scholarly journals Constructing infra-nilmanifolds admitting an Anosov diffeomorphism

2011 ◽  
Vol 228 (6) ◽  
pp. 3300-3319 ◽  
Author(s):  
Karel Dekimpe ◽  
Kelly Verheyen
Keyword(s):  
Anosov Diffeomorphism

On the integrability of intermediate distributions for Anosov diffeomorphisms

1995 ◽  
Vol 15 (2) ◽  
pp. 317-331 ◽  
Author(s):  
M. Jiang ◽  
Ya B. Pesin ◽  
R. de la Llave
Keyword(s):  
Finite Number ◽  
Lyapunov Exponents ◽  
Periodic Orbits ◽  
Three Dimensional ◽  
Dimensional Torus ◽  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms ◽  
Stable Foliation

AbstractWe study the integrability of intermediate distributions for Anosov diffeomorphisms and provide an example of a C∞-Anosov diffeomorphism on a three-dimensional torus whose intermediate stable foliation has leaves that admit only a finite number of derivatives. We also show that this phenomenon is quite abundant. In dimension four or higher this can happen even if the Lyapunov exponents at periodic orbits are constant.

Simultaneous dense and non-dense orbits for certain partially hyperbolic diffeomorphisms

2018 ◽  
Vol 40 (4) ◽  
pp. 1083-1107
Author(s):  
WEISHENG WU
Keyword(s):  
Hausdorff Dimension ◽  
Tangent Space ◽  
Anosov Diffeomorphism ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Dense Orbits ◽  
Set Of Points ◽  
Partially Hyperbolic Diffeomorphism ◽  
Stable Subspace

Let$g:M\rightarrow M$be a$C^{1+\unicode[STIX]{x1D6FC}}$-partially hyperbolic diffeomorphism preserving an ergodic normalized volume on$M$. We show that, if$f:M\rightarrow M$is a$C^{1+\unicode[STIX]{x1D6FC}}$-Anosov diffeomorphism such that the stable subspaces of$f$and$g$span the whole tangent space at some point on$M$, the set of points that equidistribute under$g$but have non-dense orbits under$f$has full Hausdorff dimension. The same result is also obtained when$M$is the torus and$f$is a toral endomorphism whose center-stable subspace does not contain the stable subspace of$g$at some point.

A C 1 Anosov diffeomorphism with a horseshoe that attracts almost any point

2017 ◽  
Vol 51 (2) ◽  
pp. 144-147
Author(s):  
C. Bonatti ◽  
S. S. Minkov ◽  
A. V. Okunev ◽  
I. S. Shilin
Keyword(s):  
Anosov Diffeomorphism

scholarly journals Foliations and conjugacy, II: the Mendes conjecture for time-one maps of flows

2020 ◽  
pp. 1-18
Author(s):  
JORGE GROISMAN ◽  
ZBIGNIEW NITECKI
Keyword(s):  
Conjugacy Classes ◽  
Topological Conjugacy ◽  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms

Abstract A diffeomorphism of theplane is Anosov if it has a hyperbolic splitting at every point of the plane. In addition to linear hyperbolic automorphisms, translations of the plane also carry an Anosov structure (the existence of Anosov structures for plane translations was originally shown by White). Mendes conjectured that these are the only topological conjugacy classes for Anosov diffeomorphisms in the plane. Very recently, Matsumoto gave an example of an Anosov diffeomorphism of the plane, which is a Brouwer translation but not topologically conjugate to a translation, disproving Mendes’ conjecture. In this paper we prove that Mendes’ claim holds when the Anosov diffeomorphism is the time-one map of a flow, via a theorem about foliations invariant under a time-one map. In particular, this shows that the kind of counterexample constructed by Matsumoto cannot be obtained from a flow on the plane.

scholarly journals Lipeomorphisms close to an Anosov diffeomorphism

1974 ◽  
Vol 53 ◽  
pp. 71-82 ◽  
Author(s):  
Kentaro Takaki
Keyword(s):  
Compact Manifold ◽  
Anosov Diffeomorphism ◽  
Structurally Stable

It is well-known that an Anosov diffeomorphism f on a compact manifold is structurally stable in the space of all C1-diffeomorphisms, with the C1-topology (Anosov [1]). In this paper we show that f is also structurally stable in the space of all lipeomorphisms, with a lipschitz topology. The proof is similar to that of the C1-case by J. Moser [4].

Rigidity of symplectic Anosov diffeomorphisms on low dimensional tori

1991 ◽  
Vol 11 (3) ◽  
pp. 427-441 ◽  
Author(s):  
L. Flaminio ◽  
A. Katok
Keyword(s):  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms ◽  
Hyperbolic Automorphism ◽  
Low Dimensional

AbstractWe show that any symplectic Anosov diffeomorphism of a four torus T4 with sufficiently smooth stable and unstable foliations is smoothly conjugate to a linear hyperbolic automorphism of T4.

EVALUATING DETERMINISTIC STRUCTURE IN MAPS DEDUCED FROM DISCRETE-TIME MEASUREMENTS

1993 ◽  
Vol 03 (03) ◽  
pp. 617-623 ◽  
Author(s):  
DANIEL T. KAPLAN
Keyword(s):  
Discrete Time ◽  
Experimental System ◽  
Nonlinear Oscillators ◽  
Anosov Diffeomorphism ◽  
Coupled Nonlinear Oscillators

A method is presented for quantifying the determinism of dynamics reconstructed from discrete-time measurements. To illustrate the use of the method, it is applied to three systems: a cosine map, an Anosov diffeomorphism, and an experimental system of coupled nonlinear oscillators.

scholarly journals On mostly expanding diffeomorphisms

2017 ◽  
Vol 38 (8) ◽  
pp. 2838-2859 ◽  
Author(s):  
MARTIN ANDERSSON ◽  
CARLOS H. VÁSQUEZ
Keyword(s):  
Anosov Diffeomorphism ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Physical Measures ◽  
Partially Hyperbolic Diffeomorphisms

In this work, we study the class of mostly expanding partially hyperbolic diffeomorphisms. We prove that such a class is$C^{r}$-open,$r>1$, among the partially hyperbolic diffeomorphisms and we prove that the mostly expanding condition guarantees the existence of physical measures and provides more information about the statistics of the system. Mañé’s classical derived-from-Anosov diffeomorphism on$\mathbb{T}^{3}$belongs to this set.

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