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Julia sets of complex Hénon maps.
- Source :
-
International Journal of Mathematics . Jun2018, Vol. 29 Issue 7, p-1. 22p. - Publication Year :
- 2018
-
Abstract
- There are two natural definitions of the Julia set for complex Hénon maps: the sets and . Whether these two sets are always equal is one of the main open questions in the field. We prove equality when the map acts hyperbolically on the a priori smaller set , under the additional hypothesis of substantial dissipativity. This result was claimed, without using the additional assumption, in [J. E. Fornæss, The julia set of hénon maps, Math. Ann. 334(2) (2006) 457-464], but the proof is incomplete. Our proof closely follows ideas from [J. E. Fornæss, The julia set of hénon maps, Math. Ann. 334(2) (2006) 457-464], deviating at two points, where substantial dissipativity is used. We show that also holds when hyperbolicity is replaced by one of the two weaker conditions. The first is quasi-hyperbolicity, introduced in [E. Bedford and J. Smillie, Polynomial diffeomorphisms of . VIII. Quasi-expansion. Amer. J. Math. 124(2) (2002) 221-271], a natural generalization of the one-dimensional notion of semi-hyperbolicity. The second is the existence of a dominated splitting on . Substantially dissipative, Hénon maps admitting a dominated splitting on the possibly larger set were recently studied in [M. Lyubich and H. Peters, Structure of partially hyperbolic hénon maps, ArXiv e-prints (2017)]. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 0129167X
- Volume :
- 29
- Issue :
- 7
- Database :
- Academic Search Index
- Journal :
- International Journal of Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 130520920
- Full Text :
- https://doi.org/10.1142/S0129167X18500477