scholarly journals Hyperbolicity in the volume-preserving scenario

2012 ◽  
Vol 33 (6) ◽  
pp. 1644-1666 ◽  
Author(s):  
ALEXANDER ARBIETO ◽  
THIAGO CATALAN
Keyword(s):  
Analogous Result ◽  
Periodic Points ◽  
Axiom A ◽  
Volume Preserving

AbstractExtending a result of Mañé, Hayashi proved that every diffeomorphism$f$which has a$C^1$-neighborhood$\mathcal {U}$where all periodic points of any$g\in \mathcal {U}$are hyperbolic is an Axiom A diffeomorphism. Here we prove the analogous result in the volume-preserving scenario.

scholarly journals Vector fields satisfying the barycenter property

2018 ◽  
Vol 16 (1) ◽  
pp. 429-436 ◽  
Author(s):  
Manseob Lee
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Axiom A ◽  
Divergence Free Vector ◽  
Divergence Free ◽  
Volume Preserving

AbstractWe show that if a vector fieldXhas theC1robustly barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, if a genericC1-vector field has the barycenter property then it does not have singularities and it is Axiom A without cycles. Moreover, we apply the results to the divergence free vector fields. It is an extension of the results of the barycenter property for generic diffeomorphisms and volume preserving diffeomorphisms [1].

scholarly journals A lower bound for topological entropy of generic non-Anosov symplectic diffeomorphisms

2013 ◽  
Vol 34 (5) ◽  
pp. 1503-1524 ◽  
Author(s):  
THIAGO CATALAN ◽  
ALI TAHZIBI
Keyword(s):  
Lyapunov Exponent ◽  
Lower Bound ◽  
Topological Entropy ◽  
Upper Semicontinuity ◽  
Periodic Points ◽  
Surface Diffeomorphisms ◽  
Symplectic Diffeomorphisms ◽  
Symbolic Extension ◽  
Symplectic Diffeomorphism ◽  
Volume Preserving

AbstractWe prove that a${C}^{1} $generic symplectic diffeomorphism is either Anosov or its topological entropy is bounded from below by the supremum over the smallest positive Lyapunov exponent of its periodic points. We also prove that${C}^{1} $generic symplectic diffeomorphisms outside the Anosov ones do not admit symbolic extension and, finally, we give examples of volume preserving surface diffeomorphisms which are not points of upper semicontinuity of the entropy function in the${C}^{1} $topology.

Continuous flows generate few homeomorphisms

2020 ◽  
Vol 63 (4) ◽  
pp. 971-983
Author(s):  
Wescley Bonomo ◽  
Paulo Varandas
Keyword(s):  
Topological Entropy ◽  
Continuous Flow ◽  
Compact Manifold ◽  
Necessary Conditions ◽  
Periodic Points ◽  
Compact Manifolds ◽  
Continuous Flows ◽  
Topological Obstructions ◽  
Volume Preserving ◽  
Dense Set

We describe topological obstructions (involving periodic points, topological entropy and rotation sets) for a homeomorphism on a compact manifold to embed in a continuous flow. We prove that homeomorphisms in a $C^{0}$-open and dense set of homeomorphisms isotopic to the identity in compact manifolds of dimension at least two are not the time-1 map of a continuous flow. Such property is also true for volume-preserving homeomorphisms in compact manifolds of dimension at least five. In the case of conservative homeomorphisms of the torus $\mathbb {T}^{d} (d\ge 2)$ isotopic to identity, we describe necessary conditions for a homeomorphism to be flowable in terms of the rotation sets.

scholarly journals On Anosov diffeomorphisms with asymptotically conformal periodic data

2009 ◽  
Vol 29 (1) ◽  
pp. 117-136 ◽  
Author(s):  
BORIS KALININ ◽  
VICTORIA SADOVSKAYA
Keyword(s):  
Periodic Points ◽  
Finite Cover ◽  
Rigidity Result ◽  
Anosov Diffeomorphisms ◽  
Local Rigidity ◽  
Return Maps ◽  
Periodic Data ◽  
Conformal Structures ◽  
Derivatives Of ◽  
Volume Preserving

AbstractWe consider transitive Anosov diffeomorphisms for which every periodic orbit has only one positive and one negative Lyapunov exponent. We prove various properties of such systems, including strong pinching, C1+β smoothness of the Anosov splitting, and C1 smoothness of measurable invariant conformal structures and distributions. We apply these results to volume-preserving diffeomorphisms with two-dimensional stable and unstable distributions and diagonalizable derivatives of the return maps at periodic points. We show that a finite cover of such a diffeomorphism is smoothly conjugate to an Anosov automorphism of 𝕋4; as a corollary, we obtain local rigidity for such diffeomorphisms. We also establish a local rigidity result for Anosov diffeomorphisms in dimension three.

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