Surface Diffeomorphisms Research Articles

An obstruction to the existence of certain dynamics in surface diffeomorphisms

AbstractLet M be a two-dimensional, compact manifold and g:Μ→ΜM be a diffeomorphism with a hyperbolic chain recurrent set. We find restrictions on the reduced zeta function p(t) of anyzero-dimensional basic set of g. If deg (p(t)) is odd, then p(1) = 0 (in ). Since there are infinitely many subshifts of finite type whose reduced zeta functions do not satisfy these restrictions, there are infinitely many subshifts which cannot be basic sets for any diffeomorphism of any surface.

Group completions and limit sets of Kleinian groups

Finitely generated groups and actions of finitely generated groups often come up in studying topology and geometry. While the most important example may be as fundamental groups of compact manifolds, questions involving finitely generated groups also arise in transformation groups, dynamical systems, and Kleinian groups. A very striking example is Mostow's theorem [17], which says that, for dimensions greater than or equal to three, closed hyperbolic manifolds are determined up to isometry by their fundamental groups. Jakob Nielsen, in a series of papers ([18-21]), used the Poincar6 disk model of hyperbolic 2-space, H 2, as int(D 2) to study surfaces and their diffeomorphisms. Given a closed surface M 2 of genus g > 2 and a diffeomorphism f : M--+M, he lifted f to a homeomorphism f ' : H 2 ~ H e and showed that f extends to a homeomorphism of the circle S 1 =P,D 2. Furthermore, the map on the circle does not depend on the particular diffeomorphism J; but only on its homotopy type. Nielsen made use of the extension o f f ' to D 2 and of the COl'responding actions of H~(M) and Aut(HI(M)) on S 1 to systematically study topological properties of diffeomorphisms of surfaces. The proof of Mostow's theorem also uses the action of Hi(M" ) on S"Given two closed hyperbolic n-manifolds M" and N" with 11 > 3 and an isomorphism q>: HI(M)--,Il l(N), there is a homotopy equivalence f : M--+N inducing 4> (since M and N are K(H, 1)'s). f can be lifted to the universal covers to !" H"~H" , and f extends to a homeomorphism f ' : S"-I--+S "1. The essence of M 9 ostow s proof is to show that f ' is conformal; this is done by first showing lhat J" is quasi-conformal and then using ergodicity of the action of H~(M) on S"-1 to show conformality. It was realized by Margulis and at least implicity by Mostow that the homotopy equivalence f : M ~ N used to construct f ' was not

On the nonwandering sets of diffeomorphisms of surfaces

Let M be a compact manifold without boundary. Let f: M → M be a C1 diffeomorphism. Then the nonwandering set Ω(f) is defined to be the closed invariant set consisting of x ∈ M such that for any neighborhood U of x, there exists an integer n ≠ 0 satisfying fn(U) ∩ U ≠ ø. In particular, the set Per (f) of all periodic points is included in Ω(f).

Invariants of the NPT isotopy classes of Morse-Smale diffeomorphisms of surfaces

Invariants of the NPT isotopy classes of Morse-Smale diffeomorphisms of surfaces

Wild unstable manifolds

LET M be a compact smooth manifold, f: M + M a diffeomorphism. An energy function for f is a Morse function M + R which decreases along non-periodic orbits and has critical points only at the periodic points off. An analogous concept for vector fields was introduced by Meyer [8], who proved that Morse-Smale vector fields on compact manifolds always have energy functions. Shub[lZ] and Takens [15] have since asserted, without proof, that the theorem is also true for Morse-Smale diffeomorphisms. In this paper we show, using an example on the three-sphere, that this conjecture is false. The example is given in 02, after a section consisting of definitions and a construction of energy functions for Morse-Smale diffeomorphisms of surfaces. The distinguishing characteristic of the counterexample on S’ is an unstable manifold whose closure is a wildly embedded arc: under certain hypotheses it is shown that such phenomena preclude the existence of energy functions. In investigating further examples of diffeomorphisms with such wild phase portraits we see that no possible extension of the notion of labeled diagram will serve to classify conjugacy classes of Morse-Smale diffeomorphisms in dimension 3. These examples are discussed in 63. All our examples are gradient-like: there is no complicated behavior in the sense of [lo]. Note. The major results of Shub and Takens in the articles referenced above are still true. An alternative proof of Shub’s theorem has appeared in [13] and elsewhere, and Takens’ arguments work just as well using non-degenerate Liapunov functions, as defined in 91.

SOURCES AND SINKS OFA-DIFFEOMORPHISMS OF SURFACES

In the present paper the mechanism of the appearance of zero-dimensional sinks and sources in the presence of one-dimensional basic sets of diffeomorphisms of two-dimensional surfaces, satisfying Axiom A, is studied. New examples are constructed of one-dimensional basic sets of structurally stable diffeomorphisms of the two-dimensional sphere. The existence is proved of zero-dimensional sinks and sources of diffeomorphisms of orientable surfaces of genus less than two, which are not Y-diffeomorphisms. An estimate is given of the number of amply situated basic sets of A-diffeomorphisms of orientable surfaces by means of topological invariants of the surfaces. Figures: 2. Bibliography: 17 items.

On the $L^p$-geometry of autonomous Hamiltonian diffeomorphisms of surfaces

We prove a number of results on the interrelation between the $L^p$-metric on the group of Hamiltonian diffeomorphisms of surfaces and the subset of all autonomous Hamiltonian diffeomorphisms. More precisely, we show that there are Hamiltonian diffeomorphisms of all surfaces of genus $g\neq 1$ lying arbitrarily $L^p$-far from this subset; answering a variant of a question of Polterovich for the $L^p$-metric.