Is a typical bi-Perron number a pseudo-Anosov dilatation?

In this note, we deduce a partial answer to the question in the title. In particular, we show that asymptotically almost all bi-Perron algebraic unit whose characteristic polynomial has degree at most $2n$ do not correspond to dilatations of pseudo-Anosov maps on a closed orientable surface of genus $n$ for $n\geq 10$. As an application of the argument, we also obtain a statement on the number of closed geodesics of the same length in the moduli space of area one abelian differentials for low genus cases

Asymptotic linearity of the mapping class group and a homological version of the Nielsen-Thurston classification

We study the action of the mapping class group on the real homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite-dimensional representation of the mapping class group. We show that this representation detects the Nielsen-Thurston classification of each mapping class. We then discuss some examples that occur in the theory of braid groups and develop an analogous theory for automorphisms of free groups. We close with some open problems.Comment: Revision, 27 page

Energy Conserving Truncations in Thermal Convection

.28> Ø ; Tg = r 2 T + @ Ø @x ; (1) where Ø is the stream function (with v = (\Gamma@ z Ø ; @ x Ø )), T is the temperature deviation from a linear profile, is the kinematic viscosity, and is the thermal conductivity. The Poisson bracket is defined as fA ; Bg j @ x A @ z B \Gamma @ z A @ x B. In the dissipationless limit, where and are set equal to zero, (1) conserves the total energy: E = K + U = 1 2 D (r Ø ) 2 E \Gamma h zT i ; (2) where the angle brackets denote an a

Topology, braids and mixing in fluids

Copyright © Royal Society 2006Stirring of fluid with moving rods is necessary in many practical applications to achieve homogeneity. These rods are topological obstacles that force stretching of fluid elements. The resulting stretching and folding is commonly observed as filaments and striations, and is a precursor to mixing. In a space-time diagram, the trajectories of the rods form a braid, and the properties of this braid impose a minimal complexity in the flow. We review the topological viewpoint of fluid mixing, and discuss how braids can be used to diagnose mixing and construct efficient mixing devices. We introduce a new, realisable design for a mixing device, the silver mixer, based on these principles.Jean-Luc Thiffeault and Matthew D. Fin

Topological chaos in flows on surfaces of arbitrary genus

http://ictam2008.adelaide.edu.au