Anosov Maps Research Articles

Pseudo-Anosov extensions and degree one maps between hyperbolic surface bundles

Let F′,F be any two closed orientable surfaces of genus g′ > g≥ 1, and f:F→ F be any pseudo-Anosov map. Then we can “extend” f to be a pseudo- Anosov map f′:F′→ F′ so that there is a fiber preserving degree one map M(F′,f′)→ M(F,f) between the hyperbolic surface bundles. Moreover the extension f′ can be chosen so that the surface bundles M(F′,f′) and M(F,f) have the same first Betti numbers.

Hyperbolic pseudo-Anosov maps almost everywhere embed into a toral automorphism

Fathi and Franks showed that a pseudo-Anosov diffeomorphism -to-one ramified covering of another pseudo-Anosov (or Anosov) map on a surface of smaller genus. As a corollary, any pseudo-Anosov diffeomorphism with orientable foliations and hyperbolic action on the first homologies almost everywhere embeds into a hyperbolic toral automorphism.

Relaxation Time of Quantized Toral Maps

We introduce the notion of relaxation time for noisy quantum maps on the 2d-dimensional torus – generalization of previously studied dissipation time. We show that the relaxation time is sensitive to the chaotic behavior of the corresponding classical system if one simultaneously considers the semiclassical limit $$({\hbar } \to 0)$$ together with the limit of small noise strength (ε → 0). Focusing on quantized smooth Anosov maps, we exhibit a semiclassical régime $${\hbar } < \epsilon ^{E} \ll 1$$ (where E > 1) in which classical and quantum relaxation times share the same asymptotics: in this régime, a quantized Anosov map relaxes to equilibrium fast, as the classical map does. As an intermediate result, we obtain rigorous estimates of the quantum-classical correspondence for noisy maps on the torus, up to times logarithmic in $${\hbar }^{{ - 1}} $$ On the other hand, we show that in the “quantum régime” $$\epsilon \ll {\hbar } \ll 1,$$ quantum and classical relaxation times behave very differently. In the special case of ergodic toral symplectomorphisms (generalized “Arnold’s cat” maps), we obtain the exact asymptotics of the quantum relaxation time and precise the régime of correspondence between quantum and classical relaxations. Communicated by Jens Marklof

Banach spaces adapted to Anosov systems

We study the spectral properties of the Ruelle-Perron-Frobenius operator associated to an Anosov map on classes of functions with high smoothness. To this end we construct anisotropic Banach spaces of distributions on which the transfer operator has a small essential spectrum. In the C ∞ case, the essential spectral radius is arbitrarily small, which yieldsa descriptionof the correlationswith arbitraryprecision. Moreover,we obtain sharp spectral stability results for deterministic and random perturbations. In particular, we obtain differentiability results for spectral data (which imply differentiability of the Sinai-Ruelle-Bowenmeasure, the variancefor the centrallimit theorem, the rates of decay for smooth observable, etc.).

Anosov families, renormalization and non-stationary subshifts

We introduce the notion of an Anosov family, a generalization of an Anosov map of a manifold. This is a sequence of diffeomorphisms along compact Riemannian manifolds such that the tangent bundles split into expanding and contracting subspaces. We develop the general theory, studying sequences of maps up to a notion of isomorphism and with respect to an equivalence relation generated by two natural operations, gathering and dispersal. Then we concentrate on linear Anosov families on the 2-torus. We study in detail a basic class of examples, the multiplicative families, and a canonical dispersal of these, the additive families. These form a natural completion to the collection of all linear Anosov maps. A renormalization procedure constructs a sequence of Markov partitions consisting of two rectangles for a given additive family. This codes the family by the non-stationary subshift of finite type determined by exactly the same sequence of matrices. Any linear positive Anosov family on the torus has a dispersal which is an additive family. The additive coding then yields a combinatorial model for the linear family, by telescoping the additive Bratteli diagram. The resulting combinatorial space is then determined by the same sequence of non-negative matrices, as a non-stationary edge shift. This generalizes and provides a new proof for theorems of Adler and Manning.

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.

Ruelle–Perron–Frobenius spectrum for Anosov maps

We extend a number of results from one-dimensional dynamics based on spectral properties of the Ruelle–Perron–Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows us to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasi-compact. (Information on the existence of a Sinai–Ruelle–Bowen measure, its smoothness properties and statistical properties readily follow from such a result.) In dimension d = 2 we show that the transfer operator associated with smooth random perturbations of the map is close, in a proper sense, to the unperturbed transfer operator. This allows us to obtain easily very strong spectral stability results, which, in turn, imply spectral stability results for smooth deterministic perturbations as well. Finally, we are able to implement an Ulam-type finite rank approximation scheme thus reducing the study of the spectral properties of the transfer operator to a finite-dimensional problem.

Renormalization of quantum Anosov maps: reduction to fixed boundary conditions.

A renormalization scheme is introduced to study quantum Anosov maps (QAMs) on a torus for general boundary conditions (BCs), whose number ( k) is always finite. It is shown that the quasienergy eigenvalue problem of a QAM for all k BCs is exactly equivalent to that of the renormalized QAM (with Planck's constant Planck's over 2pi(') = Planck's over 2pi/k) at some fixed BCs that can be of four types. The quantum cat maps are, up to time reversal, fixed points of the renormalization transformation. Several results at fixed BCs, in particular the existence of a complete basis of "crystalline" eigenstates in a classical limit, can then be derived and understood in a simple and transparent way in the general-BCs framework.

Pseudo-symmetries of Anosov maps and spectral statistics

The statistics of the quantum eigenvalues of certain families of nonlinear maps on the 2-torus are found not to belong to the universality classes one would expect from the symmetries of the (classical) dynamics the maps generate. These anomalies are shown to be caused by arithmetical quantum symmetries which do not have a classical limit. They are related to the dynamics generated by associated linear torus maps on particular rational lattices that form the support of the quantum Wigner functions.

Topological entropy and the complete invariant for expansive maps

The topological entropy of an expansive map is equal to that of the corresponding symbolic system. The topological entropy and ergodic period are a complete invariant index (h,b) for an equivalence relation, almost topological conjugacy, in the setting of ergodically supported expansive maps with shadowing property, including Anosov maps.

The ghosts of the cat

Let $A\in SL(2,{\Bbb Z})$ be given and $n\in{\Bbb N}$. $A$ induces a map $A_n$ on ${\Bbb Z}^2_n$ by $(x,y)\mapsto(a_{11}x+a_{12}y,a_{21}x+a_{22}y)$ mod $n$. $A_n$ can be thought of as a discrete approximation of the similarly defined map on the unit square. It is known that the $A_n$ have a surprisingly small period: there always is an $r\le3n$ such that $A^r_n$ is the identity on ${\Bbb Z}^2_n$.One can visualize the properties of the iterates of $A_n$ by considering the sets $P,A_nP,A^2_nP,\ldots$ for a fixed subset $P$ of ${\Bbb Z}^2_n$. (Here the cat comes into play: $P$ has often been chosen to be the digitalized picture of a cat, in particular when treating the special case of the Anosov map $A={2\ 1\atopwithdelims() 1\ 1}$.) It turns out that in the sequence $(A^k_nP)_k$ mostly ‘chaotic’ pictures occur as one would expect. However, some kind of regular behaviour can also be observed for certain powers $A^k_n$ before the end of the period.It is the aim of this paper to describe these phenomena and to explain why and how they occur. In order to quantify the chaotic behaviour of the maps on ${\Bbb Z}^2_n$ under consideration, we introduce suitable functions.Our methods are elementary. Only basic results from the geometry of numbers will be used.

Periods of discretized linear Anosov maps

Integer $m\times m$ matrices $A$ with determinant $1$ define diffeomorphisms of the $m$-dimensional torus $T^m=({\Bbb R}/{\Bbb Z})^m$ into itself. Likewise, they define bijective self-maps of the discretized tori $({\Bbb Z}/n{\Bbb Z})^m=({\Bbb z}_n)^m$. We present estimates of the surprisingly low order (or period) $\Per_A(n)$ of the iteration $A^r, r=1,2,3,\ldots,$ on the discretized torus $({\Bbb z}_n)^m$. We obtain $\Per_A(n)\leq 3n$ for dimension $m=2$. In the special case of the Anosov map $A={2\ 1 \atopwithdelims() 1\ 1}$, this result is due to Dyson and Talk [DT92]. For arbitrary dimensions $m&gt;2$ we obtain $$\Per_A(n)\leq {\rm constant}\cdot n^{m-1},$$ provided $n$ is a power of a prime number. For general $n$, number theoretic problems arise.

Periodic number for Anosov maps

LetX be a compact metric space and letf: X→X be an Anosov map,i.e., an expansive selfmap with the pseudoorbit tracing property (abbr. POTP) (see Lemma 1). IfNn(f) denotes the number of fixed points offn which we name here then-periodic number then we prove in the case asn tends to infinity thatnM≤Nn(f)≤Hn, whereM andH are two positive integers.

Reversing symmetry group of and matrices with connections to cat maps and trace maps

Dynamical systems can have both symmetries and time-reversing symmetries. Together these two types of symmetries form a group called the reversing symmetry group with the symmetries forming a normal subgroup of . We give a complete characterization of (and hence ) in the dynamical systems associated with the groups of integral matrices and . To do this, we use well known methods of number theory, such as Dirichlet's unit theorem for quadratic fields and Gaus' results on the equivalence of integer quadratic forms, and employ the algebraic structure of the modular group as a free product. We show how some recently discussed generalizations of the reversing symmetry group are also nicely illustrated when we consider affine extensions of these matrix groups. Our results are applicable to hyperbolic toral automorphisms (Anosov or cat maps), pseudo-Anosov maps, and the group of three-dimensional (3D) trace maps that preserve the Fricke - Vogt invariant.

A generalized notion of variation applied to Markov chains and Anosov maps

Extending the operator formalism of [3] we show that there exists a large class of functions which possess an exponential decay of correlations and fulfill a central limit theorem under a certain type of Markov chains. This result can be applied to the symbolic dynamics of Anosov maps, showing that in the case of a absolutely continuous invariant measure there is a large class of functions with good ergodic properties-larger than the usual class of Holder continuous functions.

Periodic number for Anosov maps

Defined on a compact metric space X an Anosov map f:X→X is a continuoussurjective selfmap which possesses the local product structure. We have shown that anAnosov map is equivalent to an expansive selfmap which has the pseudo-orbit tracingproperty. About the period and the periodic points of the Anosov map f it was knownthat N_n(f)<+∞ and cl(P(f))=Ω(f), where P(·), Ω(·), and N_n(·) indicate the periodicpoint set, nonwandering point set and the n-period number, respectively. In the present let-ter we announce a result from which one can bound the n-periodic number by a powerfunction and an exponential function.

Quantization of Anosov Maps

Cat maps, linear automorphisms of the torus, are standard examples of classically chaotic systems, but they are periodic when quantized, leading to many untypical consequences. Anosov maps are topologically equivalent to cat maps despite being nonlinear. Generalizing the original quantization of cat maps, we establish that quantum Anosov maps have typical spectral fluctuations for classically chaotic systems. The periodic orbit theory for the spectra of quantum Anosov maps is not exact, as it is for cat maps. Nonetheless our calculations verify that the semiclassical trace of the propagator is accurate well beyond any "log time" cutoff.

Stability of Anosov Maps

Anosov maps of Banach manifolds are shown to be inverse limit stable.

Stability of Anosov maps

Anosov maps of Banach manifolds are shown to be inverse limit stable.

Fixed Points of Anosov Maps of Certain Manifolds

Lemma. If $H$ is a graded exterior algebra on odd generators with augmentation ideal $J$ and $h:H \to H$ is an algebra homomorphism inducing $J/{J^2} \to J/{J^2}$ with eigenvalues $\{ {\lambda _i}\}$, then the Lefschetz number $L(h) = \Pi (1 - {\lambda _i})$. The lemma is applied to an Anosov map or diffeomorphism of a compact manifold with real cohomology $H$ to give sufficient conditions that none of the eigenvalues ${\lambda _i}$ be a root of unity and that there exist a fixed point. In particular, every Anosov diffeomorphism of a compact connected Lie group has a fixed point.