Entropy criterion for birational maps from cluster recurrences

Let $b_{1,j+1}$ be the coefficients defining a birational map

$$corresponding to a recurrence$$

, and let $[x]_+=\max(x,0)$. Define

$$P_{\pm}(\lambda)=\lambda^N+1-\sum_{j=1}^{N-1}[\pm b_{1,j+1}]_+\lambda^j.$$

Let $\lambda_{\max}$ be the root of largest magnitude among the roots of these two polynomials. Entropy criterion conjecture. The algebraic entropy is $\mathcal{E}=\log|\lambda_{\max}|$, and it is positive if and only if

$$\max\left(\sum_{j=1}^{N-1}[b_{1,j+1}]_+,\sum_{j=1}^{N-1}[-b_{1,j+1}]_+\right)\geq3.$$

This would give an explicit test for exponential degree growth and distinguish positive-entropy recurrences from zero-entropy cases; its general validity is presented as the main conjecture and remains open.