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

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

Jan 26, 2024 138 likes •646 views

Genus 2 translation surface obtained by identifying parallel sides of the regular octagon Conical singularity, cone angle 6 Figures from Zorich, 2008 Some low-dimensional dynamical systems induce a dynamical system of the same class on

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

  • f the

regular

  • ctagon
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Some low-dimensional dynamical systems induce a dynamical system of the same class

  • n a restriction of the original domain

Figures from Viana, CIM-COIMBRA

Rauzy induction on IETs

Iterating this renormalization operator creates a dynamics on the class of dynamical systems Renormalization of the flow yields a renormalization of the symbolic dynamics as well Correspondence between flow renormalization and combinatorial renormalization well understood for IETs

Figures from Vsmillie-Ulcigrai

Torus

(classical)

Regular Octagon Regular 2n-gon

(Smillie- Ulcigrai)

What about the (π/n)-isosceles triangles that tile regular 2n-gons?

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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

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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)

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This presents a potential problem for direct translation of combinatorial renormalization rules from the punctured 2n-gon to the isosceles triangle

In our case, the required subword length on the 2n-gon will be at most 5, Independent of n. However, the required subword length for the translated rule

  • n the apex-π/n isosecles

grows quadratically in n

We will return to this later if time permits.…

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Need stronger notion of admissibility

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

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Generation ‘Inverting ‘ Derivation: Interpolating Words

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

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The proof is somewhat complicated. See Section 4.3.1

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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.

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Extras and Appendices

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