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Source. Theorem 4.1, Section 4, p. 9 of the author's version named on the source card; proof pp. 9--12. The heuristic discussion that follows Theorem 4.2 is on pp. 12--13. Read on the PDF page images.

Statement

Setting as on the Theorem 3.1 page; $q$ and $r$ are odd.

Theorem 4.1 (p. 9). Let $s^{2^N}=1$. Then

$$B_{n+1,q,r}(s)=\frac{q+1}{4}B_{n,q,r}(s)\qquad\text{for all }n\ge N .$$

If $s\ne1$, then also

$$A_{n+1,q,r}(s)=\frac{q+1}{4}A_{n,q,r}(s)\qquad\text{for all }n\ge N ,$$

and for all $n$

$$A_{n,q,r}(1)=\begin{cases} \dfrac{r}{3-q}\left(\dfrac{q+1}{4}\right)^n-\dfrac{r}{3-q}, & q\ne3,\\[2ex] -\dfrac{rn}{4}, & q=3. \end{cases}$$

So for $q=3$ the coefficients $B_{n,3,r}(s)$ for every $s$, and $A_{n,3,r}(s)$ for $s\ne1$, do not change once $n\ge N$, while $A_{n,3,r}(1)$ changes with $n$. For $q\ge5$ the factor $(q+1)/4$ exceeds $1$, so these coefficients grow geometrically unless they vanish.

Read depth. Claims checked: the three formulas and their ranges were read clause by clause on the page image. The proof was read for structure only, and nothing here is independently reviewed.

Proof pointer

Split the sum over $j\le2^{n+1}$ defining $B_{n+1,q,r}(s)$ into $j\le2^n$ and $j>2^n$, use Theorem 2.1 (with negative arguments) to pair each $j$ with $j-2^n$, and observe that $T_{q,r}^{(n)}(j)$ and $T_{q,r}^{(n)}(j-2^n)$ have opposite parity, so each pair contributes $(q+1)q^{O_{q,r}^{(n)}(j)}s^j$ (pp. 9--10). The same pairing gives $4^{n+1}A_{n+1,q,r}(s)=(q+1)4^nA_{n,q,r}(s)-2^nr\sum_{j=1}^{2^n}s^j$ (pp. 10--11; the first line of that display prints $A_n$ for $A_{n+1}$). The last sum vanishes for $s\ne1$; for $s=1$ it gives $A_{n+1,q,r}(1)=\frac{q+1}{4}A_{n,q,r}(1)-\frac r4$, solved with $A_{0,q,r}(1)=0$ (p. 12).

Dependencies

Theorem 3.1 for the coefficients, and Theorem 2.1.

Bears on

Problem 1135: with $(q,r)=(3,1)$ the problem's map has polar coefficients that stay fixed in $n$, except at $x=1$, whose residue term $A_{n,3,1}(1)=-n/4$ changes with $n$ while $B_{n,3,1}(1)$ stays fixed. The paper uses this, with display (4) and Theorem 4.2, only to argue heuristically (pp. 12--13) that every orbit of the $3x+r$ map is bounded and eventually cyclic; the step from fixed coefficients to a uniform bound on compact subsets of the unit disc is not proved. Nothing is proved about whether orbits reach $1$.