Teichmüller mapping class group of the universal hyperbolic solenoid

Universal Approximator Property of the Space of Hyperbolic Tangent Functions

In this paper, first the space of hyperbolic tangent functions is introduced and then the universal approximator property of this space is proved. In fact, by using this space, any nonlinear continuous function can be uniformly approximated with any degree of accuracy. Also, as an application, this space of functions is utilized to design feedback control for a nonlinear dynamical system.

Actions of the Mapping Class Group

Let S be a closed oriented surface S of genus g ≥ 0 with m ≥ 0 marked points (punctures) and 3g − 3 + m ≥ 2. This is a survey of recent results on actions of the mapping class group of S which led to a geometric understanding of this group. Mathematics Subject Classification (2010). Primary 30F60, Secondary 20F28, 20F65, 20F69

Cohomology of the Stable Mapping Class Group

The stable mapping class group is the group of isotopy classes of automorphisms of a connected oriented surface of “large” genus. The Mumford conjecture postulates that its rational cohomology is a polynomial ring generated by certain classes κi of dimension 2i, for i > 0. Tillmann’s insight [38] that the plus construction makes the classifying space of the stable mapping class group into an in...

The Mapping Class Group of a Genus

Solenoid functions for hyperbolic sets on surfaces

We describe a construction of a moduli space of solenoid functions for the C -conjugacy classes of hyperbolic dynamical systems f on surfaces with hyperbolic basic sets f . We explain that if the holonomies are sufficiently smooth then the diffeomorphism f is rigid in the sense that it is C 1C conjugate to a hyperbolic affine model. We present a moduli space of measure solenoid functions for al...