Browsing by Author "Veech, William A."

Now showing 1 - 6 of 6
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    Applications of Rauzy Induction on the generic ergodic theory of interval exchange transformations
    (2006) Wu, Yue; Veech, William A.
    Rauzy Induction, raised by Rauzy and Veech, has been served as an important technique to study interval exchange transformations. In this thesis, based on the classical theory ([RAU][VEE1][4]), we utilized Rauzy Induction, introduced new ideas, constructed new techniques, and achieved new generic results on interval exchange transformations. In Chapter 3, we reached this final theorem (Corollary 3.3.3) for any irreducible m-permutation, the measure theoretically generic interval exchange transformation T satisfies: the topologically generic transformation in the commutant of T is rank one. This is a corollary of Theorem 3.0.1, which establishes that for any irreducible m-permutation, the measure theoretically generic interval exchange transformation T has all of its nonzero powers rank one. The connection between these two results is made by applying a theorem of J. King [KIN1]: a rank one transformation generates a dense subgroup of its commutant. When all powers are rank one and rigid, the commutant contains a dense Gdelta-set of rank one transformations. In Chapter 4, we proved that measure theoretically typically all the symmetric (with permutation (3 2 1)) three interval exchange transformations are whirly. Thus by [GLA, TSI, WEI], near Borel action of the commutant of the corresponding in terval exchange transformation admits no nontrivial spatial factors. We relied on a relation of the return times associated with the Rauzy-Veech induction, and also the idea to study the multiples of the time when the iteration of T is close to identity map. Based on the whirly property of some intervals, we complete the work by a density point argument. Whether the corresponding statement of general cases (m > 3) is true is still an open problem. In Chapter 5, further discussion on interval exchange transformations were described. A class of pseudo-Anosov inverval exchange transformations were verified to be weak mixing. They are concrete weak mixing examples. Then we also introduced the computational results with Matlab about the Rauzy classes. In Chapter 6, we studied del Junco-Rudolph's example, proved that all powers of it are rank one. We also reached a proposition about the cylinder sets, which is a positive step toward determining whether the map is whirly or not.
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    Fuchsian groups and polygonal billiards
    (1996) Ward, Clayton Collin; Veech, William A.
    Let P be a simple, closed polygon in the plane, all interior angles of which are rational multiples of $\pi$. We consider the possible paths of a point, rebounding in the interior of P with constant speed and elastic reflections. Such a dynamical system is known as "billiards in P". By means of a well-known construction, "billiard" trajectories in such a polygon P are identified with geodesic paths on a closed Riemann surface $X\sp{P}$, where the Riemannian metric is one of zero curvature with isolated singularities, and is given by a holomorphic one-form $\omega$ on the surface. To this holomorphic one-form one can canonically associate a discrete subgroup $\Gamma$ of $PSL(2,\IR$). If $\Gamma$ happens to be a lattice (has cofinite volume), then it is known that all geodesic paths in the zero-curvature metric given by $\omega$ must either be closed or uniformly distributed in the surface $X\sp{P}$. As a corollary, all billiard paths in the original polygon P must either be finite or uniformly distributed in P. A new class of examples of polygons P, whose associated group $\Gamma$ is, in fact, a lattice have been discovered. At the same time, we have discovered the first examples of triangles P, as above, for which the associated groups $\Gamma$ are not lattices (i.e. have infinite covolume). Finally, it is shown how to derive, in an explicit way, algebraic equations which specify the Riemann surface $X\sp{P}$ and one-form $\omega$, which before were only described geometrically.
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    Linear dimensions of (written el) P and LP spaces
    (1971) Lin, Tzy-Ping; Veech, William A.
    Isomorphisms between Lq and Lp spaces exist for some ratios between p and q. For some of other ratios between p and q, no isomorphism exists. Similar results exist for Lp and Lq.
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    Self-Inverses in Rauzy Classes
    (2011) Fickenscher, Jonathan Michael; Veech, William A.
    Thanks to works by M. Kontsevich and A. Zorich followed by C. Boissy, we have a classification of all Rauzy Classes of any given genus. It follows from these works that Rauzy Classes are closed under the operation of inverting the permutation. In this paper, we shall prove the existence of self-inverse permutations in every Rauzy Class by giving an explicit construction of such an element satisfying the sufficient conditions. As a corollary, we will give another proof that every Rauzy Class is closed under taking inverses. In the case of generalized permutations, generalized Rauzy Classes have been classified by works of M. Kontsevich, H. Masur and J. Smillie, E. Lanneau, and again C. Boissy. We state the definition of self-inverse for generalized permutations and prove a necessary and sufficient condition for a generalized Rauzy Class to contain self-inverse elements.
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    The behavior of orbits of some flows with two fixed points on the torus
    (1979) Mitchell, Randolph Calvin; Veech, William A.
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    Using glueing diagrams to find boundary curves of incompressible surfaces in a hyperbolic knot space
    (1984) Cohn, Aaron I.; Veech, William A.; Culler, Marc; Shalen, Peter B.; Hempel, John
    We calculate some boundary curves of incompressible surfaces in some knot spaces. This method is based on a theorem of Marc Culler and Peter Shaien, and we make use of some calculations done by William Menasco.