I'm trying to learn some stuff about divergence of geodesics. Let $\gamma$ be a geodesic in a metric space $X$. The divergence of $\gamma$ is a function $f(r)$ for $r \ge 0$ such that $f(r)$ is the length of a shortest path between $\gamma(-r)$ and $\gamma(r)$ in the complement $X \setminus B(\gamma(0), r)$ of a ball at $\gamma(0)$.

The first question I wanted to ask is the following:

Is there some geodesic in mapping class groups such that the divergence of it is at least quadratic?

It is known (due to Behrstock) that the orbit of a pseudo-anasov element is a quasi-geodesic $p$ and satisfying Morse property. Morse property of a quasi-geodesic $p$ means that any quasi-geodesic with endpoints at $p$ lies in a uniform neighborhood of $p$.

The second question is following:

Let $g$ be pA element. We connect $g^{-n}$ and $g^n$ by a geodesic $p_n$. Let $n \to \infty$. The $p_n$ will converge to a geodesic $p_\infty$ lying the uniform neighborhood of $\{g^n: n \in \mathbb N\}$. Does the geodesic $p_\infty$ have the quadratic divergence?

There seems lots of literatures about the divergence. If this is previously known, please be kind to tell me the reference.

Thanks.