TITLE: Relation between Hausdorff dimension and Bowen's equation
QUESTION [6 upvotes]: I am reading the paper  Hausdorff dimension for Horseshoes, by McCluskey and Manning. In the following theorem

 Theorem: 
Let $\Lambda$ be a basic set for a $C^1$ axiom A diffeomorphism $f:M^2\to M^2$ with $(1,1)$ splitting
$$
T_{\Lambda}M=E^s\oplus E^u.
$$
Define $\phi:W^u(\Lambda)\to\mathbb R$ by
$$
\phi(x)=-\log\|Df_x|_{E^u_x}\|.
$$
Then the Hausdorff dimension of $W^u(x)\cap\Lambda$ is given by the unique $\delta$ for which
$$
P_{f|_{\Lambda}}(\delta\phi)=0.     \qquad \qquad (1)
$$
The authors compute the Hausdorff dimension of $\Lambda\cap W^u_x$ by using the Bowen's equation (1). I read the proof but I was not able to figure out the intuition behind the Bowen's equation in this theorem. Could you give me explanation (do not need to be rigorous) about that or point out a reference ?

REPLY [3 votes]: In my opinion, Vaughn Climenhaga gave a nice answer and one of his papers named "Bowen's equation in the non-uniform setting" is a good one . You can also refer to a note given by Godofredo Iommi, i.e., ``The Bowen Formula: Dimension Theory and Thermodynamic Formalism '' which is available at:http://www.mat.puc.cl/~giommi/