Dear Harshvardhan,
  There are a number of directions that are possible to take. In terms of 
what I wrote for the initial project description there are a number of 
small issues that need to be resolved

- .rational_periodic_points() for finite fields does not currently work

- add early exit to is_conjugate()/conjugating_set() by checking sigma 
invariants.

- use dynatomic polynomial for points of small period in 
ration_periodic_points() instead of p-adic lifting

- allow to domain to be a subscheme for rational_periodic() and 
_prepepriodic() points as well as periodic_points()


a couple things of bigger scope:

- morphisms of products of projective space are lacking much of the basic 
functionality such as even an "==" comparison. Much of the 
periodic/cyclegraph etc functionality can probably be moved to there as well

- if you're familiar with algebraic number theory, creating the fields of 
definition for the nth periodic points would be helpful.

- speed up some of the complex dynamics calculations through divide and 
conquer and possibly some other projects here. See 23720. 23740 are almost 
done by that author, but not quite and gives you a good idea of the current 
status of complex dynamics functionality.

----
as a place to start, here are a couple bugs, two simple and one a little 
more involved that I am aware of but haven't gotten around to fixing yet. 
Fixing one of these would be a good demonstration of you familiarity with 
sage and ability to navigate the trac ticket system. 

1)
sage: P.<x,y> = ProjectiveSpace(QQ, 1)
sage: F = DynamicalSystem([y^2, 2*x^2 - 4*x*y + 2*y^2])
sage: F.is_polynomial()


the code uses : G = R(G/p) instead of G = R(G/p**e)


2)
R.<x> = K[]
f = x^2 + 1
f.change_ring(QQ.embeddings(CC)[0])

the code uses: if isinstance(R, Morphism) instead of: if isinstance(R, Map):

3) more involved

K.<w> = QuadraticField(5)
P.<x,y> = ProjectiveSpace(K, 1)
f = DynamicalSystem([w*x^2 + (1/5*w)*y^2, w*y^2])
f.normalize_coordinates();f

This fails to remove the common factor of 'w'.

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