Axiom A diffeomorphisms derived from Anosov flows
Christian Bonatti Nancy Guelman
Journal of Modern Dynamics 2010, 4(1): 1-63 doi: 10.3934/jmd.2010.4.1
Let $M$ be a closed $3$-manifold, and let $X_t$ be a transitive Anosov flow. We construct a diffeomorphism of the form $f(p)=Y_{t(p)}(p)$, where $Y$ is an Anosov flow equivalent to $X$. The diffeomorphism $f$ is structurally stable (satisfies Axiom A and the strong transversality condition); the non-wandering set of $f$ is the union of a transitive attractor and a transitive repeller; and $f$ is also partially hyperbolic (the direction $\RR.Y$ is the central bundle).
keywords: partial hyperbolicity AxiomA diffeomorphism Birkhoff sections Anosov flows perturbations.
$C^1$-generic conservative diffeomorphisms have trivial centralizer
Christian Bonatti Sylvain Crovisier Amie Wilkinson
Journal of Modern Dynamics 2008, 2(2): 359-373 doi: 10.3934/jmd.2008.2.359
We prove that the spaces of $C^1$ symplectomorphisms and of $C^1$ volume-preserving diffeomorphisms of connected manifolds contain residual subsets of diffeomorphisms whose centralizers are trivial.
keywords: Trivial centralizer trivial symmetries $C^1$ generic properties.
Super-exponential growth of the number of periodic orbits inside homoclinic classes
Christian Bonatti Lorenzo J. Díaz Todd Fisher
Discrete & Continuous Dynamical Systems - A 2008, 20(3): 589-604 doi: 10.3934/dcds.2008.20.589
We show that there is a residual subset $\S (M)$ of Diff$^1$ (M) such that, for every $f\in \S(M)$, any homoclinic class of $f$ containing periodic saddles $p$ and $q$ of indices $\alpha$ and $\beta$ respectively, where $\alpha< \beta$, has superexponential growth of the number of periodic points inside the homoclinic class. Furthermore, it is shown that the super-exponential growth occurs for hyperbolic periodic points of index $\gamma$ inside the homoclinic class for every $\gamma\in[\alpha,\beta]$.
keywords: index of a saddle Artin-Mazur diffeomorphism heterodimensional cycle chain recurrence class homoclinic class symbolic extensions.
The $C^{1+\alpha }$ hypothesis in Pesin Theory revisited
Christian Bonatti Sylvain Crovisier Katsutoshi Shinohara
Journal of Modern Dynamics 2013, 7(4): 605-618 doi: 10.3934/jmd.2013.7.605
We show that for every compact $3$-manifold $M$ there exists an open subset of $Diff^1(M)$ in which every generic diffeomorphism admits uncountably many ergodic probability measures that are hyperbolic while their supports are disjoint and admit a basis of attracting neighborhoods and a basis of repelling neighborhoods. As a consequence, the points in the support of these measures have no stable and no unstable manifolds. This contrasts with the higher-regularity case, where Pesin Theory gives stable and unstable manifolds with complementary dimensions at almost every point. We also give such an example in dimension two, without local genericity.
keywords: wild diffeomorphism dominated splitting Pesin theory Lyapunov exponents.
On the hyperbolicity of homoclinic classes
Christian Bonatti Shaobo Gan Dawei Yang
Discrete & Continuous Dynamical Systems - A 2009, 25(4): 1143-1162 doi: 10.3934/dcds.2009.25.1143
We give a sufficient criterion for the hyperbolicity of a homoclinic class. More precisely, if the homoclinic class $H(p)$ admits a partially hyperbolic splitting $T_{H(p)}M=E^s\oplus_{_<}F$, where $E^s$ is uniformly contracting and $\dim E^s= \ $ind$(p)$, and all periodic points homoclinically related with $p$ are uniformly $E^u$-expanding at the period, then $H(p)$ is hyperbolic. We also give some consequences of this result.
keywords: hyperbolic time shadowing lemma. homoclinic class

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