Presentation Title

Linear Trajectories on Homothety Surfaces

Faculty Mentor

Joshua Bowman

Start Date

18-11-2017 10:45 AM

End Date

18-11-2017 11:00 AM

Location

9-285

Session

Physical Sciences 2

Type of Presentation

Oral Talk

Subject Area

physical_mathematical_sciences

Abstract

A homothety surface is constructed by gluing the sides of polygons in the plane by homotheties—compositions of scalings and translations. Homothety surfaces generalize translation surfaces, which have been well-studied for several decades. We examine long-term behaviors of periodic and non-periodic linear trajectories on a one-parameter family of genus-2 homothety surfaces and compare these trajectories with those on the square torus.

Summary of research results to be presented

We find that an inverted "devil's staircase" function describes how slopes of linear trajectories on the family of homothety surfaces in question relate to the slopes of linear trajectories on the square torus. We also give examples of specific trajectories on the surface whose closures of affine interval exchange maps give sets homeomorphic to the Cantor set.

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Nov 18th, 10:45 AM Nov 18th, 11:00 AM

Linear Trajectories on Homothety Surfaces

9-285

A homothety surface is constructed by gluing the sides of polygons in the plane by homotheties—compositions of scalings and translations. Homothety surfaces generalize translation surfaces, which have been well-studied for several decades. We examine long-term behaviors of periodic and non-periodic linear trajectories on a one-parameter family of genus-2 homothety surfaces and compare these trajectories with those on the square torus.