We can have same attractor from different iterated function systems. So i wonder about self-similarity dimension concept is for IFS or its attractor. We know that when IFS satisfies the open set condition then Hausdorff dimension coincide with self-similarity dimension. And from my understanding Hausdorff dimension is unique for a set.
For example Cantor $1/3$ set, I have two IFS which are as:
First IFS: $ (f_1,f_2)$ such that $ f_1 (x) = x/3 , f_2 (x) = (x+2)/3 $ and
Second IFS: $ (g_1,g_2,g_3,g_4,g_5,g_6)$ such that $ g_i (x) = (x+(j-1))/9 , \, i=1,2,3,4,5,6, \, j=1,2,3,7,8,9 $
From first IFS I am getting self-similarity dimension is $ \log{2}/\log{3} $, and from second IFS it is $ \log{6}/\log{9}. $
And both IFS satisfies the open set conditions, so this means $1/3 $ Cantor set has two different Hausdorff dimensions. I don't know where I miss something.