Dynamical uniform boundedness conjecture

Start with a rational number and repeatedly square it, then subtract one. Starting at 0 gives 0, −1, 0, −1 forever; starting at 1 joins that cycle after one step. These numbers have finite forward orbits. Can a fixed rule complexity and a fixed number-system complexity limit how many such starting points there can be, even while the rule’s coefficients change?

The conjecture allows any rational function of degree at least two, and includes the point at infinity. Its number system is a number field, a finite extension of the rational numbers. The proposed bound depends only on the function’s degree and the field’s degree, not on the particular function or field.[1]

Repeated evaluation can turn an infinite set of possible starting numbers into a small set of finite orbit patterns. For $f(x)=x^2-1$, the points $0$ and $-1$ alternate, and $1$ reaches $0$. By contrast, $2$ gives $2,3,8,63,\ldots$, with continued growth. Does fixing the degree of the map and of the number field impose a universal ceiling on the number of field-rational starting points with finite orbits?

A preperiodic point is one whose iterates eventually enter a cycle. Write $\operatorname{PrePer}(f,K)$ for these points in $\mathbb P^1(K)$; the projective line includes infinity, which is fixed in this polynomial example. The precise target is: for each $d\ge2$ and $D\ge1$, there exists $B(d,D)$ such that

$$\lvert\operatorname{PrePer}(f,K)\rvert\le B(d,D)$$

for every degree-$d$ rational map over every number field with $[K:\mathbb Q]=D$. Coefficients and fields may vary within those degrees. The bound counts points, not merely cycles or the number of iterations performed.[1]

Some special families have uniform bounds, and a bound for polynomials follows under an additional unproved arithmetic conjecture. The remaining problem is one bound covering all rational functions of the specified degree over all number fields of the specified degree. Bounding one function at a time does not meet that requirement.[1]

Conjecture 1.16 in the cited preprint states a broader projective-space version, of which this page keeps only the projective-line case. Known Lattès cases use arithmetic torsion bounds; conditional polynomial cases assume the abcd conjecture. The paper’s geometric function-field and iterated-preimage results do not establish the full number-field count stated here.[1]

Some iteration rules come from multiplication on an elliptic curve, a smooth algebraic curve with an addition law. Their finite orbits link to finite-order points, which return to the identity after repeated addition. Known bounds for these points already give orbit bounds in that special family. This connection makes the larger problem a meeting point between arithmetic finiteness and repeated rules, rather than simply a request to count more cycles.[1]

A finite orbit has two structures: a cycle and the arithmetic points that feed into it. Uniform boundedness would constrain their combined size even as coefficients change, making the degree pair a genuine organizing principle for finite arithmetic dynamics. The Lattès case is informative because its iteration comes from elliptic-curve multiplication, where finite-order points are controlled by torsion theorems. This is a specific bridge between arithmetic and dynamics, not an assertion that the projective-line conjecture alone proves all higher-dimensional torsion results.[1]