regular slog

[bo198214]

Forgive me for being slow, but have you shown that this satisfies Szekeres' definition of regularity? and if so, where have you shown this?

In [1] Szekeres defines \( f(x) \) being regular (in the case \( 0<f'(0)=a<1 \)) if it has a family of Schroeder iterates \( f^{\circ t}(x)=\sigma^{-1}(a^t\sigma(x)) \) (where \( \sigma \) is a Schroeder function) such that
\( \lim_{x\downarrow 0} \frac{f^{\circ t}(x)}{x}=a^t \)

Such a family of Schroeder iterates is then unique (and we usually call it the regular iterates at fixed point 0).
Szekeres shows in [1] that \( f \) is regular if the principal Schroeder function
\( \sigma(x)=\lim_{n\to\infty} \frac{f^{\circ n}(x)}{a^n} \)
is used for the Schroeder iterates. (Given that it exists, and satisfies some further conditions as strict monotony, differentiability etc.)

In the case of analytic functions with asymptotic development at 0, the formal iterates are the regular iterates.

[1] Szekeres: Regular iteration of real and complex functions, 1958.