# Hamiltonian dynamics on cotangent bundle

I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $$N$$ is a differential manifold and $$S_0 ,S_1\subseteq N$$ two submanifolds of $$N$$, and we consider the corresponding conormal bundles $$\nu^{\ast} S_0$$ and $$\nu^{\ast} S_1$$. Let $$H\colon T^{\ast} N\times I\to\mathbb{R}$$ be a smooth function and $$X_H$$ the corresponding Hamiltonian vector field, and $$\phi_H^t$$ the corresponding Hamiltonian flow. We define a function $$g\colon\nu^{\ast} S_1\to\mathbb{R}$$ by $$g(x)=\mathcal{A}_H (z_H^x),$$ where $$\mathcal{A}_H$$ is the corresponding action functional, and $$z_H^x$$ the path given by $$z_H^x (t)=\phi_H^t\circ(\phi_H^1)^{-1}(x)$$. Then he claim that:

1. The intersection $$\phi_H^1 (\nu^{\ast} S_0)\cap\nu^{\ast} S_1$$ is compact in $$T^{\ast} N$$.

It's obviously not the case: if we pick $$H=0$$, then the condition is equivalent to saying that $$S_0$$ and $$S_1$$ are disjoint. But this is a procedure to his proof of the fact that the action spectrum is compact and has measure zero. So I'm wondering if

1'. The image $$g(\phi_H^1 (\nu^{\ast} S_0)\cap\nu^{\ast} S_1)$$ is compact in $$\mathbb{R}$$.

is true.

Another claim is about the linearization of the ODO $$\frac{\partial }{\partial t} -X_H$$, namely $$\nabla_{\dot{\gamma }} -DX_H (\gamma)$$. We consider the path space with boundary conditions $$\Omega_{S_0 S_1} (T^{\ast} N,\mathrm{d}\lambda)$$, given by all paths starting from $$\nu^{\ast} S_0$$ and ends at $$\nu^{\ast} S_1$$. Then the tangent space of this path space at a path $$\gamma$$ is given by $$\Gamma_{S_0S_1} (\gamma^{\ast} (TT^{\ast} N))$$, the space of vector fields on $$\gamma$$ with endpoints in the given tangent spaces. The linearization is then a linear operator $$\nabla_{\dot{\gamma}} -\mathrm{D} X_H\colon\Gamma_{S_0S_1} (\gamma^{\ast} (TT^{\ast}N))\to\Gamma (\gamma^{\ast} (TT^{\ast} N))$$. Then the book claims that

1. The intersection $$\phi_H^1 (\nu^{\ast} S_0)\cap\nu^{\ast} S_1$$ is transversal if and only if the linearization $$\nabla_{\dot{\gamma }} -\mathrm{D}X_H$$ is surjective.

I have no idea why this is true. I don't know how to relate the surjectivity of the linearization to the transversality of intersections. Any comments or answers are helpful.