Eventual stability at 
of a rational function 
of degree at least two, where
is a field, means that the number of irreducible
factors over 
of 
,
counted with multiplicity, remains bounded as 
. Here 
is the 
th iterate written with coprime polynomials.
For 
,
the polynomial considered is 
(Tucker and Zhu 2026).
Jones and Levy (2017) conjectured that every such pair over a number field is eventually stable when 
is not periodic. This hypothesis does not require its
forward orbit to be infinite. Tucker and Zhu (2026) give the counterexample 
, 
, and 
:
 |
where 
is the cyclotomic polynomial. There are
exactly 
distinct irreducible factors. The orbit
is not periodic at its starting point, although it is eventually periodic. Thus the
example refutes the stated nonperiodic-point conjecture, not a version restricted
to points with infinite forward orbit.
