Figures from Zorich, 2008 Some low-dimensional dynamical systems - PowerPoint PPT Presentation

Genus 2 translation surface obtained by identifying parallel sides of the regular octagon Conical singularity, cone angle 6 π Figures from Zorich, 2008

Some low-dimensional dynamical systems induce a dynamical system of the same class on a restriction of the original domain Rauzy induction on IETs Iterating this renormalization operator creates a dynamics on the class of IETs dynamical systems Torus Renormalization of the flow (classical ) yields a renormalization of the Regular symbolic dynamics as well Octagon (Smillie- Regular Ulcigrai) Correspondence between 2n-gon flow renormalization and combinatorial renormalization well understood for What about the ( π /n)-isosceles triangles that tile regular 2n-gons? Figures from Viana, CIM-COIMBRA Figures from Vsmillie-Ulcigrai

Fixing one exterior side of the 2n-gon as the lift of the triangle base gives a well defined map of linear trajectories from the 2n-gon to the triangle and a well-defined lift of billiard trajectories from the triangle to the 2n-gon

Can go back and forth between codings for ∑ 0 trajectories on 2n- gon and isosceles using λ , ρ -augmented transition diagrams (and adding / dropping subscripts and superscripts )

This presents a potential problem for direct translation of combinatorial renormalization rules from In our case, the However, the required subword the punctured 2n-gon to the required subword length for the translated rule isosceles triangle length on the 2n-gon on the apex- π /n isosecles will be at most 5, grows quadratically in n Independent of n . We will return to this later if time permits. …

Need stronger notion of admissibility

Unlike in the genus 1 case: not all infinitely deriveable words are realizable as cutting sequences

Generation ‘Inverting ‘ Derivation: Interpolating Words

The realizable infinitely deriveable words also satisfy a set of coherence conditions

The proof is somewhat complicated. See Section 4.3.1

Proof again somewhat complicated, uses properties of the sequence of sectors that successive derivations of an Infinitely deriveable word are admissible in. See Section 4.3.1.

Extras and Appendices