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Register a new userVeech surfaces with non-periodic directions in the trace field
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We show that each of Veechs original examples of translation surfaces with ``optimal dynamics whose trace field is of degree greater than two has non-periodic directions of vanishing SAF-invariant. Furthermore, we give explicit examples of pseudo-Anosov diffeomorphisms whose contracting direction has zero SAF-invariant.
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This paper is motivated by an interesting problem studied more than 50 years ago by Veech and which can be considered a parity, or mod 2, version of the classical equidistribution problem concerning the irrational rotation sequence. The Veech discrete 2-circle problem can also be visualized as a continuous flat dynamical system, in the form of 1-direction geodesic flow on a 2-square-b surface, a surface obtained by modifying the surface comprising two side-by-side squares by the inclusion of barriers and gates on the vertical edges, with appropriate modification of the edge identifications. A famous result of Gutkin and Veech says that 1-direction geodesic flow on any flat finite polysquare translation surface exhibits optimal behavior, in the form of an elegant uniform-periodic dichotomy. However, for irrational values of b, the 2-square-b surface is not a polysquare surface, and Veech and others have highlighted serious violations of the uniform-periodic dichotomy. Here we extend some of the results of Veech to consider cases previously not covered, and also obtain some time-quantitative description of these violations. Furthermore, we establish a far-reaching generalization of some earlier results to the class of flat finite polysquare-b-rational translation surfaces, obtained from flat finite polysquare translation surfaces in a similar way that the 2-square-b surface is constructed.
An Abelian differential gives rise to a flat structure (translation surface) on the underlying Riemann surface. In some directions the directional flow on the flat surface may contain a periodic region that is made up of maximal cylinders filled by parallel geodesics of the same length. The growth rate of the number of such regions counted with weights, as a function of the length, is quadratic with a coefficient, called Siegel-Veech constant, that is shared by almost all translation surfaces in the ambient stratum. We evaluate various Siegel-Veech constants associated to the geometry of configurations of periodic cylinders and their area, and study extremal properties of such configurations in a fixed stratum and in all strata of a fixed genus.
For each stratum of the space of translation surfaces, we introduce an infinite translation surface containing in an appropriate manner a copy of every translation surface of the stratum. Given a translation surface $(X, omega)$ in the stratum, a matrix is in its Veech group $mathrm{SL}(X,omega)$ if and only if an associated affine automorphism of the infinite surface sends each of a finite set, the ``marked {em Voronoi staples}, arising from orientation-paired segments appropriately perpendicular to Voronoi 1-cells, to another pair of orientation-paired ``marked segments. We prove a result of independent interest. For each real $age sqrt{2}$ there is an explicit hyperbolic ball such that for any Fuchsian group trivially stabilizing $i$, the Dirichlet domain centered at $i$ of the group already agrees within the ball with the intersection of the hyperbolic half-planes determined by the group elements whose Frobenius norm is at most $a$. %When $mathrm{SL}(X,omega)$ is a lattice we use this to give a condition guaranteeing that the full group $mathrm{SL}(X,omega)$ has been computed. Together, these results give rise to a new algorithm for computing Veech groups.
We provide a family of isolated tangent to the identity germs $f:(mathbb{C}^3,0) to (mathbb{C}^3,0)$ which possess only degenerate characteristic directions, and for which the lift of $f$ to any modification (with suitable properties) has only degenerate characteristic directions. This is in sharp contrast with the situation in dimension $2$, where any isolated tangent to the identity germ $f$ admits a modification where the lift of $f$ has a non-degenerate characteristic direction. We compare this situation with the resolution of singularities of the infinitesimal generator of $f$, showing that this phenomenon is not related to the non-existence of complex separatrices for vector fields of Gomez-Mont and Luengo. Finally, we describe the set of formal $f$-invariant curves, and the associated parabolic manifolds, using the techniques recently developed by Lopez-Hernanz, Raissy, Ribon, Sanz Sanchez, Vivas.
By the Lyapunov-Perron method,we prove the existence of random inertial manifolds for a class of equations driven simultaneously by non-autonomous deterministic and stochastic forcing. These invariant manifolds contain tempered pullback random attractors if such attractors exist. We also prove pathwise periodicity and almost periodicity of inertial manifolds when non-autonomous deterministic forcing is periodic and almost periodic in time, respectively.
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