Dynamics on surfaces

## Project description

The surface_dynamics package for SageMath adds functionality related to interval exchange transformations, translation surfaces, mapping classes and more. It is distributed as an external Python package. It installs on top of an existing Sage installation.

This package is based on SageMath and relies heavily on:

• gmp or mpir for arbitrary precision arithmetic
• PARI/GP for number field computations
• GAP for finite groups representation and permutation groups
• PPL (Parma Polyhedra Library) and LattE (Lattice point Enumeration) for Polytope computations

## Prerequisites

Installing surface_dynamics requires a working Sage installation (with Cython and gcc).

## Installation

The module is distributed on PyPI. You just need to run the following command:

```\$ sage -pip install surface_dynamics [--user]
```

The –user option is optional and allows to install the module in your user space (and does not require administrator rights). Alternatively, you can install the development version using

\$ sage -pip install git+https://github.com/videlec/flatsurf-package [–user]

## Check

After installing surface_dynamics, check that it works by launching Sage and typing the following commands. You should get the same output as below.:

```sage: from surface_dynamics.all import *
sage: o = Origami('(1,2)','(1,3)')
sage: print o
(1,2)(3)
(1,3)(2)
sage: o.sum_of_lyapunov_exponents()
4/3
sage: o.lyapunov_exponents_approx()
[0.33441823619678734]
sage: o.veech_group()
Arithmetic subgroup with permutations of right cosets
S2=(2,3)
S3=(1,2,3)
L=(1,2)
R=(1,3)
H_5(2^4)^odd

sage: AbelianStrata(genus=3).list()
[H_3(4), H_3(3, 1), H_3(2^2), H_3(2, 1^2), H_3(1^4)]

sage: O = OrigamiDatabase()
sage: q = O.query(("stratum","=",AbelianStratum(2)), ("nb_squares","=",5))
sage: q.number_of()
2
sage: for o in q: print o, "\n"
(1)(2)(3)(4,5)
(1,2,3,4)(5)

(1)(2)(3,4,5)
(1,2,3)(4)(5)

sage: Q12_reg.lyapunov_exponents_H_plus()
[0.6671, 0.4506, 0.2372, 0.08841]
sage: Q12_reg.lyapunov_exponents_H_minus()
[1.001, 0.6669, 0.45018, 0.3139, 0.23218, 0.12143, 0.08594]
```

## Source code

The complete source code is available at

https://github.com/videlec/flatsurf-package

Once you developed the source code you might want to install the package with

\$ sage -pip install . [–user] [–upgrade]

wher the options –user and –upgrade are optional. It can also be compiled inplace using

\$ sage -python setup.py build_ext –inplace

## Contact

Your comments and help are welcome: vincent.delecroix@labri.fr For problems with Mac OS X: samuel.lelievre@gmail.com

## Authors

• Vincent Delecroix: maintainer
• Samuel Lelièvre: contribution for origamis and permutation representative of quadratic strata
• Charles Fougeron: Lyapunov exponents for strata coverings

## Versions

• flatsurf 0.3 is in version beta
• flatsurf 0.2 was released on 2015-11-15 (as a Sage spkg).
• flatsurf 0.1 was released on 2015-07-30 (as a Sage spkg).