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Journal of Modern Dynamics

2011 , Volume 5 , Issue 2

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Local rigidity of homogeneous parabolic actions: I. A model case
Danijela Damjanovic and  Anatole Katok
2011, 5(2): 203-235 doi: 10.3934/jmd.2011.5.203 +[Abstract](39) +[PDF](362.3KB)
We show a weak form of local differentiable rigidity for the rank $2$ abelian action of upper unipotents on $SL(2,R)$ $\times$ $SL(2,R)$ $/\Gamma$. Namely, for a $2$-parameter family of sufficiently small perturbations of the action, satisfying certain transversality conditions, there exists a parameter for which the perturbation is smoothly conjugate to the action up to an automorphism of the acting group. This weak form of rigidity for the parabolic action in question is optimal since the action lives in a family of dynamically different actions. The method of proof is based on a KAM-type iteration and we discuss in the paper several other potential applications of our approach.
Measures invariant under horospherical subgroups in positive characteristic
Amir Mohammadi
2011, 5(2): 237-254 doi: 10.3934/jmd.2011.5.237 +[Abstract](47) +[PDF](270.7KB)
We prove measure rigidity for the action of maximal horospherical subgroups on homogeneous spaces over a field of positive characteristic. In the case when the lattice is uniform we prove the action of any horospherical subgroup is uniquely ergodic.
Outer billiards and the pinwheel map
Richard Evan Schwartz
2011, 5(2): 255-283 doi: 10.3934/jmd.2011.5.255 +[Abstract](42) +[PDF](1053.5KB)
In this paper we establish an equivalence between an outer billiards system based on a convex polygon $P$ and an auxiliary system, which we call the pinwheel map, that is based on $P$ in a different way. The pinwheel map is akin to a first-return map of the outer billiards map. The virtue of our result is that most of the main questions about outer billiards can be formulated in terms of the pinwheel map, and the pinwheel map is simpler and seems more amenable to fruitful analysis.
Square-tiled cyclic covers
Giovanni Forni , Carlos Matheus and  Anton Zorich
2011, 5(2): 285-318 doi: 10.3934/jmd.2011.5.285 +[Abstract](48) +[PDF](390.3KB)
A cyclic cover of the complex projective line branched at four appropriate points has a natural structure of a square-tiled surface. We describe the combinatorics of such a square-tiled surface, the geometry of the corresponding Teichmüller curve, and compute the Lyapunov exponents of the determinant bundle over the Teichmüller curve with respect to the geodesic flow. This paper includes a new example (announced by G. Forni and C. Matheus in [17] of a Teichmüller curve of a square-tiled cyclic cover in a stratum of Abelian differentials in genus four with a maximally degenerate Kontsevich--Zorich spectrum (the only known example in genus three found previously by Forni also corresponds to a square-tiled cyclic cover [15]. We present several new examples of Teichmüller curves in strata of holomorphic and meromorphic quadratic differentials with a maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichmüller curves with maximally degenerate spectra. We prove that this is indeed the case within the class of square-tiled cyclic covers.
Lyapunov spectrum of square-tiled cyclic covers
Alex Eskin , Maxim Kontsevich and  Anton Zorich
2011, 5(2): 319-353 doi: 10.3934/jmd.2011.5.319 +[Abstract](56) +[PDF](396.9KB)
A cyclic cover over $CP^1$ branched at four points inherits a natural flat structure from the "pillow" flat structure on the basic sphere. We give an explicit formula for all individual Lyapunov exponents of the Hodge bundle over the corresponding arithmetic Teichmüller curve. The key technical element is evaluation of degrees of line subbundles of the Hodge bundle, corresponding to eigenspaces of the induced action of deck transformations.
A geometric criterion for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle
Giovanni Forni
2011, 5(2): 355-395 doi: 10.3934/jmd.2011.5.355 +[Abstract](1120) +[PDF](435.6KB)
We establish a geometric criterion on a $SL(2, R)$-invariant ergodic probability measure on the moduli space of holomorphic abelian differentials on Riemann surfaces for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle on the real Hodge bundle. Applications include measures supported on the $SL(2, R)$-orbits of all algebraically primitive Veech surfaces (see also [7]) and of all Prym eigenforms discovered in [34], as well as all canonical absolutely continuous measures on connected components of strata of the moduli space of abelian differentials (see also [4, 17]). The argument simplifies and generalizes our proof for the case of canonical measures [17]. In the Appendix, Carlos Matheus discusses several relevant examples which further illustrate the power and the limitations of our criterion.
New light on solving the sextic by iteration: An algorithm using reliable dynamics
Scott Crass
2011, 5(2): 397-408 doi: 10.3934/jmd.2011.5.397 +[Abstract](36) +[PDF](296.5KB)
In recent work on holomorphic maps that are symmetric under certain complex reflection groups---generated by complex reflections through a set of hyperplanes, the author announced a general conjecture related to reflection groups. The claim is that for each reflection group $G$, there is a $G$-equivariant holomorphic map that is critical exactly on the set of reflecting hyperplanes.
    One such group is the Valentiner action $\mathcal{V}$---isomorphic to the alternating group $\mathcal{A}_6$---on the complex projective plane. A previous algorithm that solved sixth-degree equations harnessed the dynamics of a $\mathcal{V}$-equivariant. However, important global dynamical properties of this map were unproven. Revisiting the question in light of the reflection group conjecture led to the discovery of a degree-31 map that is critical on the 45 lines of reflection for $\mathcal{V}$. The map's critical finiteness provides a means of proving its possession of the previous elusive global properties. Finally, a sextic-solving procedure that employs this map's reliable dynamics is developed.

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