Including code to compute Lyapunov exponents for translation surfaces.

[sagetrac-Fougeroc]

Code for studying lyapunov exponents of translation surfaces and their covers.

UPDATE:

• See compatibility with other existing code for interval echanges and strata
  compatible with code for quadratic and abelian stratum components

sage: q = QuadraticStratum([7,1,-1,-1,-1,-1]).components()[0]
sage: q.lyapunov_exponents()
[1.0006353514168902,
 0.44659813628011014,
 0.43073741652656045,
 0.2278515712988079,
 0.18675189999010267,
 0.07722243283694208]
sage: q.lyapunov_exponents_H_plus()
[0.44777037675789444, 0.185001855706418]

BUG:

• There is a bug for zero lyapunov exponents when decomposing in isotropic components :

sage: import sage.dynamics.flat_surfaces.lyapunov_exponents.interval_exchange as intex
sage: R = intex.cyclic_cover_iet(4, [1, 1, 1, 1])
sage: R.lyapunov_exponents_H_plus()
[0.9996553085103, 0.0007776980910571506, 0.00022201024035355403]
sage: R.lyapunov_exponents_H_plus_isotopic()
[[], [1.0002381322489762], [-0.09753642448496574], [-0.09750440918016302]]

CC: @sagetrac-Fougeroc @videlec

Component: geometry

Keywords: lyapunov_exponents, translation_surfaces, sagedays57

Reviewer: Vincent Delecroix

Issue created by migration from https://trac.sagemath.org/ticket/16102

[videlec]

comment:2

Hi Charles,

Here are some remarks.

About syntax:

• remove trailing whitespaces
• documentation lines should be at most 80 characters long
• all functions and methods must have an EXAMPLES block. In order
  to see how much work you have to do, you can use
  sage -coverage MY_FILE.
• The block RETURN should be OUTPUT

General design:

• It would be more coherent to have an object "covering of stratum".
  In a concrete way it would be a permutation of interval exchange
  (that may be considered as a marking) together with a covering data.
  We do not care that much about a concrete interval exchange in your code.
• avoid using import from sage.all, to know from which module you need
  to import an object you can use import_statements(MY_OBJECT). For more
  info look at import_statements?.

About code:

• there are already several things in Sage for interval exchange
  transformations and strata. It would be simpler to use them.
• you must put a comment about precision = 128 and
  diff_sum_seuil = 2**(-precision+20). Even better, remove them.
• in order to fit nicely with other components of Sage, you should
  separate the code (like your class IntExchange) and the examples
  (like the functions EKZ_example).

[videlec]

Reviewer: Vincent Delecroix

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

20f5170	Merge branch 'develop' into lyapunov_exponents
bb60056	Merge branch '16102/lyapunov_exponents' into lyapunov_exponents
5a93d69	Merge branch 'develop' of trac.sagemath.org:sage into lyapunov_exponents
e2315e7	merging with develop
ab009e5	removed useless files
e1129fc	remove files
ab3a1c4	remove file
9a37545	remove files
a47c68c	Include test for speed
6172656	Merge branch 'integrated_lyapunov' into 16102/lyapunov_exponents

[sagetrac-git]

Changed commit from 048431e to 6172656

[sagetrac-Fougeroc]

Changed author from Fougeroc to Charles Fougeron

[sagetrac-Fougeroc]

Attachment: compare1_100_.png

Compare computation of lyapunov exponents wrt number of iteration

[sagetrac-Fougeroc]

Compare computation of lyapunov exponents wrt number of iteration (zoom)

[sagetrac-Fougeroc]

Attachment: compare40_100_.png

Attachment: compare_speed.png

Compare log(T)/nb_iterations for diffrent genus and nb_iterations

[sagetrac-Fougeroc]

comment:6

I updated the code which integrate the computation of lyapunov exponents for quadratic strata.
You can now use the function :

stratum_component = QuadraticStratum([1]*8 + [-1]*4).components()[0]
stratum_component.orientable_cover().lyapunov_exponents_H_plus()

I noticed a strange behaviour while doing some computeur simulations for asymptotic of lyapunov exponents. When genus is getting bigger (and consequently the number of intervals), the speed is decreasing. While increasing the number of iteration, log(t)/n is also increasing. A surprising behaviour (possibly due to float approximation) is that the there is a limit value for this function for a large value of genus.

This behaviour seems to be responsible for a wrong approximation of lyapunov exponents asymptotically:

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

02de200	Add computation of lyapunov exponents for orientable cover over quadratic stratum.
227ce37	Merge trac.sagemath.org:sage into 16102/lyapunov_exponents

[sagetrac-git]

Changed commit from 6172656 to 227ce37

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

23ee7e7	fix bug
e91fca7	fix bug

[sagetrac-git]

Changed commit from 227ce37 to e91fca7

[fchapoton]

comment:12

does not apply, needs rebase

[fchapoton]

Dependencies: #16159

[sagetrac-git]

Changed commit from e91fca7 to f21ab92

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

f21ab92

Merge branch 'develop' into lyapunov

[videlec]

comment:15

I guess that we should close this ticket with mileston "wontfix" as I did with the other tickets #14683, #16159, #16170, #16169. This means that the ticket would be closed without integrating the branch into Sage.

[fchapoton]

comment:16

Vincent, do you confirm this should be closed as invalid ?

[videlec]

Changed dependencies from #16159 to none

[videlec]

comment:17

Yes! Charles' code is now part of the package surface_dynamics (see #20695).

[videlec]

Changed branch from u/Fougeroc/Lyapunov_exponents_for_translation_surfaces to none

[videlec]

Changed author from Charles Fougeron to none

[videlec]

Changed commit from f21ab92 to none

[embray]

comment:19

Determined to be invalid/duplicate/wontfix (closing as "wontfix" as a catch-all resolution).