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Journal of Modern Dynamics

2011 , Volume 5 , Issue 3

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Contact homology of orbit complements and implied existence
Al Momin
2011, 5(3): 409-472 doi: 10.3934/jmd.2011.5.409 +[Abstract](507) +[PDF](841.1KB)
For Reeb vector fields on closed 3-manifolds, cylindrical contact homology is used to show that the existence of a set of closed Reeb orbit with certain knotting/linking properties implies the existence of other Reeb orbits with other knotting/linking properties relative to the original set. We work out a few examples on the 3-sphere to illustrate the theory, and describe an application to closed geodesics on $S^2$ (a version of a result of Angenent in [1]).
Outer billiards on the Penrose kite: Compactification and renormalization
Richard Evan Schwartz
2011, 5(3): 473-581 doi: 10.3934/jmd.2011.5.473 +[Abstract](387) +[PDF](1247.0KB)
We give a fairly complete analysis of outer billiards on the Penrose kite. Our analysis reveals that this $2$-dimensional dynamical system has a $3$-dimensional compactification, a certain polyhedron exchange map defined on the $3$-torus, and that this $3$-dimensional system admits a renormalization scheme. The two features allow us to make sharp statements concerning the distribution, large- and fine-scale geometry, and hidden algebraic symmetry, of the orbits. One concrete result is that the union of the unbounded orbits has Hausdorff dimension $1$. We establish many of the results with computer-aided proofs that involve only integer arithmetic.
A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map
Marie-Claude Arnaud
2011, 5(3): 583-591 doi: 10.3934/jmd.2011.5.583 +[Abstract](395) +[PDF](178.5KB)
We construct a $C^1$ symplectic twist map $f$ of the annulus that has an essential invariant curve $\Gamma$ such that:
    •$\Gamma$ is not differentiable;
    •$f$ ↾ $\Gamma$ is conjugate to a Denjoy counterexample.
Bernoulli equilibrium states for surface diffeomorphisms
Omri M. Sarig
2011, 5(3): 593-608 doi: 10.3934/jmd.2011.5.593 +[Abstract](432) +[PDF](257.9KB)
Suppose $f\colon M\to M$ is a $C^{1+\alpha}$ $(\alpha>0)$ diffeomorphism on a compact smooth orientable manifold $M$ of dimension 2, and let $\mu_\Psi$ be an equilibrium measure for a Hölder-continuous potential $\Psi\colon M\to \mathbb R$. We show that if $\mu_\Psi$ has positive measure-theoretic entropy, then $f$ is measure-theoretically isomorphic mod $\mu_\Psi$ to the product of a Bernoulli scheme and a finite rotation.
On distortion in groups of homeomorphisms
Światosław R. Gal and Jarek Kędra
2011, 5(3): 609-622 doi: 10.3934/jmd.2011.5.609 +[Abstract](440) +[PDF](230.5KB)
Let $X$ be a path-connected topological space admitting a universal cover. Let Homeo$(X, a)$ denote the group of homeomorphisms of $X$ preserving a degree one cohomology class $ a$.
    We investigate the distortion in Homeo$(X, a)$. Let $g\in$ Homeo$(X, a)$. We define a Nielsen-type equivalence relation on the space of $g$-invariant Borel probability measures on $X$ and prove that if a homeomorphism $g$ admits two nonequivalent invariant measures then it is undistorted. We also define a local rotation number of a homeomorphism generalizing the notion of the rotation of a homeomorphism of the circle. Then we prove that a homeomorphism is undistorted if its rotation number is nonconstant.

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