I have a question regarding the SRB measure.

As Lai-Sang Young puts it, the SRB-measure is the invariant ergodic invariant measure most compatible with volume. (see [ http://www.cims.nyu.edu/~lsy/papers/SRBsurvey.pdf ]).

More precisely, let $\Phi\colon \Gamma\to\Gamma$ be a $C^1$ Anosov diffeomorphism of a compact Riemannian manifold $\Gamma$ to itself. In that case, at any $x \in \Gamma$ the tangent space $\mathrm{T}x$ can be split into $\mathrm D \Phi$-invariant stable and unstable subspaces $E_u$ and $E_s$, such that $\mathrm{D}\Phi\vert_{E_u}$ and $\mathrm{D}\Phi\vert_{E_s}$ are uniformly expanding and contracting, respectively.

Now the unstable manifold $W^\epsilon_u(x)$ of a point $x$ is the set of all points $y$ such that for any $n \in \mathbb N$ we have $d(\Phi^{-n}x,\Phi^{-n}y)<\epsilon$, where $d$ is the Riemannian metric. In other words, it consists of points with a similar "past" than $x$.

Let $w$ be a local unstable manifold, $\lambda_w$ the volume induced on $w$ by the Riemannian volume on $\Gamma$. The SRB measure $\mu$ is defined as the unique invariant measure such that $\mu_\omega$ is absolutely continuous with respect to $\lambda_w$, where $\mu_\omega$ is its disintegration or conditioning on the unstable manifolds.

My question now is: Are there any results to what the density $\frac{\mathrm d \mu_w}{\mathrm d \lambda_w}$ should be?

In my opinion it should be something like $\vert\det\mathrm D\Phi\vert_w\vert$, which is the determinant of the flow conditioned on the unstable directions. So basically the product of the positive eigenvalues of $\mathrm D \Phi$, which should be well-defined and the same for all $x \in w$. I also know examples where this holds, but I don't know if something is known in general. (I found a counter-example to that. Actually, the examples where this holds were some affine-linear maps of Baker type were eigenvalues were constant.)

I would be grateful for any help, Cheers, Bernhard


Yes, there is for instance a formula in Y. Pesin, "Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties", Ergodic Theory and dynamical systems 1992, 12, 123–151. Check in particular pages 130-131.

This is not the first time this (or a similar) formula appears, but I just happen to know this paper well.


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