Intuition behind the Duistermaat-Guillemin version of Weyl's law The theorem in question (see this paper), after a modification by Ivrii (see this paper) states the following: 
Let $(M, g)$ be a compact Riemannian manifold of dimension $n \geq 2$. Assume that the set $G := \{(x, \xi) \in S^*(M) : \Phi_t(x, \xi) = (x, \xi), \text{   for some   } t > 0\}$ has zero Liouville measure (this means that the set of periodic geodesics has measure zero). Then the conclusion is:
$$N(\lambda) = \frac{1}{(2\pi)^n}\text{Vol }(B^*M)\lambda^n + o(\lambda^{n - 1}),$$
where $N(\lambda)$ represents the number of Laplace eigenvalues $\leq \lambda$. 
I understand the proof in the sense that I understand how every step of the proof is mathematically true. But I cannot get any insight as to why people would expect it to be true. Particularly, I am not getting any intuition into the periodic geodesic requirement. Let me ask a couple of questions on this:
(a) What are some non-trivial examples of compact manifolds where the set of periodic geodesics has measure zero?
(b) Also, what is the idea behind this geodesic assumption entering the picture at all? In other words, why would you expect that having fewer periodic geodesics would give an improvement over the traditional Weyl law?
(c) Reference request: have there been other improvements of the Weyl's law under other assumptions?
 A: Small error: $N(\lambda)$ is a number of eigenvalues $< \lambda^m$ (where $m$ is an order of operator, obviously $m=2$ here). It is not a geometry but a Hamiltonian dynamics (which becomes a Riemannian geometry for Laplacian). To better understand one should consider semiclassical asymptotics. For example, consider $N_h(E)$ the number of eigenvalues of the quantum Hamiltonian $H= -h^2\Delta +V(x)$ which are $<E$. This is more general problem; for $V=0$, $E=1$ and $h=\lambda^{-1}$ it becomes an original problem.
Quantum properties (of $H$) like a better remainder estimate in the distribution of eigenvalues and equidistribution of eigenfunctions are intimately related to classical properties like not too many periodic trajectories or ergodicity of the Hamiltonian flow. 
Is there a better remainder estimate? Yes, a little: with the remainder $O(\lambda^{d-1}/\ln(\lambda)$ or even $\lambda^{d-1-\delta}$ with a small exponent $\delta>0$. But they require much stronger assumptions (to the Hamiltonian flow). F.e. the former holds if sectional curvature is always negative while the latter usually if the case of completely integrable Hamiltonian flow (+ the usual non-periodicity).
However, if we consider the case of all geodesics periodic, there are plenty of such manifolds, f.e. Zoll-Tanner manifolds, then with general lower order terms instead of one eigenvalue of a high multiplicity we have a narrow cluster of eigenvalues and for generic lower order terms we can get asymptotic distribution of the eigenvalues inside of the cluster (there are $\asymp \lambda^{d-1}$ of them) with an error estimate $O(\lambda^{d-2})$; so we can get an asymptotics of $N(\lambda)$ with an error estimate $O(\lambda^{d-2})$––but it will not be a Weyl asymptotics but
\begin{equation}
N(\lambda) = c_0\lambda ^d + (c_1 +Q(\lambda)) \lambda^{d-1}+ O(\lambda ^{d-2})
\end{equation}
with a periodic function $Q(\lambda)$ (there are formulae for it).
There are also asymptotics (under certain assumptions) when not all geodesics are periodic but the set of periodic geodesics has a positive measure––the case first considered by Yu. Safarov. 
