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Sage Days 112 Online, February-March 2021

Sage is an open source software for mathematics. This is the webpage for the Sage workshop for ANR CODYS. The aim is to:

introduce Sage to people in the ANR
implement Loick Lhote's code to compute spectrum of transfer operators on continued fraction algorithms.
help people to implement their own projects

The workshop will take place on two Thusrday mornings to be determined replacing the weekly sage seminar thursdaysbdx.

Please fill up the poll.

Material

The following projects have been discussed during the last ANR meeting in December.

Implement computation of spectra for transfer operators associated to :
1. Gauss map
2. Jacobi-Perron
3. Ostrowski
4. Triangle map
5. ...
Check Alkauskas theorem.
Fast approximation of top Lyapunov exponent (Pollicott paper Maximal Lyapunov exponents for random matrix products via periodic orbits and determinant formula)
- See whether it can be extended to simplicial systems... several problems
  - use canonical measure instead of Bernoulli measure
  - non-negative matrices instead of positive
Proven enclosure using ball arithmetic and remainder estimates
- spectrum of transfer operators
- top Lyapunov exponent
- resonances of zeta function (eg https://arxiv.org/abs/2002.03334)
- Hausdorff dimensions (eg the recent https://arxiv.org/abs/2012.07083)

Here are the project that where mentionned during the first meeting

"Certify" (or at least check) invariant measures with SageMath [Valerie]
First Lyapunov exponent using Pollicott https://doi.org/10.1007/s00222-010-0246-y
Faster algorithm to compute only the k first eigenvalues (instead of the whole spectrum)
Iterative methods for the computation of the k first eigenvalues and eignfunctions of the spectrum (the (n+1) x (n+1) matrix is obtained from the (n) x (n) matrix by adding one row and one column)... one issue is already to obtain the accuracy at n-th step.
Higher dimensional Daudet-Flajolet-Vallée (for example Brun in dimension 2, Jacobi-Perron) : Loïck : I can try to apply the method to Brun Algorithm...)

This workshop is supported by ANR CODYS.

Février

Mars

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-== Schedule ==
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-Presentation of Daudet-Flajolet-Vallée algorithm on Gauss map [[https://wiki.sagemath.org/days112?action=AttachFile&do=get&target=Gauss+spectrum.ipynb|notebook]].
+  * Daudet-Flajolet-Vallée algorithm on Gauss map [[https://wiki.sagemath.org/days112?action=AttachFile&do=get&target=Gauss+spectrum.ipynb|notebook]].
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+  * Pollicott algorithm for Lyapunov exponents of random matrix products [[https://wiki.sagemath.org/days112?action=AttachFile&do=get&target=Pollicott%27s+algorithm.ipynb|notebook]].