Axiom A Research Articles

An analytical construction of the SRB measures for Baker-type maps.

For a class of dynamical systems, called the axiom-A systems, Sinai, Ruelle and Bowen showed the existence of an invariant measure (SRB measure) weakly attracting the temporal average of any initial distribution that is absolutely continuous with respect to the Lebesgue measure. Recently, the SRB measures were found to be related to the nonequilibrium stationary state distribution functions for thermostated or open systems. Inspite of the importance of these SRB measures, it is difficult to handle them analytically because they are often singular functions. In this article, for three kinds of Baker-type maps, the SRB measures are analytically constructed with the aid of a functional equation, which was proposed by de Rham in order to deal with a class of singular functions. We first briefly review the properties of singular functions including those of de Rham. Then, the Baker-type maps are described, one of which is nonconservative but time reversible, the second has a Cantor-like invariant set, and the third is a model of a simple chemical reaction R<-->I<-->P. For the second example, the cases with and without escape are considered. For the last example, we consider the reaction processes in a closed system and in an open system under a flux boundary condition. In all cases, we show that the evolution equation of the distribution functions partially integrated over the unstable direction is very similar to de Rham's functional equation and, employing this analogy, we explicitly construct the SRB measures. (c) 1998 American Institute of Physics.

Conditions of topological conjugacy of gradient-like diffeomorphisms on irreducible 3-manifolds

In this paper, we obtain the topological classification of gradient-like diffeomorphisms and the conditions of topological conjugacy of Morse-Smale diffeomorphisms with finite sets of heteroclinic trajectories on three-dimensional manifolds.

On the topological equivalence of Morse-Smale diffeomorphisms with a finite set of heteroclinic trajectories on irreducible 3-manifolds

On the topological equivalence of Morse-Smale diffeomorphisms with a finite set of heteroclinic trajectories on irreducible 3-manifolds

Formula omitted]* plus Axiom A Does Not Imply No-Cycle

An example is presented to indicate that X * plus Axiom A does not imply the no-cycle condition.

Centralizers and iterate radicals of morse-smale diffeomorphisms of the circle

In this paper, we will use the embedding flows in [1], [2] to give a complete description of the smooth centralizers and iterate radicals of allC r (r ≥ 2) Morse-Smale diffeomorphisms of the circleS1. As a result, we prove that every centralizer is a solvable subgroup of Diff r (S1).

Codimension-two reflection and non-hyperbolic invariant lines

The generic unfolding of codimension-two reflection periodic orbit (two eigenvalues +or-1 and the others with norm different from 1) embedded in a Morse-Smale diffeomorphism is analysed. Local and global bifurcations are described. Non-hyperbolic invariant manifolds are constructed as an essential tool to prove local and global stability of the unfolding.

Topological classification of Morse-Smale diffeomorphisms with finite set of heteroclinic trajectories on surfaces

Topological classification of Morse-Smale diffeomorphisms with finite set of heteroclinic trajectories on surfaces

Characteristics of Index One Orbits of Morse-Smale Diffeomorphisms

We use Lyapunov functions to find restrictions on the periodic data of index one orbits of Morse-Smale diffeomorphisms. The resulting structure is then used to find connections between index one orbits.

Characteristics of index one orbits of Morse-Smale diffeomorphisms

We use Lyapunov functions to find restrictions on the periodic data of index one orbits of Morse-Smale diffeomorphisms. The resulting structure is then used to find connections between index one orbits.

Differentiable conjugacies of Morse-Smale diffeomorphisms

We provide in this paper a complete set of invariants for differentiable conjugacies of Morse-Smale diffeomorphisms. In the setting of topological conjugacies, these diffeomorphisms play an important role similar to that of gradients whose singularities are non-degenerate and have their stable and unstable manifolds in general position: they exhibit a simple and yet stable dynamics, and they exist on every manifold [PS]. Our main global invariant corresponds to a generalization of Mather's invariant for diffeomorphisms of the interval.

Cohomology of class groups of cyclotomic fields; an application to Morse-Smale diffeomorphisms

The obstruction group SSF came up in the topological investigations of Shub and Franks. It was studied by Bass and Lenstra; the latter showed that SSF ≅ ⊕ n⩾1 Pic Z ζ n, 1 n . In this paper the structure of SSF as an abelian group is determined. It is proved that SSF ≅ ⊕ n⩾1 Z⧸/n Z confirming a conjecture of Lenstra.

Chaos after bifurcation of a morse-smale diffeomorphism through a one-cycle saddle-node and interations of multivalued mappings of an interval and a circle

Chaos after bifurcation of a morse-smale diffeomorphism through a one-cycle saddle-node and interations of multivalued mappings of an interval and a circle

Resonances of chaotic dynamical systems.

We present analytic properties of the power spectrum for a class of chaotic dynamical systems (Axiom-A systems). The power spectrum is meromorphic in a strip; the position of the poles (resonances) depends on the system considered, but only their residues depend on the observable monitored. In relation with these results we also discuss the exponential or nonexponential decay of correlation functions at infinity. In conclusion, it appears desirable to analyze the decay of correlation functions and the possible analyticity of power spectra for physical time evolutions, and for computer generated simple dynamical systems (non-Axiom-A in general).

Virtual Permutations of Z [ Z n ] Complexes

We extend the characterization of virtual permutation endomorphisms in the case where I17J(M) = Zn. We show that for endomorphisms of Z[Zn] complexes the appropriate eigenvalue condition is that all eigenvalues be roots of units of the group ring Z[Zn]. Among these endomorphisms the virtual permutations are detected by Ko. The main application is in identifying Morse-Smale isotopy classes on these manifolds. Morse-Smale diffeomorphisms are the simplest structurally stable dynamical systems. An important problem is to determine when a given isotopy class contains a Morse-Smale diffeomorphism. This problem was reduced by Shub and Sullivan to the algebraic problem of representing homology maps by special matrices called virtual permutations. In this note we report progress in the nonsimply connected case.

Rate of approach to minima and sinks—the Morse-Smale case

The dynamical systems herein are Morse-Smale diffeomorphisms and flows on C ∞ {C^\infty } compact manifolds. We show the asymptotic rate of approach of orbits to the sinks of the systems to be bounded by an expression of the form K exp ⁡ ( − D N ) K\;\exp ( - DN) , where D D may be any number smaller than C = min p { 1 / m log Jac D P f m | W u ( P ) } C = {\min _p}\{ 1/m\;\log \;\operatorname {Jac}\;{D_P}\,{f^m}|{W^u}(P)\} . Here the minimum is taken over all nonsink P P in the nonwandering set of f f , and m m is the period of P P . We extend our theorems to the entire manifold, so that there is no restriction on the location of the initial points of trajectories.

Bifurcations and stability of families of diffeomorphisms

We consider one parameter families or arcs of diffeomorphisms. For families starting with Morse-Smale diffeomorphisms we characterize various types of (structural) stability at or near the first bifurcation point. We also give a complete description of the stable arcs of diffeomorphisms whose limit sets consist of finitely many orbits. Universal models for the local unfoldings of the bifurcating periodic orbits (especially saddle-nodes) are established, as well as several results on the global dynamical structure of the bifurcating diffeomorphisms. Moduli of stability related to saddle-connections are introduced.

Immediate conditional hyperbolicity in dynamical systems

AbstractFor many diffeomorphisms of a compact manifold X, eventual conditional hyperbolicity implies immediate conditional hyperbolicity in some (possibly new) Finsler structures. That is, if A and B are vector bundle isomorphisms over the mapping ƒ of the base X, such that uniformly on X, then there exist new norms for A and B such that uniformly on X, whenever the mapping ƒ satisfies the condition that there exist infinitely many N ≥ 1 such that any ƒ-invariant. For example, this condition on ƒ holds if any one of the following conditions holds: (1) ƒ is periodic; (2) ƒ is periodic on its non-wandering set; (3) ƒ has a finite non-wandering set (for example, ƒ is a Morse-Smale diffeomorphism); (4) ƒ is an almost periodic mapping of a connected base X; (5) ƒ is a mapping of the circle with no periodic points; or (6) ƒ and all its powers are uniquely ergodic. We consider various types of eventually conditionally hyperbolic systems and describe sufficient conditions on ƒ to have immediate conditional hyperbolicity of these systems in some new Finsler structures. Thus, for a sizable class of dynamical systems, we settle, in the affirmative, a question raised by Hirsch, Pugh, and Shub.

Orientation reversing Morse-Smale diffeomorphisms ofS 2

Orientation reversing Morse-Smale diffeomorphisms ofS 2

The existence of Morse–Smale diffeomorphisms

PERHAPS THE simplest smooth discrete time dynamical systems are the Morse-Smale diffeomorphisms. Among structurally stable systems they exhibit the simplest recurrent behavior-a finite set of hyperbolic periodic pionts and no other recurrence. These systems have been the object of considerable research. Palis and Smale[l3] proved that they are structurally stable. More recent work has dealt with the relationship of the homotopy class of a diffeomorphism to the kind of dynamics it exhibts [5,9, 11, 161 and the question of the existence of a Morse-Smale diffeomorphism in a given homotopy class. This latter topic is the subject of this article. In[l6], Shub and Sullivan showed among other things that a necessary condition for the existence of a Morse-Smale diffeomorphism is that all eigenvalues of the induced maps on homology be roots of unity. In the case of simply connected manifolds of dimension greater than five. they reduced the question of existence to an algebraic condition on the chain level for the diffeomorphism (1.2, below) and using this condition pionted out the existence of additional obstructions related to the ideal class groups of the cyclotomic fields. In this article we identify the group in which the obstruction lies, express it in terms of the algebraic K-theory of the induced endomorphisms f*: H,(M)+H,(M) and show that there are no further obstructions. More precisely, we consider the category QI of abelian groups with quasi-idempotent endomorphisms (i.e. having all eigenvalues zero or roots of unity) and show the obstruction lies in the torsion subgroup G of Ko(QI). The obstruction depends in fact only on the elements d(Lf*k]) in G, where Lf*k] denotes the class in Ko(QI) of the quasi-idempotent endomorphism f*L: H,(M, Z) + H,(M, Z) and c$: K,(QI)+ G is a projection onto the summand G. Of course if f is a diffeomorphism f *Ir will be quasi-unipotent (having only roots of unity as eigenvalues) since it is an automorphism. Our main result is the following.

Orientation-reversing Morse-Smale diffeomorphisms on the torus

For orientation-reversing diffeomorphisms on the torus necessary and sufficient conditions are given for an isotopy class to admit a Morse-Smale diffeomorphism with a specified periodic behavior.