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 ?


I think there are several different ways to make intuitive sense of this, so I'll have a bit of a go at each of them, and hope you find it helpful.

First explanation: At the global level for similarity maps.

Let $M$ be a manifold, and consider a conformal map $f\colon M\to M$. (By conformal we mean that $Df_x$ is a scalar multiple of an isometry.) The simplest case to consider is the one where $\|Df_x\|$ is constant everywhere, say $\|Df_x\| = e^\lambda$ for some $\lambda > 0$; for example, let's consider the case where $M$ is the $n$-dimensional torus $\mathbb{R}^n/\mathbb{Z}^n$ and $f(x)=e^\lambda x$.

In this case, $\lambda$ is the Lyapunov exponent of every point in $M$, and you can easily check that if $B(x,n,\delta)$ is the Bowen ball of radius $\delta$ and order $n$, then $B(x,n,\delta) = B(x,\delta e^{-\lambda n})$. The metric balls $B(x,\delta e^{-\lambda n})$ are used in the definition of Hausdorff dimension, while the Bowen balls $B(x,n,\delta)$ are used in the definition of entropy. If instead of the classical definition of entropy one uses Bowen's definition as a Caratheodory dimension characteristic (see Pesin's book, for example), then the definition of entropy is a verbatim transcription of the definition of Hausdorff dimension, with Bowen balls replacing metric balls. In particular, it is pretty straightforward to see that $$ \text{Hausdorff dimension} = \frac{\text{topological entropy}}{\text{Lyapunov exponent}}. \qquad \qquad \text{(1)} $$ At some level, everything else is just a fancy generalisation of (1). Indeed, in the case mentioned, the potential $\log \|Df_x\|$ is equal to $\lambda$ everywhere, and so the pressure function is given by $$ P_\Lambda (-t\log \|Df_x) = h_\mathrm{top}(\Lambda, f) - t \lambda $$ using basic properties of pressure. The unique root of this equation is $t=h_\mathrm{top}(\Lambda, f) / \lambda$, which by (1) is exactly the Hausdorff dimension.

Second explanation: At the local level for arbitrary maps.

Now consider an arbitrary conformal map, so $\|Df_x\|$ may vary from point to point. One way of computing both Hausdorff dimension and topological entropy/pressure is by using (invariant) measures, so let $\mu$ be an invariant measure. Then if you know something about the pointwise dimensions and local entropies of $\mu$, you can use that knowledge to calculate (or at least estimate) global dimensional quantities like Hausdorff dimension, entropy, and pressure. But at the local level, subject to some bounded distortion estimates, it's not too difficult to show that $$ d_\mu(x) = \frac{h_\mu(x)}{\lambda(x)}; $$ that is, that $$ \text{pointwise dimension} = \frac{\text{local entropy}}{\text{Lyapunov exponent}}, $$ a nice counterpart to (1) in this more general setting.

Third explanation: As a generalisation of Moran's equation.

If you build a Cantor set on the interval (or indeed, in $\mathbb{R}^n$) by using $k$ basic sets at every stage of the construction, each of which is scaled down by a factor of $\lambda_k$ from the basic set at the previous stage that contains it, then Moran's theorem says that the Hausdorff dimension of the resulting Cantor set is the unique value of $t$ such that $$ \lambda_1^t + \cdots + \lambda_k^t = 1. \qquad \qquad \text{(2)} $$ If you take the log of both sides, you get a special case of Bowen's equation. Probably the easiest case to see this in is when $k=2$ and $\lambda=1/3$; then the unique solution of (2) is $t=\log 2/\log 3$, as expected. So one can think of Bowen's equation as the natural generalisation of Moran's equation to the setting where the ratio coefficients can vary at each stage of the construction of a Cantor set.

References: We included a discussion and proof of Moran's Theorem in chapter 2 of Lectures on Fractal Geometry and Dynamical Systems, by Yakov Pesin and Vaughn Climenhaga, and the relationship (1) and its siblings appear in various places in the first four chapters of that book. And if it's not too gauche to continue to refer to my own work, I'll point out that "Bowen's equation in the non-uniform setting", which is here on the arXiv, contains some more precise and detailed formulations of a bunch of the things I said above.

  • $\begingroup$ Hi Vaughn, thank you very much for this nice answer. $\endgroup$ – Leandro Aug 4 '10 at 6:32

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/

  • $\begingroup$ Hi ljjpfx, thank you for the references, I don't knew them. $\endgroup$ – Leandro Jun 18 '11 at 5:32

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