Sinai–Ruelle–Bowen measures for N-dimensional derived from Anosov diffeomorphisms

1993 ◽  
Vol 13 (1) ◽  
pp. 21-44 ◽  
Author(s):  
Maria Carvalho
Keyword(s):  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms ◽  
Axiom A ◽  
Srb Measures ◽  
Saddle Node ◽  
Basic Set ◽  
Hyperbolic Attractors ◽  
Generic Bifurcation

AbstractThis paper is about the existence of transitive non-hyperbolic attractors with corresponding SRB measures for arcs of diffeomorphisms crossing the boundary of the Axiom A systems, obtained through an elementary generic bifurcation (Hopf, saddle-node or flip) on a transitive Anosov diffeomorphism or an attracting basic set.

scholarly journals SRB measures for partially hyperbolic attractors of local diffeomorphisms

2018 ◽  
Vol 40 (6) ◽  
pp. 1545-1593
Author(s):  
ANDERSON CRUZ ◽  
PAULO VARANDAS
Keyword(s):  
Thermodynamic Formalism ◽  
Positive Lebesgue Measure ◽  
Stable Bundle ◽  
Local Diffeomorphism ◽  
Local Diffeomorphisms ◽  
Axiom A ◽  
Srb Measures ◽  
Cone Field ◽  
Hyperbolic Attractors ◽  
Partially Hyperbolic

We contribute to the thermodynamic formalism of partially hyperbolic attractors for local diffeomorphisms admitting an invariant stable bundle and a positively invariant cone field with non-uniform cone expansion at a positive Lebesgue measure set of points. These include the case of attractors for Axiom A endomorphisms and partially hyperbolic endomorphisms derived from Anosov. We prove these attractors have finitely many SRB measures, that these are hyperbolic, and that the SRB measure is unique provided the dynamics is transitive. Moreover, we show that the SRB measures are statistically stable (in the weak$^{\ast }$ topology) and that their entropy varies continuously with respect to the local 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.

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.

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.

INVERSE TOPOLOGICAL PRESSURE WITH APPLICATIONS TO HOLOMORPHIC DYNAMICS OF SEVERAL COMPLEX VARIABLES

2004 ◽  
Vol 06 (04) ◽  
pp. 653-679 ◽  
Author(s):  
EUGEN MIHAILESCU ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Technical Condition ◽  
Topological Pressure ◽  
Holomorphic Dynamics ◽  
Critical Set ◽  
Axiom A ◽  
Type Definition ◽  
Basic Set ◽  
Unique Zero ◽  
Stable Dimension ◽  
The One

In this paper, we introduce a few notions of inverse topological pressure [Formula: see text], defined in terms of backward orbits (prehistories) instead of forward orbits. This inverse topological pressure has some properties similar to the regular (forward) pressure but, in general, if the map is not a homeomorphism, they do not coincide. In fact, there are several ways to define inverse topological pressure; for instance, we show that the Bowen type definition coincides with the one using spanning sets. Then we consider the case of a holomorphic map [Formula: see text] which is Axiom A and such that its critical set does not intersect a particular basic set of saddle type Λ. We will prove that, under a technical condition, the Hausdorff dimension of the intersection between the local stable manifold and the basic set is equal to ts, i.e. [Formula: see text], for all points x belonging to Λ. Here ts represents the unique zero of the function t→P-(tϕs), with P- denoting the inverse topological pressure and [Formula: see text], y∈Λ. In general, [Formula: see text] will be estimated above by ts and below by [Formula: see text], where [Formula: see text] is the unique zero of the map t→P_(tϕs). As a corollary we obtain that, if the stable dimension is non-zero, then Λ must be a non-Jordan curve, and also, if f|Λ happens to be a homeomorphism (like in the examples from [13]), then the stable dimension cannot be zero.

scholarly journals SRB measures for Axiom A endomorphisms

2004 ◽  
Vol 11 (6) ◽  
pp. 785-797 ◽  
Author(s):  
Mariusz Urbanski ◽  
Christian Wolf
Keyword(s):  
Axiom A ◽  
Srb Measures

scholarly journals Homology for one-dimensional solenoids

2017 ◽  
Vol 121 (2) ◽  
pp. 219 ◽  
Author(s):  
Massoud Amini ◽  
Ian F. Putnam ◽  
Sarah Saeidi Gholikandi
Keyword(s):  
Finite Type ◽  
Homology Theory ◽  
One Dimensional ◽  
Axiom A ◽  
Group Invariant ◽  
Dimension Group ◽  
Basic Set ◽  
The One ◽  
Topological Dynamical Systems ◽  
Smale Spaces

Smale spaces are a particular class of hyperbolic topological dynamical systems, defined by David Ruelle. The definition was introduced to give an axiomatic description of the dynamical properties of Smale's Axiom A systems when restricted to a basic set. They include Anosov diffeomeorphisms, shifts of finite type and various solenoids constructed by R. F. Williams. The second author constructed a homology theory for Smale spaces which is based on (and extends) Krieger's dimension group invariant for shifts of finite type. In this paper, we compute this homology for the one-dimensional generalized solenoids of R. F. Williams.

On the Sensitivity of Non-Generic Bifurcation of Non-Linear Normal Modes

2007 ◽  
Author(s):  
C. H. Pak ◽  
Y. S. Choi
Keyword(s):  
Normal Mode ◽  
Natural Frequencies ◽  
Normal Modes ◽  
New Normal ◽  
Perturbed System ◽  
Saddle Node ◽  
Non Linear ◽  
The Difference ◽  
Generic Bifurcation ◽  
The Stability

It is shown that a non-generic bifurcation of non-linear normal modes may occur if the ratio of linear natural frequencies is near r-to-one, r = 1, 3, 5 ·······. Non-generic bifurcations are explicitly obtained in the systems having certain symmetry, as observed frequently in literatures. It is found that there are two kinds of non-generic bifurcations, super-critical and sub-critical. The normal mode generated by the former kind is extended to large amplitude, but that by the latter kind is limited to small amplitude which depends on the difference between two linear natural frequencies and disappears when two frequencies are equal. Since a non-generic bifurcation is not generic, it is expected generically that if a system having a non-generic bifurcation is perturbed then the non-generic bifurcation disappears and generic bifurcation appear in the perturbed system. Examples are given to verify the change in bifurcations and to obtain the stability behavior of normal modes. It is found that if a system having a super-critical non-generic bifurcation is perturbed, then two new normal modes are generated, one is stable, but the other unstable, implying a saddle-node bifurcation. If the system having a sub-critical non-generic bifurcation is perturbed, then no new normal mode is generated, but there is an interval of instability on a normal mode, implying two saddle-node bifurcations on the mode. Application of this study is discussed.

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