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| = Falt surfaces examples = === Installation === You need to install [[http://wiki.sagemath.org/combinat|sage-combinat]] which is done with the following command {{{ $ sage -combinat install }}} The latter installs a modified version of Sage. For more on what this command does or how to reverse it, you can read this [[http://wiki.sagemath.org/combinat/MercurialStepByStep|step by step tutorial]]. === Link to documentations and tutorials === To get introduced to Sage, look at the documentation on [[http://sagemath.org]] |
= Flat surfaces examples = |
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| Build a permutation and its Rauzy diagram | |
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| sage: r = p.rauzy_diagram() sage: print r Rauzy diagram with 7 permutations sage: r.path |
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| === Square tiled surfaces === | Build a path in the Rauzy diagram {{{ sage: path=r.path(p,'t','t','b','t','b','b','t','b') sage: path.is_full() # all intervals are seen during Rauzy induction True sage: path.is_loop() True }}} Build an interval exchange map associated to this path {{{ sage: l,v,m=path.matrix().eigenvectors_right()[3] # l is the eigenvalue, v the vector and m the multiplicity sage m == 1 sage: t= iet.IntervalExchangeTransformation(p,v[0]) sage: print t Interval exchange transformation of [0, 4.390256884515514?[ with permutation a b c d d c b a }}} And we now check that the interval exchange map is self-similar {{{ sage: tt = sage: t.rauzy_move(iterations=8) sage: print tt Interval exchange transformation of [0, 1[ with permutation a b c d d c b a sage: tt.normalize(l) == t True }}} === Square-tiled surfaces === |
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| And now, we build its Teichmueller curve and compute some of its invariant (rk: it is not yet clear which property should be attached to the teichmueller curve and which one should be attached to the Veech group) | And now, we build its Teichmueller curve and compute some of its invariants (rk: it is not clear yet which properties should be attached to the Teichmueller curve and which should be attached to the Veech group) |
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| sage: G.nu2() #elliptic points of order 2 | sage: G.nu2() # elliptic points of order 2 |
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| sage: G.nu3() #elliptic points of order3 | sage: G.nu3() # elliptic points of order3 |
Flat surfaces examples
Using interval exchange transformations
Permutations of interval exchange transformations are created
Build a permutation and its Rauzy diagram
sage: p = iet.Permutation('a b c d','d c b a')
sage: p.connected_component()
H_hyp(2)
sage: r = p.rauzy_diagram()
sage: print r
Rauzy diagram with 7 permutations
sage: r.pathBuild a path in the Rauzy diagram
sage: path=r.path(p,'t','t','b','t','b','b','t','b') sage: path.is_full() # all intervals are seen during Rauzy induction True sage: path.is_loop() True
Build an interval exchange map associated to this path
sage: l,v,m=path.matrix().eigenvectors_right()[3] # l is the eigenvalue, v the vector and m the multiplicity sage m == 1 sage: t= iet.IntervalExchangeTransformation(p,v[0]) sage: print t Interval exchange transformation of [0, 4.390256884515514?[ with permutation a b c d d c b a
And we now check that the interval exchange map is self-similar
sage: tt = sage: t.rauzy_move(iterations=8) sage: print tt Interval exchange transformation of [0, 1[ with permutation a b c d d c b a sage: tt.normalize(l) == t True
Square-tiled surfaces
Let us build the genus 2 origami with three squares
sage: o = Origami('(1,2)', '(1,3)')
sage: print o
(1, 2)
(1, 3)And now, we build its Teichmueller curve and compute some of its invariants (rk: it is not clear yet which properties should be attached to the Teichmueller curve and which should be attached to the Veech group)
sage: t = o.teichmueller_curve() sage: G = t.veech_group() sage: G.index() 3 sage: G.nu2() # elliptic points of order 2 1 sage: G.nu3() # elliptic points of order3 1 sage: G.ncusps() # number of cusps 2 sage: t.sum_of_lyapunov_exponents() 4/3