PseudoAnosov Package

The PseudoAnosov package contains Mathematica and C++ functions for extracting properties of characteristic polynomials of pseudo-Anosov maps of surfaces. The goal is to determine whether a given polynomial is allowable on a given stratum. A stratum is a collection of singularities for a closed surface of genus g. A polynomial is allowable if it can (at least in principle) arise as the characteristic polynomial associated with a pseudo-Anosov map on that surface, where the map preserves the singularities up to permutations.

Most of the functions in the package assume that the pseudo-Anosov map stabilizes an orientable foliation, so the singularities must all have even degree. If the pseudo-Anosov map exists, then its dilatation is given by the largest root of the polynomial, also called its Perron root.

This package is an extension of the code used in papers by Erwan Lanneau and Jean-Luc Thiffeault to prove lower bounds on the smallest dilatation on closed surfaces and braids. A minimal version was included as part of the braids paper as PseudoAnosovLite.

The PseudoAnosov package is maintained by Jean-Luc Thiffeault.

See the complete list of functions in the PseudoAnosov package. There are also several sample notebooks in the repository.

A sample Mathematica session

First we load the package:

In[1]:= <<PseudoAnosov.m

Let's start with the torus (genus g=1), where we are dealing with Anosov maps (no singularities). We can list all the reciprocal monic polynomials of degree 2g=2 (with integer coefficients) with dilatation less than, say, 3:

In[2]:= P = ReciprocalPolynomialBoundedList[x,2,3]

                    2
Out[2]= {1 - 3 x + x }

There is only one such polynomial, and it gives the lowest dilatation on that surface.

Genus 2 surfaces are more interesting. First we can list all the possible strata (singularity data) for the case were the foliation is orientable:

In[3]:= s = OrientableStrataList[2]

Out[3]= {{4}, {2, 2}}

There are two strata, one with a degree-4 singularity (6 prongs), the other with two degree-2 singularities (4 prongs). The sum of the degrees in each case is 4=4g-4=-2(Euler characteristic), where g=2 is the genus. Next we find all monic reciprocal polynomials of degree 2g=4 with dilatations less than 1.75:

In[4]:= P = ReciprocalPolynomialBoundedList[x,4,1.75]

                  2    3    4
Out[4]= {1 - x - x  - x  + x }

There is only one such polynomial (which is why we chose the value 1.75). Its root with the largest magnitude (Perron root) is

In[5]:= PerronRoot /@ P

Out[5]= {1.72208}

Let's see if this polynomial is a good candidate for a pseudo-Anosov map. The command StratumOrbits tries to find a set of regular periodic orbits that is compatible with a polynomial on a given stratum. If we try the polynomial P[[1]] on the second stratum s[[2]],

In[6]:= StratumOrbits[s[[2]],P[[1]]]

Out[6]= {}

we get nothing, suggesting that this polynomial is not associated with any pseudo-Anosov maps on the stratum s[[2]]={2,2}. However, on the stratum s[[1]]={4} we get

In[7]:= so = StratumOrbits[s[[1]],P[[1]]]

                                 2    3    4
Out[7]= {{Polynomial -> 1 - x - x  - x  + x , Stratum -> {{4, 1}},
     SingularitiesPermutation -> {{{1}}},
     SeparatricesPermutation -> {{{2, 3, 1}}},
     SingularitiesLefschetzBlock -> {1, 1, -5},
     RegularOrbits ->
      {0, 1, 0, 1, 3, 3, 6, 9, 14, 21, 36, 54, 90, 141, 230, 369, 606, 977, 1608, 2619, 4312, 7074, 11682, 19248, 31872, 52731, 87514, 145260, 241644, 402137, 670380, 1118187, 1867560, 3121221, 5221938, 8742312, 14648958, 24562068, 41214696, 69199515, 116263056, 195445504, 328749954, 553264722, 931601482, 1569414123, 2645169030, 4460292930, 7524259626, 12698241600}}}

This is a candidate pseudo-Anosov, since there exists a collection of regular periodic orbits that together with the singularity respect the Lefschetz fixed-point theorem. We can present the information better with

In[8]:= StratumOrbitsTable[so[[1]]]

Out[8]=

The table shows the polynomial and its Perron root. The stratum is denoted 4^1 or {{4,1}}, which means degree 4 with multiplicity 1, i.e., the same as {4}. There is only one singularity, so the permutation of singularities by the map can only be {1}. The singularity has six prongs or separatrices, 3 of which are labeled 'ingoing' and the other 3 'outgoing'. These ingoing/outgoing separatrices are cyclically permuted as {2,3,1} by this (hypothetical) pseudo-Anosov map. The table then gives Lefschetz number sequences for iterates n of the map. The last row gives the number of regular orbits (#ro). We see that there are no regular fixed points (n=1), one period-2 orbit (n=2), and so on.

It turns out that a pseudo-Anosov map having this polynomial does indeed exist, and we just deduced that it must have the minimum dilatation for a genus 2 surface, as was first shown by Zhirov (1995), since there are no candidate polynomials with a lower dilatation.

License

PseudoAnosov is released under the GNU General Public License v3. See COPYING and LICENSE.

Support

The development of PseudoAnosov was supported by the US National Science Foundation, under grants DMS-0806821 and CMMI-1233935.

jeanluct/PseudoAnosov

PseudoAnosov Package

A sample Mathematica session

License

Support