Anosov Diffeomorphisms Research Articles

Invariants of two-dimensional projectively Anosov diffeomorphisms and their applications

Invariants of two-dimensional projectively Anosov diffeomorphisms and their applications

Persistence of Bowen-Ruelle-Sinai measures

We study the changes on the Bowen-Ruelle-Sinai measures along an arcthat starts at an Anosov diffeomorphism on a two-torus and reachesthe boundary of its stability component while a flat homoclinictangency or a first cubic heteroclinic tangency is happening. Theoutermost diffeomorphisms of such arcs are not hyperbolic but areconjugate to the original Anosov diffeomorphism and share similarergodic traits. In particular, the torus is a global attractor witha full supported physical measure.

Weakly Hyperbolic Actions of Kazhdan Groups on Tori

We study the ergodic and rigidity properties of weakly hyperbolic actions. First, we establish ergodicity for C2 volume preserving weakly hyperbolic group actions on closed manifolds. For the integral action generated by a single Anosov diffeomorphism this theorem is classical and originally due to Anosov. Motivated by the Franks/Manning classification of Anosov diffeomorphisms on tori, we restrict our attention to weakly hyperbolic actions on the torus. When the acting group is a lattice subgroup of a semisimple Lie group with no compact factors and all (almost) simple factors of real rank at least two, we show that weak hyperbolicity in the original action implies weak hyperbolicity for the induced action on the fundamental group. As a corollary, we obtain that any such action on the torus is continuously semiconjugate to the affine action coming from the fundamental group via a map unique in the homotopy class of the identity. Under the additional assumption that some partially hyperbolic group element has quasi-isometrically embedded lifts of unstable leaves to the universal cover, we obtain a conjugacy, resulting in a continuous classification for these actions.

Diffeomorphisms admitting SRB measures and their regularity

We are interested in the stochastic property of some "Anosov-like" system. In this paper we will treat a transitive and partially hyperbolic diffeomorphism f of a 2-dimensional torus with uniformly contracting direction, and show that if f is of C2 and admits an SRB measure, then f is an Anosov diffeomorphism. In our proof we use the Pujals-Sambarino theorem for C2 diffeomorphisms with dominated splitting. In the case of C1+α the above statement is not true in general, i.e. we can construct a C1+α counter example of Maneville-Pomeau type.

ROOTS AND CENTRALIZERS OF ANOSOV DIFFEOMORPHISMS ON TORI

The centralizer of a group element g is the group of elements commuting with g. Powers and, if existent, roots of g are contained in the centralizer. We call a centralizer trivial if it consists of the integer powers of g, only. In the group of torus homeomorphisms, we study the centralizer of torus diffeomorphisms of hyperbolic Anosov type. As a result, the centralizer can be calculated, isomorphically, in the much smaller group of affine torus automorphisms. In particular, Anosov diffeomorphisms of the 2-torus with a unique fixed point possess trivial centralizer, up to a trivial involution. The identification of individual reactors from data on reactor cascades, for example in chemical engineering, is one source of motivation for the study of roots of diffeomorphisms. Another possible source is the study of commuting diffeomorphisms in finite-dimensional spatially or spatio-temporally chaotic systems.

Transitive partially hyperbolic diffeomorphisms on 3-manifolds

The known examples of transitive partially hyperbolic diffeomorphisms on 3-manifolds belong to 3 basic classes: perturbations of skew products over an Anosov map of T 2 , perturbations of the time one map of a transitive Anosov flow, and certain derived from Anosov diffeomorphisms of the torus T 3 . In this work we characterize the two first types by a local hypothesis associated to one closed periodic curve.

The Nielsen numbers of Anosov diffeomorphisms on flat Riemannian manifolds

The Nielsen numbers of Anosov diffeomorphisms on flat Riemannian manifolds

Fredholm determinants, Anosov maps and Ruelle resonances

I show that the dynamical determinant, associated to anAnosov diffeomorphism, is the Fredholmdeterminant of the corresponding Ruelle-Perron-Frobenius transferoperator acting on appropriate Banach spaces.As a consequence it follows, for example, that the zeroes of the dynamical determinantdescribe the eigenvalues of the transfer operator and the Ruelleresonances and that, for $\C^\infty$ Anosov diffeomorphisms, thedynamical determinant is an entire function.

Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms

We study non-hyperbolic repellers of diffeomorphisms derived from transitive Anosov diffeomorphisms with unstable dimension 2 through a Hopf bifurcation. Using some recent abstract results about non-uniformly expanding maps with holes, by ourselves and by Dysman, we show that the Hausdorff dimension and the limit capacity (box dimension) of the repeller are strictly less than the dimension of the ambient manifold.

Transitivity of Euclidean extensions of Anosov diffeomorphisms

We consider the class of unbounded from above and from below is equivalent to topological transitivity. We also show that in the above class topological transitivity and stable topological transitivity are equivalent.

On Algebraic Anosov Diffeomorphisms on Nilmanifolds

This article is devoted to the algebraic approaches to Anosov diffeomorphisms. All examples of Anosov diffeomorphisms known so far are connected directly or indirectly with compact nilmanifolds. We consider some new necessary conditions for existence of these diffeomorphisms on nilmanifolds. We demonstrated the absence of Anosov diffeomorphisms on some classes of nilmanifolds. We also prove some results on Lie groups and lattices in them.

Analyticity of the SRB measure of a lattice of coupled Anosov diffeomorphisms of the torus

We consider the “thermodynamic limit” of a d-dimensional lattice of hyperbolic dynamical systems on the 2-torus, interacting via weak and nearest neighbor coupling. We prove that the SRB measure is analytic in the strength of the coupling. The proof is based on symbolic dynamics techniques that allow us to map the SRB measure into a Gibbs measure for a spin system on a (d+1)-dimensional lattice. This Gibbs measure can be studied by an extension (decimation) of the usual “cluster expansion” techniques.

Dissipation time and decay of correlations

We consider the effect of noise on the dynamics generated by volume-preserving maps on a d-dimensional torus. The quantity we use to measure the irreversibility of the dynamics is the dissipation time. We focus on the asymptotic behaviour of this time in the limit of small noise. We derive universal lower and upper bounds for the dissipation time in terms of various properties of the map and its associated propagators: spectral properties, local expansivity, and global mixing properties. We show that the dissipation is slow for a general class of non-weakly-mixing maps; on the other hand, it is fast for a large class of exponentially mixing systems, which include uniformly expanding maps and Anosov diffeomorphisms.

A dynamical lefschetz trace formula for algebraic anosov diffeomorphisms

Dynamical Lefschetz trace formulas for flows on compact manifolds were first conjectured by GUILLEMIN [5] and later but independently by PATTERSON [12]. These formulas express the alternating sum of the traces of the induced flow on cohomology by a sum over contributions from the compact orbits. Here the cohomology theory depends on the choice of a foliation which is respected by the flow. This note makes a contribution to PATTERSON's setup. He looks at Anosov flows and considers the unstable foliation. An interesting example of this situation is provided by the geodesic flow on the sphere bundle of a cocompact quotient of PSLe (]~) by a lattice. Using representation theory, PATTERSON was able to define a trace on the infinite dimensional foliation cohomologies. The dynamical Lefschetz trace formula was then shown to be a consequence of the Selberg trace formula. Incidentally this example also fits into GUILLEMIN'S framework who treated it in [5]. Much more work in the context of more general symmetric spaces was done by BUNKE, DEITMAR, JUHL, OLBRICH and SCHUBERT. We re fe r the r eade r to the book of JUHL [7] for a comprehensive overview. Among all Anosov flows there are the ones with an integrable complement or equivalently, those where the sum of the stable and the unstable bundle is integrable. One gets examples by suspending Anosov diffeomorphisms and conjecturally all Anosov flows with an integrable complement should arise in this way up to a time change by a constant factor, [13] w 3. Granting this conjecture the dynamical Lefschetz trace formula for such Anosov flows can be reduced to the following problem:

ON LOCAL AND GLOBAL RIGIDITY OF QUASI-CONFORMAL ANOSOV DIFFEOMORPHISMS

We consider a transitive uniformly quasi-conformal Anosov diffeomorphism is an infranilmanifold. We also describe necessary and sufficient conditions for smoothness of conjugacy between such a diffeomorphism and a small perturbation.AMS 2000 Mathematics subject classification: Primary 37C; 37D

ANOSOV DIFFEOMORPHISMS ON A CLASS OF 2-STEP NILMANIFOLDS

In this paper we study nilmanifolds which are modeled on a quotient of a free 2-step nilpotent Lie group by a 1-dimensional subgroup. In fact we obtain a very easy criterion to decide whether or not such a nilmanifold admits an Anosov diffeomorphism.

Expanding maps, Anosov diffeomorphisms and affine structures on infra-nilmanifolds

In this paper we present a complete description of Lie algebras admitting an expanding and/or a hyperbolic automorphism in terms of R -gradings of these Lie algebras. Using this, we then show that any infra-nilmanifold which admits an expanding map or an Anosov diffeomorphism also has a complete, affinely flat structure.

Geometric decay of infection probabilities for the anisotropic contact process

Consider the anisotropic symmetric contact process on a homogeneous tree T 2d of degree 2 d⩾2 with a single initially infected site at the root vertex of the tree. We show that, for all values of the infection vector λ, each integer n⩾1, and each vertex x∈ T 2d at distance n from the root vertex, the probability P (x is ever infected )=u x(λ) satisfies u x ( λ)⩽[ β c ( λ)] n−1 for some function β c that we will specify. This geometric decay property governs the growth and dispersal behaviour of the process and lies at the core of the method of Hueter (preprint, arXiv: math.PR/0109047), which applies the thermodynamic formalism and the theory of Gibbs states by Bowen (Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer, Berlin, 1975) to the contact process on trees. We leave open the question as to when (if at all) λ c is the maximal infection rate among the components of λ.

Examples of Anosov diffeomorphisms

We prove that if n is any graded rational Lie algebra, then the simply connected nilpotent Lie group N×N with Lie algebra (n⊗R)⊕(n⊗R) has a lattice Γ such that the corresponding nilmanifold (N×N)/Γ admits an Anosov diffeomorphism. This gives a procedure to construct easily several explicit examples of nilmanifolds admitting an Anosov diffeomorphism, and shows that a reasonable classification up to homeomorphism (or even up to commensurability) of such nilmanifolds would not be possible.

Pseudo-Anosov Flows and Incompressible Tori

We study incompressible tori in 3-manifolds supporting pseudo-Anosov flows and more generally Z ⊕ Z subgroups of the fundamental group of such a manifold. If no element in this subgroup can be represented by a closed orbit of the pseudo-Anosov flow, we prove that the flow is topologically conjugate to a suspension of an Anosov diffeomorphism of the torus. In particular it is non singular and is an Anosov flow. It follows that either a pseudo-Anosov flow is topologically conjugate to a suspension Anosov flow, or any immersed incompressible torus can be realized as a free homotopy from a closed orbit of the flow to itself. The key tool is an analysis of group actions on non-Hausdorff trees, also known as R-order trees – we produce an invariant axis in the free action case. An application of these results is the following: suppose the manifold has an R-covered foliation transverse to a pseudo-Anosov flow. If the flow is not an R-covered Anosov flow, then it follows that the manifold is atoroidal.