Periodic Orbits and Spectral Statistics of Pseudointegrable Billiards

We demonstrate for a generic pseudointegrable billiard that the number of periodic orbit families with length less than $l$ increases as $\pi b_0l^2/\langle a(l) \rangle$, where $b_0$ is a constant and $\langle a(l) \rangle$ is the average area occupied by these families. We also find that $\langle a(l) \rangle$ increases with $l$ before saturating. Finally, we show that periodic orbits provide a good estimate of spectral correlations in the corresponding quantum spectrum and thus conclude that diffraction effects are not as significant in such studies.Comment: 13 pages in RevTex including 5 figure

Length functions on currents and applications to dynamics and counting

The aim of this (mostly expository) article is twofold. We first explore a variety of length functions on the space of currents, and we survey recent work regarding applications of length functions to counting problems. Secondly, we use length functions to provide a proof of a folklore theorem which states that pseudo-Anosov homeomorphisms of closed hyperbolic surfaces act on the space of projective geodesic currents with uniform north-south dynamics.Comment: 35pp, 2 figures, comments welcome! Second version: minor corrections. To appear as a chapter in the forthcoming book "In the tradition of Thurston" edited by V. Alberge, K. Ohshika and A. Papadopoulo

Periodic Orbits in Polygonal Billiards

We review some properties of periodic orbit families in polygonal billiards and discuss in particular a sum rule that they obey. In addition, we provide algorithms to determine periodic orbit families and present numerical results that shed new light on the proliferation law and its variation with the genus of the invariant surface. Finally, we deal with correlations in the length spectrum and find that long orbits display Poisson fluctuations.Comment: 30 pages (Latex) including 11 figure

Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow

We compute the sum of the positive Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow. The computation is based on the analytic Riemann-Roch Theorem and uses a comparison of determinants of flat and hyperbolic Laplacians when the underlying Riemann surface degenerates.Comment: Minor corrections. To appear in Publications mathematiques de l'IHE

Square-tiled cyclic covers

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 Teichm\"uller curve, and compute the Lyapunov exponents of the determinant bundle over the Teichm\"uller curve with respect to the geodesic flow. This paper includes a new example (announced by G. Forni and C. Matheus in \cite{Forni:Matheus}) of a Teichm\"uller 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 found previously by Forni in genus three also corresponds to a square-tiled cyclic cover \cite{ForniSurvey}). We present several new examples of Teichm\"uller curves in strata of holomorphic and meromorphic quadratic differentials with maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichm\"uller curves with maximally degenerate spectrum. We prove that this is indeed the case within the class of square-tiled cyclic covers.Comment: 34 pages, 6 figures. Final version incorporating referees comments. In particular, a gap in the previous version was corrected. This file uses the journal's class file (jmd.cls), so that it is very similar to published versio

Automorphisms of graphs of cyclic splittings of free groups

We prove that any isometry of the graph of cyclic splittings of a finitely generated free group $F_N$ of rank $N\ge 3$ is induced by an outer automorphism of $F_N$. The same statement also applies to the graphs of maximally-cyclic splittings, and of very small splittings.Comment: 22 pages, 5 figures. Small modifications. To appear in Geometriae Dedicat

Comparative Network Analysis of Preterm vs. Full-Term Infant-Mother Interactions

Several studies have reported that interactions of mothers with preterm infants show differential characteristics compared to that of mothers with full-term infants. Interaction of preterm dyads is often reported as less harmonious. However, observations and explanations concerning the underlying mechanisms are inconsistent. In this work 30 preterm and 42 full-term mother-infant dyads were observed at one year of age. Free play interactions were videotaped and coded using a micro-analytic coding system. The video records were coded at one second resolution and studied by a novel approach using network analysis tools. The advantage of our approach is that it reveals the patterns of behavioral transitions in the interactions. We found that the most frequent behavioral transitions are the same in the two groups. However, we have identified several high and lower frequency transitions which occur significantly more often in the preterm or full-term group. Our analysis also suggests that the variability of behavioral transitions is significantly higher in the preterm group. This higher variability is mostly resulted from the diversity of transitions involving non-harmonious behaviors. We have identified a maladaptive pattern in the maternal behavior in the preterm group, involving intrusiveness and disengagement. Application of the approach reported in this paper to longitudinal data could elucidate whether these maladaptive maternal behavioral changes place the infant at risk for later emotional, cognitive and behavioral disturbance

Generic Continuous Spectrum for Ergodic Schr"odinger Operators

We consider discrete Schr"odinger operators on the line with potentials generated by a minimal homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon's Lemma that for a generic continuous sampling function, the associated Schr"odinger operators have no eigenvalues in a topological or metric sense, respectively. We present a number of applications, particularly to shifts and skew-shifts on the torus.Comment: 14 page

Shapes of polyhedra, mixed volumes and hyperbolic geometry

We generalize to higher dimensions the Bavard–Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex d -dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The Alexandrov–Fenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting cone-manifold are equal to π/2

Problematic Instagram use: the role of perceived feeling of presence and escapism

The use of social networking sites is becoming increasingly popular. Although there are many studies investigating the problematic use of social networking sites such as Facebook, little is known about problematic Instagram use (PIU) and factors related to it. The present study developed a complex model in order to examine the mediating role of perceived feeling of presence (i.e., social, spatial, and co-presence) and escapism between using different Instagram features and PIU. A total of 333 Instagram users from a high school and a state university, aged between 14 and 23 years (Mage = 17.74 years, SD = 2.37, 61% female), completed a "paper-and-pencil" questionnaire comprising measures of social presence, spatial presence, co-presence, Instagram escapism, and PIU. In addition, frequency of use of five different Instagram features (i.e., watching live streams; watching videos; looking at posted photographs; liking, commenting on others' posts; and getting likes and comments from others) were assessed using a 7-point Likert scale. Analysis indicated that watching live streams was indirectly associated with PIU via escapism, spatial presence, and co-presence. Leaving likes and comments on others' posts was both directly and indirectly associated with PIU via co-presence and escapism. Escapism mediated the relationships between social and spatial presence and co-presence and PIU. The findings of the present study appear to indicate that a minority of individuals use Instagram problematically and that problematic Instagram use is associated with the frequency of watching live streams, liking, and commenting on others’ posts on Instagram, being able to feel a higher sense of presence using Instagram, and using Instagram as an escape from reality