Let $C_\lambda$ and $C_\nu$ be the middle $\lambda$ and $\nu$ cantor sets, respectively. I want to show the map $\Pi_{\lambda,\nu}:C_\lambda\rightarrow{C_\nu}$ is $\gamma$-hölder continuous, with $\gamma=\frac{\log(\frac{\lambda+1}{2})}{\log(\frac{\nu+1}{2})}$, where redefine $\Pi_{\lambda,\nu}$ as follows:
where $i_k\in\{0,1\}$.
Im following the proof through my lecturers notes:
I'm not sure on two things: why was $N$ defined? what purpose does it serve? and secondly, how do the last two inequalities yield the desired result?
I found the answer:
firstly my lecturer made a mistake, the constant next to the sum should be $\frac{1+\lambda}{1-\lambda}$.
Secondly, regarding the second inequality we can choose a $d>0$ such that the inequality is $\le$ instead. This is done because $N$ is the first time $x$ and $y$ differ, so before that, they coincide. This means they live in the same interval on the $(N-1)^{th}$ level. hence $|\Pi(x)-\Pi(y)|\le{\frac{1-\lambda}{2}}^{N-1}$. From this, the result can be obtained.