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Generic area-preserving reversible diffeomorphisms  [PDF]
Mário Bessa,Maria Carvalho,Alexandre Rodrigues
Mathematics , 2014, DOI: 10.1088/0951-7715/28/6/1695
Abstract: Let M be a surface and R an involution in M whose set of fixed points is a submanifold with dimension 1 and such that R is an isometry. We will show that there is a residual subset of C1 area-preserving R-reversible diffeomorphisms which are either Anosov or have zero Lyapunov exponents at almost every point.
A C1 generic condition for existence of symbolic extensions of volume preserving diffeomorphisms  [PDF]
Thiago Catalan
Mathematics , 2011, DOI: 10.1088/0951-7715/25/12/3505
Abstract: We prove that a C1-generic volume preserving diffeomorphism has a symbolic extension if and only if this diffeomorphism is partial hyperbolic. This result is obtained by means of good dichotomies. In particular, we prove Bonatti's conjecture in the volume preserving scenario. More precisely, in the complement of Anosov diffeomorphisms we have densely robust heterodimensional cycles.
Generic Continuity of Metric Entropy for Volume-preserving Diffeomorphisms  [PDF]
Jiagang Yang,Yunhua Zhou
Mathematics , 2013,
Abstract: Let $M$ be a compact manifold and $\text{Diff}^1_m(M)$ be the set of $C^1$ volume-preserving diffeomorphisms of $M$. We prove that there is a residual subset $\mathcal {R}\subset \text{Diff}^1_m(M)$ such that each $f\in \mathcal{R}$ is a continuity point of the map $g\to h_m(g)$ from $\text{Diff}^1_m(M)$ to $\mathbb{R}$, where $h_m(g)$ is the metric entropy of $g$ with respect to volume measure $m$.
Transitivity of conservative diffeomorphisms isotopic to Anosov on $\mathbb{T}^3$  [PDF]
Martin Andersson,Shaobo Gan
Mathematics , 2015,
Abstract: 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.
Nonuniform Hyperbolicity, Global Dominated Splittings and Generic Properties of Volume-Preserving Diffeomorphisms  [PDF]
Artur Avila,Jairo Bochi
Mathematics , 2009,
Abstract: We study generic volume-preserving diffeomorphisms on compact manifolds. We show that the following property holds generically in the $C^1$ topology: Either there is at least one zero Lyapunov exponent at almost every point, or the set of points with only non-zero exponents forms an ergodic component. Moreover, if this nonuniformly hyperbolic component has positive measure then it is essentially dense in the manifold (that is, it has a positive measure intersection with any nonempty open set) and there is a global dominated splitting. For the proof we establish some new properties of independent interest that hold $C^r$-generically for any $r \geq 1$, namely: the continuity of the ergodic decomposition, the persistence of invariant sets, and the $L^1$-continuity of Lyapunov exponents.
Universal description of viscoelasticity with foliation preserving diffeomorphisms  [PDF]
Tatsuo Azeyanagi,Masafumi Fukuma,Hikaru Kawai,Kentaroh Yoshida
Physics , 2009, DOI: 10.1016/j.physletb.2009.10.027
Abstract: A universal description is proposed for generic viscoelastic systems with a single relaxation time. Foliation preserving diffeomorphisms are introduced as an underlying symmetry which naturally interpolates between the two extreme limits of elasticity and fluidity. The symmetry is found to be powerful enough to determine the dynamics in the first order of strains.
Homoclinic Points For Area-Preserving Surface Diffeomorphisms  [PDF]
Zhihong Xia
Mathematics , 2006,
Abstract: We show a $C^r$ connecting lemma for area-preserving surface diffeomorphisms and for periodic Hamiltonian on surfaces. We prove that for a generic $C^r$, $r=1, 2, ...$, $\infty$, area-preserving diffeomorphism on a compact orientable surface, homotopic to identity, every hyperbolic periodic point has a transversal homoclinic point. We also show that for a $C^r$, $r=1, 2, ...$, $\infty$ generic time periodic Hamiltonian vector field in a compact orientable surface, every hyperbolic periodic trajectory has a transversal homoclinic point. The proof explores the special properties of diffeomorphisms that are generated by Hamiltonian flows.
A generic dimensional property of the invariant measures for circle diffeomorphisms  [PDF]
Shigenori Matsumoto
Mathematics , 2011,
Abstract: Given any Liouville number $\alpha$, it is shown that the nullity of the Hausdorff dimension of the invariant measure is generic in the space of the orientation preserving $C^\infty$ diffeomorphisms of the circle with rotation number $\alpha$.
Transitivity and topological mixing for C1 diffeomorphisms  [PDF]
Flavio Abdenur,Sylvain Crovisier
Mathematics , 2011,
Abstract: We prove that, on connected compact manifolds, both C1-generic conservative diffeomorphisms and C1-generic transitive diffeomorphisms are topologically mixing. This is obtained through a description of the periods of a homoclinic class and by a control of the period of the periodic points given by the closing lemma.
Robust transitivity and density of periodic points of partially hyperbolic diffeomorphisms  [PDF]
Alien Herrera Torres,Ana Tercia Monteiro Oliveira
Mathematics , 2014,
Abstract: We prove results related to robust transitivity and density of periodic points of Partially Hyperbolic Diffeomorphisms under conditions involving Accessibility and a property in the tangent bundle .
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