Stably ergodic approximation: two examples

2000 ◽  
Vol 20 (3) ◽  
pp. 875-893 ◽  
Author(s):  
MICHAEL SHUB ◽  
AMIE WILKINSON
Keyword(s):  
Compact Manifold ◽  
Standard Map ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Volume Preserving

It has been conjectured that the stably ergodic diffeomorphisms are open and dense in the space of volume-preserving, partially hyperbolic diffeomorphisms of a compact manifold. In this paper we deal with two recalcitrant examples; the standard map cross Anosov and the ergodic automorphisms of the 4-torus. In both cases we show that they may be approximated by stably ergodic diffeomorphisms which have the stable accessibility property.

scholarly journals Transitivity of conservative diffeomorphisms isotopic to Anosov on

2016 ◽  
Vol 38 (1) ◽  
pp. 1-9
Author(s):  
MARTIN ANDERSSON ◽  
SHAOBO GAN
Keyword(s):  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Volume Preserving

We prove transitivity for volume-preserving $C^{1+}$ diffeomorphisms on $\mathbb{T}^{3}$ which are isotopic to a linear Anosov automorphism along a path of weakly partially hyperbolic diffeomorphisms.

scholarly journals Center foliation: absolute continuity, disintegration and rigidity

2014 ◽  
Vol 36 (1) ◽  
pp. 256-275 ◽  
Author(s):  
RÉGIS VARÃO
Keyword(s):  
Absolute Continuity ◽  
Geodesic Flows ◽  
Absolutely Continuous ◽  
Rigidity Result ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Smooth Conjugacy ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism ◽  
Volume Preserving

In this paper we address the issues of absolute continuity for the center foliation, as well as the disintegration on the non-absolute continuous case and rigidity of volume-preserving partially hyperbolic diffeomorphisms isotopic to a linear Anosov automorphism on $\mathbb{T}^{3}$. It is shown that the disintegration of volume on center leaves for these diffeomorphisms may be neither atomic nor Lebesgue, in contrast to the dichotomy (Lebesgue or atomic) obtained by Avila, Viana and Wilkinson [Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows. Preprint, 2012, arXiv:1110.2365v2] for perturbations of time-one of geodesic flow. In the case of atomic disintegration of volume on the center leaves of an Anosov diffeomorphism on $\mathbb{T}^{3}$, we show that it has to be one atom per leaf. Moreover, we show that not even a $C^{1}$ center foliation implies a rigidity result. However, for a volume-preserving partially hyperbolic diffeomorphism isotopic to a linear Anosov automorphism, assuming the center foliation is $C^{1}$ and transversely absolutely continuous with bounded Jacobians, we obtain smooth conjugacy to its linearization.

scholarly journals Rigidity for partially hyperbolic diffeomorphisms

2017 ◽  
Vol 38 (8) ◽  
pp. 3188-3200 ◽  
Author(s):  
RÉGIS VARÃO
Keyword(s):  
Absolute Continuity ◽  
Geodesic Flows ◽  
Anosov Flow ◽  
Rigidity Result ◽  
Global Rigidity ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Natural Hypothesis ◽  
Volume Preserving

In this work we completely classify $C^{\infty }$ conjugacy for smooth conservative (pointwise) partially hyperbolic diffeomorphisms homotopic to a linear Anosov automorphism on the 3-torus by its center foliation behavior. We prove that the uniform version of absolute continuity for the center foliation is the natural hypothesis to obtain $C^{\infty }$ conjugacy to its linear Anosov automorphism. Avila, Viana and Wilkinson [Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows. J. Eur. Math. Soc. (JEMS)17(6) (2015), 1435–1462] proved that for a perturbation in the volume preserving case of the time-one map of an Anosov flow absolute continuity of the center foliation implies smooth rigidity. The absolute version of absolute continuity is the appropriate scenario for our context since it is not possible to obtain a result analogous to that of Avila, Viana and Wilkinson for our class of maps, for absolute continuity alone fails miserably to imply smooth rigidity for our class of maps. Our theorem is a global rigidity result as we do not assume the diffeomorphism to be at some distance from the linear Anosov automorphism. We also do not assume ergodicity. In particular, a metric condition on the center foliation implies ergodicity and $C^{\infty }$ center foliation.

Absolute continuity, Lyapunov exponents, and rigidity II: systems with compact center leaves

2021 ◽  
pp. 1-54
Author(s):  
A. AVILA ◽  
MARCELO VIANA ◽  
A. WILKINSON
Keyword(s):  
Hyperbolic Systems ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Hyperbolic Diffeomorphisms ◽  
Volume Measure ◽  
Rigid Model ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partial Extension ◽  
Volume Preserving

Abstract We explore new connections between the dynamics of conservative partially hyperbolic systems and the geometric measure-theoretic properties of their invariant foliations. Our methods are applied to two main classes of volume-preserving diffeomorphisms: fibered partially hyperbolic diffeomorphisms and center-fixing partially hyperbolic systems. When the center is one-dimensional, assuming the diffeomorphism is accessible, we prove that the disintegration of the volume measure along the center foliation is either atomic or Lebesgue. Moreover, the latter case is rigid in dimension three (this does not require accessibility): the center foliation is actually smooth and the diffeomorphism is smoothly conjugate to an explicit rigid model. A partial extension to fibered partially hyperbolic systems with compact fibers of any dimension is also obtained. A common feature of these classes of diffeomorphisms is that the center leaves either are compact or can be made compact by taking an appropriate dynamically defined quotient. For volume-preserving partially hyperbolic diffeomorphisms whose center foliation is absolutely continuous, if the generic center leaf is a circle, then every center leaf is compact.

scholarly journals On the stable ergodicity of Berger–Carrasco’s example

2018 ◽  
Vol 40 (4) ◽  
pp. 1008-1056
Author(s):  
DAVI OBATA
Keyword(s):  
Lyapunov Exponents ◽  
Two Dimensional ◽  
Dominated Splitting ◽  
Standard Map ◽  
Hyperbolic Diffeomorphisms ◽  
Stable Ergodicity ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphism ◽  
Volume Preserving ◽  
Math Phys

We prove the stable ergodicity of an example of a volume-preserving, partially hyperbolic diffeomorphism introduced by Berger and Carrasco in [Berger and Carrasco. Non-uniformly hyperbolic diffeomorphisms derived from the standard map. Comm. Math. Phys.329 (2014), 239–262]. This example is robustly non-uniformly hyperbolic, with a two-dimensional center; almost every point has both positive and negative Lyapunov exponents along the center direction and does not admit a dominated splitting of the center direction. The main novelty of our proof is that we do not use accessibility.

scholarly journals Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

2020 ◽  
pp. 1-17
Author(s):  
THOMAS BARTHELMÉ ◽  
SERGIO R. FENLEY ◽  
STEVEN FRANKEL ◽  
RAFAEL POTRIE
Keyword(s):  
Large Class ◽  
Universal Cover ◽  
Isotopy Class ◽  
Seifert Manifold ◽  
Hyperbolic Diffeomorphisms ◽  
Surface Homeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism

Abstract We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

scholarly journals Quasi-shadowing for partially hyperbolic diffeomorphisms

2014 ◽  
Vol 35 (2) ◽  
pp. 412-430 ◽  
Author(s):  
HUYI HU ◽  
YUNHUA ZHOU ◽  
YUJUN ZHU
Keyword(s):  
Spectral Decomposition ◽  
Hyperbolic Systems ◽  
Decomposition Theorem ◽  
Closing Lemma ◽  
Shadowing Property ◽  
Hyperbolic Diffeomorphisms ◽  
Uniformly Hyperbolic Systems ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism

AbstractA partially hyperbolic diffeomorphism $f$ has the quasi-shadowing property if for any pseudo orbit $\{x_{k}\}_{k\in \mathbb{Z}}$, there is a sequence of points $\{y_{k}\}_{k\in \mathbb{Z}}$ tracing it in which $y_{k+1}$ is obtained from $f(y_{k})$ by a motion ${\it\tau}$ along the center direction. We show that any partially hyperbolic diffeomorphism has the quasi-shadowing property, and if $f$ has a $C^{1}$ center foliation then we can require ${\it\tau}$ to move the points along the center foliation. As applications, we show that any partially hyperbolic diffeomorphism is topologically quasi-stable under $C^{0}$-perturbation. When $f$ has a uniformly compact $C^{1}$ center foliation, we also give partially hyperbolic diffeomorphism versions of some theorems which hold for uniformly hyperbolic systems, such as the Anosov closing lemma, the cloud lemma and the spectral decomposition theorem.

scholarly journals On the non-robustness of intermingled basins

2016 ◽  
Vol 38 (1) ◽  
pp. 384-400 ◽  
Author(s):  
RAÚL URES ◽  
CARLOS H. VÁSQUEZ
Keyword(s):  
Natural Question ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Physical Measures ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic Diffeomorphism

It is well known that it is possible to construct a partially hyperbolic diffeomorphism on the 3-torus in a similar way to Kan’s example. It has two hyperbolic physical measures with intermingled basins supported on two embedded tori with Anosov dynamics. A natural question is how robust is the intermingled basin phenomenon for diffeomorphisms defined on boundaryless manifolds? In this work we study partially hyperbolic diffeomorphisms on the 3-torus and show that the intermingled basin phenomenon is not robust.

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