x ∈ Pk (C) et on d´efinit lov(f ) = lim sup n→+∞ 1 1 log(Vol(Γn )) = lim sup log n n→+∞ n Γn ωn∧k , n o` u ωn est la forme de K¨ ahler sur Pk (C) induite par la forme de Fubini-Study sur chaque facteur. 6. — Si f : Pk (C) → Pk (C) est un endomorphisme holomorphe de degr´e alg´ebrique d, on a lov(f ) = k log d. D´emonstration. — C’est un calcul cohomologique. On a Vol(Γn ) = Γn ωn∧k = (ω + f ∗ ω + . . ,n−1}k = (1 + d + · · · + dn−1 )k = Par cons´equent, lov f = lim sup n→+∞ dn − 1 d−1 k .

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