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Journal of Modern Dynamics

2012 , Volume 6 , Issue 3

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No planar billiard possesses an open set of quadrilateral trajectories
Alexey Glutsyuk and Yury Kudryashov
2012, 6(3): 287-326 doi: 10.3934/jmd.2012.6.287 +[Abstract](382) +[PDF](1188.8KB)
Abstract:
The article is devoted to a particular case of Ivriĭ's conjecture on periodic orbits of billiards. The general conjecture states that the set of periodic orbits of the billiard in a domain with smooth boundary in the Euclidean space has measure zero. In this article we prove that for any domain with piecewise $C^4$-smooth boundary in the plane the set of quadrilateral trajectories of the corresponding billiard has measure zero.
The Julia set of a post-critically finite endomorphism of $\mathbb{PC}^2$
Volodymyr Nekrashevych
2012, 6(3): 327-375 doi: 10.3934/jmd.2012.6.327 +[Abstract](435) +[PDF](999.4KB)
Abstract:
We construct a combinatorial model of the Julia set of the endomorphism $f(z, w)=((1-2z/w)^2, (1-2/w)^2)$ of $\mathbb{PC}^2$.
Compact asymptotically harmonic manifolds
Andrew M. Zimmer
2012, 6(3): 377-403 doi: 10.3934/jmd.2012.6.377 +[Abstract](372) +[PDF](304.7KB)
Abstract:
A complete Riemannian manifold without conjugate points is said to be asymptotically harmonic if the mean curvature of its horospheres is a universal constant. Examples of asymptotically harmonic manifolds include flat spaces and rank-one locally symmetric spaces of noncompact type. In this paper we show that this list exhausts the compact asymptotically harmonic manifolds under a variety of assumptions including nonpositive curvature or Gromov-hyperbolic fundamental group. We then present a new characterization of symmetric spaces amongst the set of all visibility manifolds.
Schwarz triangle mappings and Teichmüller curves: Abelian square-tiled surfaces
Alex Wright
2012, 6(3): 405-426 doi: 10.3934/jmd.2012.6.405 +[Abstract](350) +[PDF](319.2KB)
Abstract:
We consider normal covers of $\mathbb{C}P^1$ with abelian deck group and branched over at most four points. Families of such covers yield arithmetic Teichmüller curves, whose period mapping may be described geometrically in terms of Schwarz triangle mappings. These Teichmüller curves are generated by abelian square-tiled surfaces.
    We compute all individual Lyapunov exponents for abelian square-tiled surfaces, and demonstrate a direct and transparent dependence on the geometry of the period mapping. For this we develop a result of independent interest, which, for certain rank two bundles, expresses Lyapunov exponents in terms of the period mapping. In the case of abelian square-tiled surfaces, the Lyapunov exponents are ratios of areas of hyperbolic triangles.

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