scholarly journals Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting

2018 ◽  
Vol 38 (4) ◽  
pp. 2125-2140 ◽  
Author(s):  
Xinsheng Wang ◽  
◽  
Lin Wang ◽  
Yujun Zhu ◽  
◽  
...  
Keyword(s):  
Dominated Splitting

scholarly journals Vortices over Riemann surfaces and dominated splittings

2021 ◽  
pp. 1-26
Author(s):  
THOMAS METTLER ◽  
GABRIEL P. PATERNAIN
Keyword(s):  
Euler Characteristic ◽  
Tangent Bundle ◽  
Riemann Surfaces ◽  
Dominated Splitting ◽  
Vortex Equations ◽  
Special Cases ◽  
Unit Tangent Bundle ◽  
Anosov Flows

Abstract We associate a flow $\phi $ with a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi $ always admits a dominated splitting and identify special cases in which $\phi $ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$ .

scholarly journals The entropy of $C^1$-diffeomorphisms without a dominated splitting

2018 ◽  
Vol 370 (9) ◽  
pp. 6685-6734 ◽  
Author(s):  
Jérôme Buzzi ◽  
Sylvain Crovisier ◽  
Todd Fisher
Keyword(s):  
Dominated Splitting

scholarly journals Julia sets of complex Hénon maps

2018 ◽  
Vol 29 (07) ◽  
pp. 1850047
Author(s):  
Lorenzo Guerini ◽  
Han Peters
Keyword(s):  
Julia Set ◽  
A Priori ◽  
Natural Generalization ◽  
Dominated Splitting ◽  
Henon Maps ◽  
Additional Hypothesis ◽  
Hénon Maps ◽  
Open Questions ◽  
The One ◽  
Weaker Conditions

There are two natural definitions of the Julia set for complex Hénon maps: the sets [Formula: see text] and [Formula: see text]. 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 [Formula: see text], 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 [Formula: see text] 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 [Formula: see text]. 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 [Formula: see text]. Substantially dissipative, Hénon maps admitting a dominated splitting on the possibly larger set [Formula: see text] were recently studied in [M. Lyubich and H. Peters, Structure of partially hyperbolic hénon maps, ArXiv e-prints (2017)].

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