Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow

2007 ◽  
Keyword(s):  
Hyperbolic Dynamics ◽  
Partially Hyperbolic ◽  
Teichmüller Flow

scholarly journals Center Lyapunov exponents in partially hyperbolic dynamics

2015 ◽  
Vol 8 (3/4) ◽  
pp. 549-576
Author(s):  
Ali Tahzibi ◽  
Andrey Gogolev
Keyword(s):  
Lyapunov Exponents ◽  
Hyperbolic Dynamics ◽  
Partially Hyperbolic

Partially hyperbolic dynamics in dimension three

2017 ◽  
Vol 38 (8) ◽  
pp. 2801-2837 ◽  
Author(s):  
PABLO D. CARRASCO ◽  
FEDERICO RODRIGUEZ-HERTZ ◽  
JANA RODRIGUEZ-HERTZ ◽  
RAÚL URES
Keyword(s):  
State Of The Art ◽  
Natural Generalization ◽  
The State ◽  
Great Activity ◽  
Partial Hyperbolicity ◽  
Hyperbolic Dynamics ◽  
Partially Hyperbolic ◽  
The 1960S

Partial hyperbolicity appeared in the 1960s as a natural generalization of hyperbolicity. In the last 20 years, there has been great activity in this area. Here we survey the state of the art in some related topics, focusing especially on partial hyperbolicity in dimension three. The reason for this is not only that it is the smallest dimension in which non-degenerate partial hyperbolicity can occur, but also that the topology of$3$-manifolds influences the dynamics in revealing ways.

Partially hyperbolic and transitive dynamics generated by heteroclinic cycles

2001 ◽  
Vol 21 (1) ◽  
pp. 25-76 ◽  
Author(s):  
LORENZO J. DÍAZ ◽  
JORGE ROCHA
Keyword(s):  
Periodic Points ◽  
Heteroclinic Cycles ◽  
Hyperbolic Dynamics ◽  
Open Set ◽  
Transitive Set ◽  
Locally Maximal ◽  
Partially Hyperbolic ◽  
Homoclinic Tangencies

We study \mathcal{C}^k-diffeomorphisms, k\ge 1, f: M\to M, exhibiting heterodimensional cycles (i.e. cycles containing periodic points of different stable indices). We prove that if f can not be \mathcal{C}^k-approximated by diffeomorphisms with homoclinic tangencies, then f is in the closure of an open set \mathcal{U}\subset \operatorname{Diff}^k(M) consisting of diffeomorphisms g with a non-hyperbolic transitive set \Lambda_g which is locally maximal and strongly partially hyperbolic (the partially hyperbolic splitting at \Lambda_g has three non-trivial directions). As a consequence, in the case of 3-manifolds, we give new examples of open sets of \mathcal{C}^1-diffeomorphisms for which residually infinitely many sinks or sources coexist (\mathcal{C}^1-Newhouse's phenomenon). We also prove that the occurrence of non-hyperbolic dynamics has persistent character in the unfolding of heterodimensional cycles.

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