## Tag Archives: conditional measure

### Some distinguished meausres

This is a note taken from V. Kaimanovich’s paper Bowen-Margulis and Patterson measures on negatively curved compact manifolds.

Let $M$ be a simply connected negatively curved manifold (the exponential map $\exp_x:T_xM\to M$ turns out to be a diffeomorphism for every $x$ by Cartan-Hadamard Theorem). Two geodesic rays on $M$ are called asymptotic if they they have a bounded distance in the future. Denote by $\partial M$ the set of asymptotic classes of geodesic rays on $M$. Then $\overline{M}=M\sqcup\partial M$ is called the visibility compactification of $M$. Let $\partial^2 M=\partial M\times \partial^2 M\backslash \triangle$.

• For each pair $(x,\alpha)\in M\times \partial M$, there exists a unique geodesic ray $\gamma=\gamma_{x,\alpha}\in\alpha$ with $\gamma(0)=x$.
• For each pair $(\alpha,\beta)\in \partial^2 M$, there exists a unique complete geodesic $\gamma$ with $\gamma(R_+)\in\alpha$ and $\gamma(R_-)\in\beta$.
• Each unit vector $\xi\in SM$ corresponds to a complete geodesic $\gamma_\xi$, and hence two maps $\xi\mapsto \alpha(\gamma_\xi)$ and $\xi\mapsto \beta(\gamma_\xi)$. This induces a $\mathbb{R}$-fibration $SM\to \partial^2 M$, $\xi\mapsto (\alpha(\gamma_\xi),\alpha(\gamma_\xi))$.
• For each pair $(x,\alpha)$, we pick the geodesic ray $\gamma$ and define Busemann function $b_{\alpha,x}:y\in M\mapsto \lim_{t\to\infty}(d(x,\gamma(t))-t)$. The level sets of $b_{\alpha,x}$ are the horospheres on $M$ centered at $\alpha\in\partial M$.
• More generally, let $b_\alpha(y,z)=b_{\alpha,x}(y)-b_{\alpha,x}(z)$ (need to check it is independent of the choice of $x$ and $\gamma_{\alpha,x}$) (Remark). Moreover, $b_\alpha(y,z)$ gives the signed distance between the horospheres passing through $y$ and $z$ centered at $\alpha$.
Remark: Another view point is $b_\alpha(y,z)=b_{\alpha,z}(y)$. Clearly it is independent of $x$.
• Define another function $B_x:(\alpha,\beta)\in\partial ^2M \mapsto b_\alpha(x,y)+b_\beta(x,y)$, where $y$ lies on the geodesic $\gamma_{\alpha,\beta}$. (Check it is independent of the choices of $y$ on that geodesic). Geometrically, it measures the length of the segment cut out from $\gamma$ by the horospheres passing through $x$ and centered at $\alpha$ and $\beta$. Moreover we have $B_x(\alpha,\beta)-B_y(\alpha,\beta)=b_{\alpha}(x,y)+b_{\beta}(x,y)$.
4. Ledrappier-Young’s criterion for measures with Absolutely Continuous Conditional measures on Unstable manifolds. Let $f:M\to M$ be a $C^2$ diffeomorphism and $\mu$ be an invariant probability measure. For $\mu$-a.e. $x$, if there exist negative Lyapunov exponent(s) at $x$, then the set of points with exponentially approximating future of $x$ is a $C^2$-submanifold, $W^u(x)$. Then $\mu$ is said to have ACCU if for each measurable partition $\xi$ with $\xi(x)\subset W^u(x)$ and contains an unstable plaque for $\mu$-a.e. $x$, the conditional measure $\mu_{\xi(x)}\ll m_{W^u(x)}$ for $\mu$-a.e. $x$.
Theorem. $\mu$ has ACCU if and only if $h_\mu(f)=\Lambda^+(\mu)$, where $\Lambda^+(\mu)=\int\sum\lambda^+_i(x)d\mu(x)$.
Moreover, the density $\frac{d\mu_{\xi(x)}}{dm_{W^u(x)}}$ is strictly positive and $C^1$ on $\xi(x)$ for $\mu$-a.e. $x$.