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).
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    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).

    Definition. Such a measure is called the SRB measure.

    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.

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