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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 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


Installing surface_dynamics requires a working Sage installation (with Cython and gcc). Installing the optional SageMath packages gap_packages, and latte_int is recommended and will improve or extend the functionality in surface_dynamics. The optional package database_gap is also recommended if using SageMath < 8.6 (in SageMath 8.6 it was merged partly into the gap and partly into the gap_packages packages).


The module is distributed on PyPI and is easily installed through the Python package manager pip. If you downloaded a binary from the SageMath website (including the Cygwin version running on Windows) or compiled from source, 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).

If you use Debian or Ubuntu and you installed Sage through the operating system’s package manager (that is, the package sagemath), run these two commands:

$ source /usr/share/sagemath/bin/sage-env
$ pip install surface_dynamics --user

If you use Arch Linux, you need to install from source (see next section).

Install and use source version

This section provides detailed instructions on how to download, modify and install the development version of surface_dynamics. In all commands,

  • PIP has to be replaced by either pip, pip2, or sage -pip

  • PYTHON has to be replaced by either python, python2 or sage -python

If you are an Arch Linux user with the sagemath package installed, use PIP=pip2 and PYTHON=python2. If you downloaded SageMath as a tarball or installed it from source use PIP='sage -pip' and PYTHON='sage -python'.

You can install the latest development version in one line with:

$ PIP install git+ [--user]

As before, the --user option is optional and when specified will install the module in your user space.

You can also perform a two stage installation that will allow you to modify the source code. The first step is to clone the repository:

$ git clone

The above command creates a repository surface_dynamics with the source code, documentation and miscellaneous files. You can then change to the directory thus created and install the surface dynamics module with:

$ cd surface_dynamics
$ PIP install . [--user]

Do not forget the . that refers to the current directory.

When you don’t want to install the package or you are testing some modifications to the source code, a more convenient way of using surface dynamics is to do everything locally. To do so, you need to compile the module in place via:

$ PYTHON build_ext --inplace

Once done, you can import the surface_dynamics module. To check that you are actually using the right module (i.e. the local one) you can do in a SageMath session:

sage: import surface_dynamics
sage: surface_dynamics.__path__        # random

The result of the command must correspond to the path of the repository created by the command git clone given above. The compilation step PYTHON build_ext has to be redone each time you modify a C or Cython source file (i.e. with .c, .h, .pxd or .pyx extension). In other words, it is not needed if you only modify or create Python files (i.e. .py files).

If you wish to install your custom version of surface_dynamics just use PIP as indicated before.



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: o
sage: o.sum_of_lyapunov_exponents()
sage: o.lyapunov_exponents_approx()    # abs tol 0.05
sage: o.veech_group()
Arithmetic subgroup with permutations of right cosets
sage: q = QuadraticStratum(1, 1, 1, 1)
sage: q.orientation_cover()
sage: q.components()
sage: c = q.components()[0]
sage: c
sage: c.orientation_cover_component()

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()
sage: for o in q:
....:     print("%s\n- - - - - - - -" % o)
- - - - - - - -
- - - - - - - -
sage: Q12_reg = QuadraticStratum(12).regular_component()
sage: Q12_reg.lyapunov_exponents_H_plus(nb_iterations=2**20)   # abs tol 0.05
[0.6634, 0.4496, 0.2305, 0.0871]
sage: Q12_reg.lyapunov_exponents_H_minus(nb_iterations=2**20)  # abs tol 0.05
[1.0000, 0.3087, 0.1192]

Installing development version - source code

The development webpage is

Assuming you have the program git on your computer, you can install the development version with the command:

$ sage -pip install git+ [--user]


Your comments and help are welcome:

For problems with macOS:


  • Vincent Delecroix: maintainer

  • Samuel Lelièvre: origami and permutation representatives for quadratic strata

  • Charles Fougeron: Lyapunov exponents for strata coverings

  • Luke Jeffreys: single cylinder representatives for strata of Abelian differentials


To cite the software, use the following Bibtex entry:

@manual{ Sdyn,
  Author = { Delecroix, V. et al. },
  Month  = { March },
  Year   = { 2019 },
  Title  = { surface_dynamics - SageMath package, Version 0.4.1 },
  Doi    = { 10.5281/zenodo.3237923 },
  Url    = { }


  • 0.4.6 was released on 2021-03-13 (as a Python package on PyPI)

  • 0.4.5 was released on 2020-10-22 (as a Python package on PyPI)

  • 0.4.4 was released on 2020-01-31 (as a Python package on PyPI)

  • 0.4.3 was released on 2019-07-28 (as a Python package on PyPI)

  • 0.4.2 was released on 2019-06-21 (as a Python package on PyPI)

  • 0.4.1 was released on 2019-03-26 (as a Python package on PyPI)

  • 0.4.0 was released on 2018-05-14 (as a Python package on PyPI)

  • 0.3 was released on 2017-08-11 (as a Python package on PyPI)

  • 0.2 was released on 2015-11-15 (as a Sage spkg)

  • 0.1 was released on 2015-07-30 (as a Sage spkg)

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