Going from piecewise to genuine geodesic without decreasing number of intersections?

Question

Let $(M^2,g)$ be a complete, two-dimensional Riemannian manifold be given; also given is $\gamma: [0,\infty) \to M$, an injective geodesic in $M$.

Suppose there are two geodesic segments $\gamma_i : [a_i,b_i] \to M$, $i = 1,2$ with endpoints on $\gamma$ and \begin{equation} \gamma_1(b_1) = \gamma_2(a_2). \end{equation} Moreover, the points $\gamma_1(a_1),\gamma_1(b_1) = \gamma_2(a_2), \gamma_2(b_2)$ appear in chronological order along $\gamma$.

You may also assume that $\gamma_1,\gamma_2$ 'lie on either side of $\gamma$', by which I mean that \begin{equation} \langle \gamma_1'(b_1), n \rangle \text{ and } \langle \gamma_2'(a_2) , n \rangle \end{equation} have the same sign, writing $n \in T_{\gamma_1(b_1)}M$ for the unit normal to $\gamma$.

The concatenated, piecewise defined geodesic $\gamma_2 \! \cdot \! \gamma_1$ intersects $\gamma$ three times. Is there a genuine geodesic $\Gamma \neq \gamma$ in $M$ that intersects $\gamma$ three or more times?

Edit. I've specialized the original question (which was phrased in terms of $N$ geodesic segments), to the special case where $N = 2$.

Answer 1

Here's a counterexample.

Take a cone of unrolled angle strictly between $\frac\pi2$ and $\pi$. Smooth the vertex and take $\gamma$ to be a radial ray far from the vertex. Then you can make a piecewise geodesic which loops around and intersects $\gamma$ as many times as you like, but no genuine geodesic intersects it more than twice.