I am looking for a reference about the relation between pseudo-Anosov theory and algebraic intersection numbers.
Let $S$ be an oriented surface and $\psi : S \stackrel{\sim}{\to} S$ be a pseudo-Anosov surface automorphism. For simple closed curves $\alpha, \beta \subset S$, it is known that the geometric intersection number $i(\alpha, \psi^n(\beta))$ grows exponentially as $n$ increases. And the exponential growth rate is the stretching factor of $\psi$.
I am wondering the following:
Let $\hat{i}(\alpha, \psi^n(\beta))$ be the algebraic intersection number of $\alpha$ and $\psi^n(\beta)$. Then, does $\hat{i}(\alpha,\psi^n(\beta))$ grow exponentially as $n$ increases? If then, is the growth rate the stretching factor of $\psi$?
I am trying to find some references, but I could not find anything.
Thanks in advance.
This is not true.
In fact, if $S$ is a closed surface of genus $\ge 2$ then there exists a pseudo-Anosov homeomorphism $\psi : S \to S$ such that the induced action $\psi : H_1(S,\mathbb Z) \to H_1(S,\mathbb Z)$ is the identity, hence $\hat i(\alpha,\psi^n(\beta)) = \hat i(\alpha,\beta)$ is a constant independent of $n$.
The construction of such $\psi$ is easy. Start with an essential simple closed curve $c$ that separates $S$. Let $d$ be another simple closed curve that separates $S$ such that $d$ is transverse to $c$ and each component of $S - (c \cup d)$ is a polygon with $\ge 4$ sides. Let $\tau_c,\tau_d$ be the Dehn twists around $c$ and $d$, and note that each of them induces the identity on $H_1(S,\mathbb Z)$, and so $\psi = \tau_c \tau_d^{-1}$ also induces the identity. And by a result of Penner, the mapping class of $\psi$ is pseudo-Anosov.