| May 24th | Step skew products with circle or interval fibre |
| Alexey Okunev | |
| Loughborough University | |
An iterated function system is a tuple of smooth maps (in this talk orientation-preserving diffeomorphisms) from some manifold M to itself. The dynamics of the semigroup generated by an IFS can be naturally encoded by one map, a skew product over Bernoulli shift with the fibre M. The fibre maps of this skew product depend only on the zeroth element of the sequence in the base, such skew products are called step skew products. As the dynamics in the base is "hyperbolic", step skew products can be considered toy examples of partially hyperbolic dynamical systems. We will discuss the following phenomena exhibited by step skew products with one-dimensional fibre:
Then we will talk about the following result: generic step skew product with circle fibre either is robustly transitive or has an absorbing domain (the base times a finite disjoint union of intervals). In the second case the skew product can be reduced to a skew product with interval fibre. |
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| April 26th | Generalizing pseudo-Anosov maps |
| Philip Boyland | |
| University of Florida | |
Thurston's pseudoAnosov (pA) maps play a central role in low-dimensional topology and dynamics. After reviewing their definition and basic properties we discuss a generalization motivated by a family of sphere homeomorphisms derived from the tent family. The generalization retains most of the dynamical properties and replaces the pA invariant measured foliations with invariant measured turbulations. These consist of a full measure subset of disjoint immersed lines each carrying a nice measure. |
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| April 19th | Typical Geodesics on the Modular Surface |
| Vaibhav Gadre | |
| University of Glasgow | |
The modular surface is the quotient of the hyperbolic plane by the action of the group SL(2,Z). We will survey some results for typical geodesics on the modular surface, as a prototype of more general work. The focus will be statistics of cusp excursions of a typical geodesic. This theory admits many nice perspectives from continued fractions to dynamics of the hyperbolic geodesic flow. In our talk, we will also consider several different notions of what a typical geodesic means and outline how the results differ across these various notions. |