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John Franks

Publications and source records attributed to John Franks.

28 records · Page 2Linked to original sources
arXiv2003

Periodic points of Hamiltonian surface diffeomorphisms

The main result of this paper is that every non-trivial Hamiltonian diffeomorphism of a closed oriented surface of genus at least one has periodic points of arbitrarily high period. The same result is true for S^2 provided the diffeomorphism has at least three fixed points. In addition we show that up to isotopy relative to its fixed point set, every orientation preserving diffeomorphism F: S --> S of a closed orientable surface has a normal form. If the fixed point set is finite this is just the Thurston normal form.

math.DS↗
arXiv2003

Area preserving group actions on surfaces

Suppose G is an almost simple group containing a subgroup isomorphic to the three-dimensional integer Heisenberg group. For example any finite index subgroup of SL(3,Z) is such a group. The main result of this paper is that every action of G on a closed oriented surface by area preserving diffeomorphisms factors through a finite group.

math.DS↗
arXiv2003

A Hölder continuous vector field tangent to many foliations

We construct an example of a Hölder continuous vector field on the plane which is tangent to all foliations in a continuous family of pairwise distinct $C^1$ foliations. Given any $1 \le r <\infty,$ the construction can be done in such a way that each leaf of each foliation is the graph of a $C^r$ function from $\R$ to $\R.$ We also show the existence of a continuous vector field $X$ on $\R^2$ and two foliations $\cal{F}$ and $\cal{G}$ on $\R^2$ each tangent to $X$ with a dense subset $\cal E$ of $\R^2$ such that at every point $x\in \cal E$ the leaves $F_x$ and $G_x$ of the foliation $\cal{F}$ and $\cal{G}$ through $x$ are topologically transverse.

math.DS↗
arXiv2003

Rotation Numbers and Instability Sets

Translation and rotation numbers have played an interesting and important role in the qualitative description of various dynamical systems. In this exposition we are especially interested in applications which lead to proofs of periodic motions in various kinds of dynamics on the annulus. The applications include billiards and geodesic flows. Going beyond this simple qualitative invariant in the study of the dynamics of area preserving annulus maps, G.D. Birkhoff was led to the concept of ``regions of instability'' for twist maps. We discuss the closely related notion of instability sets for a generic area preserving surface diffeomorphism and develop their properties.

math.DS↗
arXiv2001

Groups of homeomorphisms of one-manifolds, I: actions of nonlinear groups

This self-contained paper is part of a series \cite{FF2,FF3} on actions by diffeomorphisms of infinite groups on compact manifolds. The two main results presented here are: 1) Any homomorphism of (almost any) mapping class group or automorphism group of a free group into $\Diff_+^r(S^1), r\geq 2$ is trivial. For r=0 Nielsen showed that in many cases nontrivial (even faithful) representations exist. Somewhat weaker results are proven for finite index subgroups. 2) We construct a finitely-presented group of real-analytic diffeomorphisms of $\R$ which is not residually finite.

math.DS↗
arXiv2000

Regions of instability for non-twist maps

In this paper we consider an analog of the regions of instability for twist maps in the context of area preserving diffeomorphisms which are not twist maps. Several properties analogous to those of classical regions of instability are proved.

math.DS↗
arXiv1999

Shift Equivalence and the Conley index

In this paper we introduce filtration pairs for isolated invariant sets of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Lastly, we define partially ordered Morse decompositions and prove the existence of Morse set filtrations for such decompositions.

math.DS↗
arXiv1996

The rotation set and periodic points for torus homeomorphisms

We consider the rotation set $ρ(F)$ for a lift $F$ of an area preserving homeomorphism $f: \t^2\to \t^2$, which is homotopic to the identity. The relationship between this set and the existence of periodic points for $f$ is least well understood in the case when this set is a line segment. We show that in this case if a vector $v$ lies in $ρ(F)$ and has both co-ordinates rational, then there is a periodic point $x\in \t^2$ with the property that $$\frac{F^q(x_0)-x_0}q = v$$ where $x_0\in \re^2$ is any lift of $x$ and $q$ is the least period of $x$.

math.DS↗