I am having some difficulties understanding an argument in a proof. Here is an excerpt from Lyubich–Peters - Classification of invariant Fatou components for dissipative Henon maps, first geometric proof in the Appendix:
The following argument is a more geometric presentation of the argument given in the main body of the paper. To fix the idea, let us assume that the image of $f$ is the standard cusp $\{w^2$ = $z^3\}$. Let us include into holomorphic foliation $\mathbb{F}$ with leaves $L_\lambda = \{w^2 = \lambda z^3\}$, $\lambda \in \hat{\mathbb{C}}$. Let us puncture out $0$, and consider the space $\mathbb{O}$ of leaves in the punctured neighborhood of the origin. This space has a natural Riemann orbifold structure (supported on the sphere) whose local charts are obtained by taking local transversals to $\mathbb F$ and slicing the leaves to it. There are two orbifold points on $\mathbb O$: the leaf $w = 0$ is an orbifold point of order $3$ and the leaf $z = 0$ is an orbifold point of order $2$. So, the Euler characteristic of $\mathbb O$ is equal to $1/2+1/3 < 1$.
I should mention that I have an introductory understanding of Orbifolds, Riemannian Orbifolds, and Foliations. I am trying to understand how $\mathbb{O}$ is acquiring the structure, and what "supported on a sphere" refers to in this case. If possible, an intuitive description of this process is what I would like to have as an answer.
P.S. By cusp they mean $\{(z,w): z^{2} - w^{3}= 0 \} \subset \mathbb{C}^{2}$.
