Arithmetic and Complex Dynamics

The goal of sage-dynamics is to improve the open source mathematical software Sage for computer exploration in dynamical systems and foster code sharing between researchers in this area. This portion focuses on the Arithmetic (Number Theoretic) and Complex aspects of dynamical systems.

News

Past News

• Sage Days Currently holding Sage Days 55 (November 7-10, 2013) at Florida Institute of Technology.

• January 30, 2012 - May 4, 2012 ICERM semeser program on Complex and Arithmetic Dynamics

How to participate and contribute

• sage-dynamics: Google group

  • anyone may subscribe by sending an e-mail to: sage-dynamics+subscribe at googlegroups dot com

Documentation and Tutorials

• Sage developer's guide

• sage-combinat has many excellent tutorials combinat docs

Road Map

The arithmetic and complex dynamics functionality in Sage is currently in its infancy. A significant amount of functionality was developped at the ICERM semester in Spring 2012 and now we have started the process of moving this into Sage through a series of patches (trac tickets). Most of that functionality is current in experimental for that been greatly expanded upon at Sage Days 55. Much remains to be done. Below you will find a road map of what has been implemented, what is in the process of being implemented, and ideas for future functionality.

In Progress

• (#15382) needs_work: MacCaulay Resultant - Soli Vishkautsan, Hao Chen

• (#15389) needs-work: Krumm-Doyle Small Points Algorithm - David Krumm, John Doyle

• (#15392) needs-review: Bruin-Molnar Algorithm for minimal models - Brian Stout, Ben Hutz

• (#15393) in-progress: FMV Algorithm for automorphism groups - Bianca Thompson, Ben Hutz, Joao Faria

• Eigenvalues (see #14990 and #15390) for an implementation of the algebraic closure of finite field) - Vincent Delecroix , Ben Hutz

• PostCriticallyFiniteMorphisms - Holly Krieger, Adam Towsley, Vincent Delecroix, Ben Hutz, Patrick Ingram

• (#15394) Lattes map from an Elliptic Curve - Patrick Ingram

• (#15378) Composition of Morphisms - Vincent Delecroix, Donald Richardson, Soli Vishkautsan

• Enabled canonical height for maps of \PP^N over number fields. Here is an early draft worksheet: sws, for N>1 it requires the macaulay_resultant patch from above. - Adam Towsley, Paul Fili

• (#15396) needs-review: Implement .an_element() for ProjectiveSpace - Ben Hutz

Complete

• #13130 closed sage-5.8.beta3: Basic architecture changes : support for projective spaces over rings - Ben Hutz

• #14217 closed sage-5.10.beta3: Basic iteration functionality for projective and affine spaces - new directory structure in schemes - Ben Hutz

• #14218 closed sage-5.13.beta2: Height and canonical heights for points and morphisms - Ben Hutz

• #14219 closed sage-5.13.beta4:- Rational preperiodic points - Ben Hutz

• #15377 closed sage-5.13.beta3: improve documentation of normalize_coordinates - Ben Hutz

• #15376 closed sage-5.13.beta4: canonical heights for points with integer fix - Paul Fili

• #15373 closed sage-5.13.beta3: Global height for integer fix - Paul Fili

Wishlist

• Check if for a given algebraic parameter c the map z -> z^2 + c is hyperbolic... and more generally for rational maps of P1 determine the existence (and list) of attracting cycles

• is_conjugate() for morphisms and iterator over morphisms of fixed degree up to conjugacy (medium)
• cyclegraph() and orbit_structure() to work with Zmod and other finite spaces not just finite fields (medium)
• primes_of_bad_reduction() and is_morphism() made to work for endomorphisms of subschemes (easy)
• products of projective space (Ben Hutz)
• dynamics on Wehler K3 surfaces (Joao de Faria)

• error_bound computation for canonical height in dimension > 1

• update affine space to include the appropriate functionality from projective space
• enumeration of points of small height over number fields for affine and projective spaces
• fix all the white space issues in the projective and affine folders (easy)
• specific functionality for regular polynomial endomorphisms of P^N (Patrick might start implementing this someday)
• Chebyshev creator (if it doesn't already exist)
• moduli space invariants - symmetric functions in multipliers of periodic points, others...
• use real interval field for floating point computations (in heights and possibly rational preperiodic point functions)

• issue with dynatomic polynomials (see (5) from (#14219)

• Ponies (Patrick)
• PLEASE ADD MORE...