Some Interesting Integer Polynomial Maps

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by  Alexander Borisov

SOME INTERESTING INTEGER POLYNOMIAL MAPS
ALEXANDER BORISOV
3
1
0 Abstract. Weintroducethreesimplepolynomialmapswithinte-
2
ger coefficients that have interesting dynamical properties modulo
n
a primes.
J

3

] Thefirst mapisthesimplest, andwasdiscovered whileworiking with
T
N Mark Sapir on [BS].
.
h
t
a
Example 1. (Additive Trap) Define F (x,y) = (u,v), where
m at
[

1 (u,v) = (x2y,x2y +xy2)
v
7
1
5 Note that modulo any prime p the p-th iteration of F sends every-
at
0
1. thing to (0,0). Indeed, if x 6= 0, then v = y + 1. So after no more
u x
0
3 than p − 1 iterations, we get y = 0, which forces x and y to become
1
: zero at the next step and forever afterwards. Note that this attractor
v
i
X property of (0,0) does not hold for extensions of Z/(pZ) or over Z or
r
a C. Indeed, the map F is dominant, so by [BS] its periodic orbits over
at
an algebraic closure of any finite field are Zariski dense.
Example 2. (Multiplicative Trap) Define F (x,y) = (u,v), where
mt
2 2
(u,v) = (x y(x−y),2xy (x−y))
The research of the author was supported in part by the NSA grants H98230-
08-1-0129,H98230-06-1-0034and H98230-11-1-0148.
1
2 ALEXANDERBORISOV
Note that modulo any prime p if x 6= 0 and y 6= 1, then u = 2y. If 2
x v x
is a generator of (Z/pZ)∗, then iterations of this map eventually send
everything to (0,0).
Example 3. (Power Trap) Define F (x,y) = (u,v), where
pt
3 3
(u,v) = (x y(x−y),xy (x−y))
Note that modulo any prime p if x 6= 0 and y 6= 1, then u = (y)2. If
x v x
p = 2k +1, then k+1 iterations of this map send everything to (0,0).
For general p a high enough iteration of F sends to (0,0) all pairs
pt
(x,y) with one of the coordinates zero or y having 2-primary order in
x
(Z/pZ)∗.
Obviously, with more variables one can create more complicated
maps, but it is unclear how much can be actually “programed” us-
ing integer polynomial maps. The following question is very natural in
this respect.
Question. Does there exist an integer polynomial map with two
different fixed points such modulo every prime its sufficiently high it-
eration will send any initial point to one of the fixed points, depending
on whether or not the initial first coordinate is zero?
References
[BS] Alexander Borisov, Mark Sapir, Polynomial maps over finite fields and resid-
ual finiteness of mapping tori of group endomorphisms, Invent. Math., 160, 2
(2005), 341 - 356.
E-mail address: [email protected]
Department of Mathematics, University of Pittsburgh, 301 Thack-
eray Hall, Pittsburgh, PA 15260, USA.