Zooming equation

Zooming equation is tentative name for the equation

(1) $~ ~ ~ T\big(f(z)\big)= f( s\, z)~$

where $~T~$ is some given function and $~s~$ is some given constant. Usually it is assumed that $~T~$ is holomorphic at least in some vicinity of zero, and $~s\!=\!T'(0)~$. Function $~f~$ is requested to built-up.

The physical sense of function $~T~$ is transfer function; it describes the variation of value of function $~f~$ at the scaling, zooming of its argument with factor $~s~$.

The tentative name for the solution $~f~$ of the zooming equation is zooming function.

Schroeder equaiton

The solution $~f$ of the zooming equation is related to the solution $~g~$ of the Schroeder equation

(2) $~ ~ ~ g\big(T(z)\big)= s \, g(z) $

it is assumed that $~f=g^{-1}$ and $~g=f^{-1}$, id est, in wide ranges of values of $z$, the relations below hold:

(3) $~ ~ ~ f(g(z))=z$

(4) $~ ~ ~ g(f(z))=z$

Regular iteration

Assuming that zero is fixed point of the transfer function $~T~$ is holomorphioc (regular) in vicinity of zero, the zooming function $~f~$ and the Schroeder function $~g~$ can be used to construct the non–integer iterate $~T^r~$, that is regular in vicinity of zero, id est, the regular iterate

(5) $\displaystyle ~ ~ ~ T^r(z)=f\Big(r\,s\, g(z)\Big)$

Such a rule can be extended to the cases of other fixed points $~L~$, performing the corresponding transform of the transfer function from $~T~$ to $~t~$, assuming that

(6) $~ ~ ~ t(z)=T(z\!+\!L)-L$

and treating $~t~$ as a new transfer function with fixed point zero. After to construct iterates of function $~t~$, regular at zero, the $~r$th iteration of function $~T~$, regular at $~L~$, can be expressed as

(7) $~ ~ ~ T^t(z)=t^r(z\!-\!L)+L$

If the transfer function $~T~$ has more than one fixed points, then, the non-integer iterate $~T^r~$, regular at one of these fixed points, has no need to be regular at another fixed point; another fixed point may be branchpoint . The non-integer iterates of various functions can be constructed also with the superfunction and Abel function, considering the transfer equation instead of the zooming equation .