Abstract
We study an interplay between homoclinic behavior and singularities in surface endomorphisms. We show that appropriate rescalings near homoclinic orbits intersecting fold singularities yield families of noninvertible Hénon-like maps. Then we construct positive measure sets of parameters corresponding to maps which exhibit nonuniformly hyperbolic behavior. This implies an extension of the celebrated theorem of Benedicks and Carleson, and that of Mora and Viana to surface endomorphisms.
| Original language | English |
|---|---|
| Pages (from-to) | 53-113 |
| Number of pages | 61 |
| Journal | Tokyo Journal of Mathematics |
| Volume | 34 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2011 Jan 1 |
| Externally published | Yes |
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ASJC Scopus subject areas
- Mathematics(all)
Cite this
Abundance of non-uniform hyperbolicity in bifurcations of surface endomorphisms. / Takahasi, Hiroki.
In: Tokyo Journal of Mathematics, Vol. 34, No. 1, 01.01.2011, p. 53-113.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Abundance of non-uniform hyperbolicity in bifurcations of surface endomorphisms
AU - Takahasi, Hiroki
PY - 2011/1/1
Y1 - 2011/1/1
N2 - We study an interplay between homoclinic behavior and singularities in surface endomorphisms. We show that appropriate rescalings near homoclinic orbits intersecting fold singularities yield families of noninvertible Hénon-like maps. Then we construct positive measure sets of parameters corresponding to maps which exhibit nonuniformly hyperbolic behavior. This implies an extension of the celebrated theorem of Benedicks and Carleson, and that of Mora and Viana to surface endomorphisms.
AB - We study an interplay between homoclinic behavior and singularities in surface endomorphisms. We show that appropriate rescalings near homoclinic orbits intersecting fold singularities yield families of noninvertible Hénon-like maps. Then we construct positive measure sets of parameters corresponding to maps which exhibit nonuniformly hyperbolic behavior. This implies an extension of the celebrated theorem of Benedicks and Carleson, and that of Mora and Viana to surface endomorphisms.
UR - http://www.scopus.com/inward/record.url?scp=84860386381&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84860386381&partnerID=8YFLogxK
U2 - 10.3836/tjm/1313074445
DO - 10.3836/tjm/1313074445
M3 - Article
AN - SCOPUS:84860386381
VL - 34
SP - 53
EP - 113
JO - Tokyo Journal of Mathematics
JF - Tokyo Journal of Mathematics
SN - 0387-3870
IS - 1
ER -