More Examples of Positive Measures on Manifolds

Question

Given a smooth manifold $M$, there are several ways of constructing measures on $M$. The most common procedure I've seen is by starting with a $(0,2)$ tensor field $T$ on $M$, and defining for each chart $(U,x)$, the function $\rho_x := \sqrt{|\det T_{(x), ij}|}$. These functions then give us a non-negative scalar density, $\rho$ on $M$. Using this scalar density, we can essentially (chart by chart) "pull back" the Lebesgue measure to get a well-defined positive measure $\lambda_{\rho}$ on $M$. For example if we're on a (pseudo)-Riemannian manifold, we can use the metric tensor to get the usual Riemannian volume measure. If we're on a symplectic manifold, we can use this same recipe with the symplectic form $\omega$ (or equivalently we just take $\left|\frac{\omega^n}{n!}\right|$... where we think of a scalar density as a section of an appropriate bundle).

Now, my question is, can someone provide me some interesting examples where other types of measures naturally arise; for example are there some other structures which are studied (aside from Riemannian/symplectic, since these are the only two I know) in more advanced areas of geometry/analysis from which a natural notion of a positive measure on a manifold arises. Also, could you provide a (brief) explanation of where such a construction is used/why it is useful. I'm mainly asking to just broaden my perspective. Thanks in advance.

Answer 1

In dynamics (in particular smooth ergodic theory), one has an anonymous diffeomorphism or a flow (or the action of a more sophisticated group) on a manifold and one is interested in invariant measures, and in particular smooth invariant measures (measures given by locally integrable densities $\rho$ w/r/t the Lebesgue measure class with $\rho$ bounded away from $0$ and $\infty$). Reinterpreting Riemannian/symplectic geometries as the static aspects of geodesic/Hamiltonian/symplectic flows, this is a generalization of the constructions mentioned in the OP. See Usefulness of invariant measures with full measure on some set for why invariant measures in general are useful and important in dynamics. Of course, emphasizing the ambient manifold $M$, one could consider this as understanding the symmetries $\text{Diff}(M)$ (or infinitesimal symmetries $\Gamma(TM)$) of $M$ (which is a more general group of symmetries than groups of isometries/symplectomorphisms).

Smooth invariant measures (often also abbreviated as "acip"s, for "absolutely continuous invariant measure") are in particular interesting as these measures are in a sense measures that make it easiest to describe the dynamics. For instance Pesin entropy formula gives for a $C^2$ diffeomorphism $f:M\to M$ of a compact smooth manifold, if $\mu$ is a smooth $f$-invariant Borel probability measure then

$$\text{ent}_\mu(f) = \int_M \sum_{\chi\in\text{LSpec}(f)} \delta^\chi_x\max\{\chi_x,0\}\, d\mu(x),$$

where the LHS is the Kolmogorov-Sinai entropy of $f$ w/r/t $\mu$ and the sum inside the integral is over the Lyapunov exponents of $f$ and $\delta^\chi_x\in\mathbb{Z}_{\geq1}$ is the multiplicity of $\chi$ at the point $x$. There is a similar formula for $\mu$ not necessarily smooth (due to Ledrappier and Young), but now the multiplicities $\delta^\chi_x$ are possibly fractions (each one of them being its maximum value is equivalent to another absolute continuity property, invariant measures with this property are called SRB). (See e.g. Computing Lyapunov Exponents or Lyapunov exponent for 2D map? for a description of Lyapunov exponents).

(As a sanity check, for $f$ an affine automorphism of a torus (which preserves Haar measure; see Do full rank matrices in $\mathbb Z^{d\times d}$ preserve integrals of functions on the torus?), the above formula says that the entropy of $f$ is the sum of the logarithms of moduli of eigenvalues of $f'(0)$ outside the unit circle in the complex plane, counted with multiplicity. Thus entropy is carried in directions where there is exponential expansion. This simpler formula for affine automorphisms was proved earlier by Arov and Berg.)

I should note that once a reference density on the manifold is picked, the existence of a smooth invariant measure boils down to the existence of solutions to a certain "cohomological equation"; see Intuition of cocycles and their use in dynamical systems and Proof of Liouville's theorem for volume preservation for related discussions.