Periodic rational difference equations

Definitions and mathematical statement

For an integer $k\ge1$, consider the recurrence \[ x_{j+k}=\frac{A_0+\sum_{i=1}^k A_i x_{j+i-1}} {B_0+\sum_{i=1}^k B_i x_{j+i-1}}, \] where all coefficients are nonnegative, each coefficient sum is positive, and $A_1^2+B_1^2>0$. Positive initial values give positive subsequent terms. Let $F:(0,\infty)^k\to(0,\infty)^k$ shift the state one place and append the displayed quotient. The recurrence is globally periodic if $F^p=\operatorname{id}$ for some positive integer $p$; its global period is the least such $p$. Individual initial states may have smaller periods.

The problem asks whether every globally periodic recurrence of this form comes from one of the following five models by positive rescaling of all terms and by index dilation: \[ x_{j+1}=x_j,\quad x_{j+1}=1/x_j,\quad x_{j+2}=x_{j+1}/x_j,\quad x_{j+2}=(1+x_{j+1})/x_j, \] \[ x_{j+3}=(1+x_{j+1}+x_{j+2})/x_j. \] Index dilation replaces each lag by a common positive integer multiple, interlacing independent copies of the original recurrence. Rescaling means substituting $y_j=a x_j$ with $a>0$ and simplifying the quotient.

Thus the requested classification concerns identities valid for every positive initial state, not just the occurrence of some periodic orbits. Coefficient nonnegativity and genuine dependence on the earliest term are essential restrictions: dropping them admits additional globally periodic rational maps. Exceptional states of smaller period do not invalidate global periodicity in this formulation.

Short English statement

Are the five classical models, their rescalings, and their interlacings all globally periodic recurrences in this family?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700011 (AMR-046-0011), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Gasull - Some open problems in low dimensional dynamical systems (2020), Problem environment 11: Periodic rational difference equations. Original source indexed by AMR-046-0011. https://arxiv.org/abs/2012.02524.