Why should $ s \in ( 0 , 1 ) $ for splinter to be approximately geometric?

Question

Recently I was studying functional equation, in which I came across the following paragraphs in Christopher G. Small's book Functional Equations and How to Solve Them.

3.5.1 The Koenigs algorithm for Schröder's equation

First note that if $ f ( x ) $ is any solution to Schröder's equation $$ f [ \alpha ( x ) ] = s f ( x ) $$ then so is any constant multiple of $ f ( x ) $.

If the splinter of $ \alpha $ behaves approximately as a geometric sequence, a solution can be found. The splinter $ \alpha ^ n ( x ) $ is said to be approximately geometric if there exists a number $ s \in ( 0 , 1 ) $ such that $$ \lim _ { n \to \infty } \frac { \alpha ^ n ( x ) } { s ^ n } $$ exists, is finite, and is nonzero. In this case we say that the splinter has rate $ s $.

On a domain of values $ x $ where the splinter of $ x $ is approximately geometric with rate $ s $ independent of $ x $, a solution to Schröder's equation is given by $$ f ( x ) = \lim _ { n \to \infty } \frac { \alpha ^ n ( x ) } { s ^ n } $$

Why should $ s \in ( 0 , 1 ) $?

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Small, Christopher G., Functional equations and how to solve them, Problem Books in Mathematics. New York, NY: Springer (ISBN 978-0-387-34539-0/pbk). xii, 129 p. (2007). ZBL1152.39300.

Answer 1

The condition $ s \in ( 0 , 1 ) $ is not necessary for the part you've quoted; in fact it works for any nonzero value of $ s $.

This condition is set only for convenience, as is first put forward in the second paragraph of Section 3.4 of the book. The idea is that

1. the case $ s = 0 $ has a trivial set of solutions: those functions that are identically zero on the range of $ \alpha $, and with arbitrary values at other points of the domain of interest;
2. the case $ s = 1 $ can be treated separately, as is done in Section 3.3 of the book;
3. the case $ s > 1 $ can be reduced to the case $ 0 < s < 1 $ if $ \alpha $ is invertible: $ f \left( \alpha ^ { - 1 } ( x ) \right) = \frac 1 s f ( x ) $ is another instance of Schröder's equation;
4. the case $ s < 0 $ can be solved more easily using the solution of the case $ s > 0 $: $ f \left( \alpha ^ 2 ( x ) \right) = s ^ 2 f ( x ) $ is another instance of Schröder's equation, and one can search for the solutions of the original equation among the solutions obtained from the other one.

So, as is mentioned in the second paragraph of Section 3.4 of the book, it often happens that one can transform the original problem to a case where $ s \in ( 0 , 1 ) $, and hence focusing on this case is not much restrictive.