I have recently been made to believe, that a better understanding of dynamically defined cantor sets will be useful to compute Hausdorff dimension of a specific class of sets I'm interested in. However, all literature on the matter which I've found seem to require background which I am not familiar with, and which I don't think the explanations in them are self-contained. I was wondering whether there is some very basic work giving the basic setting of dynamically defined cantor sets?
I am familiar with measure theory and topology, but my experience in dynamics is relatively scarce. I would appreciate any useful tips.
Edit:
Since the time I posted, I noticed that all my references funnel to a textbook by Palis and Takens, , mainly section 4. I'm more interested in the ability to compute and give estimates for the Hausdorff dimension, which they discuss in section 4.2, but there are some definitions which are unclear to me.
For example, they talk about a presentation $\mathcal{U}$ of a cantor set $K$ as an ordering of the bounded gaps of $K$. I'm already confused as to what this means. They then define the thickness of a gap $U$ at its boundary point, $u\in \partial U$, and denote it by $\tau(K,U,u)$. This is defined as the size of the ratio of the lengths $\frac{m(C_{u})}{m(U)}$, where $C_U$ is the maximal interval with $u$ as one of its edges and $C_U$ does not intersect any gap of length larger than $m(U)$.
Finaly they define the thickness and denseness of $K$ as $$ \tau(K):= \underset{U\in \mathcal{U}}{\sup} \underset{u\in \partial U}{\inf} \tau(K,U,u) \quad \text{and} \quad \theta(K):= \underset{U\in \mathcal{U}}{\inf} \underset{u\in \partial U}{\sup} \tau(K,U,u), $$ accordingly. It is unclear to me why these sizes are well defined as well. They then go on to show the estimate
$$ \frac{\log 2}{\log(2+\frac{1}{\tau(K)})} \leq \dim(K)\leq \frac{\log 2}{\log(2+\frac{1}{\theta(K)})} $$.
I'm trying to understand what is applicable in my case, but I am unable to follow their arguments because of a misunderstanding regarding these definitions.
P.S: I also previously thought that the notion of attractors, stable manifold and unstable manifold were crucial for understanding this estimate, but I think understanding the definitions above will be enough.