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Let A be a rational function of one complex variable of degree at least two, and 
$z_0$ its repelling fixed point with the multiplier 
$\unicode{x3bb} .$ A Poincaré function associated with 
$z_0$ is a function 
meromorphic on 
${\mathbb C}$ such that 
, 
and 
In this paper, we study the following problem: given Poincaré functions 
and 
, find out if there is an algebraic relation 
between them and, if such a relation exists, describe the corresponding algebraic curve 
$f(x,y)=0.$ We provide a solution, which can be viewed as a refinement of the classical theorem of Ritt about commuting rational functions. We also reprove and extend previous results concerning algebraic dependencies between Böttcher functions.
