Let $M$ be a smooth manifold and $J$ be an integrable almost complex structure on $M$. Let $f: M\to M$ be a diffeomorphism with $f_{*}:TM\to TM$ its tangent map. Then it is easy to see that $f_{*}Jf_{*}^{-1}$ is a new almost complex structure.
Question: Is $f_{*}Jf_{*}^{-1}$ an integrable almost complex structure on $M$?
On
Note the following two interesting phenomena:
1.$f: (M,J)\to (M,f_*Jf_*^{-1})$ is a biholomorphic map but $f : (M, J) \to (M, J)$ may be not a biholomorphic map.
2.$Id: (M,J)\to (M,f_*Jf_*^{-1})$ may be not a biholomorphic map but $Id : (M, J) \to (M, J)$ is a biholomorphic map.
The point is that although $(M,J)$ is isomorphic to $(M,f_*Jf_*^{-1})$, but they are not exactly the same complex manifolds. In other words, they have isomorphic but different complex structures.
As $f$ is a diffeomorphism, $f_* : \mathfrak{X}(M) \to \mathfrak{X}(M)$ is an isomorphism of vector spaces, and as $f_*[X_1, X_2] = [f_*X_1, f_*X_2]$, it is also an isomorphism of Lie algebras; the same is true for $f_*^{-1}$.
Let $Y_1, Y_2 \in \mathfrak{X}(M)$. As $f_*$ is an isomorphism, there are $X_1, X_2 \in \mathfrak{X}(M)$ such that $f_*X_i = Y_i$ (and hence $f_*^{-1}Y_i = X_i$).
Now we have
\begin{align*} &N_{f_*Jf_*^{-1}}(Y_1, Y_2)\\ =&\ N_{f_*Jf_*^{-1}}(f_*X_1, f_*X_2)\\ =&\ [f_*X_1, f_*X_2] + f_*Jf_*^{-1}([f_*Jf_*^{-1}f_*X_1, f_*X_2] + [f_*X_1, f_*Jf_*^{-1}f_*X_2])\\ &- [f_*Jf_*^{-1}f_*X_1, f_*Jf_*^{-1}f_*X_2]\\ =&\ [f_*X_1, f_*X_2] + f_*Jf_*^{-1}([f_*JX_1, f_*X_2] + [f_*X_1, f_*JX_2]) - [f_*JX_1, f_*JX_2]\\ =&\ f_*[X_1, X_2] + f_*Jf_*^{-1}(f_*[JX_1, X_2] + f_*[X_1, JX_2]) - f_*[JX_1, JX_2]\\ =&\ f_*[X_1, X_2] + f_*Jf_*^{-1}f_*([JX_1, X_2] + [X_1, JX_2]) - f_*[JX_1, JX_2]\\ =&\ f_*[X_1, X_2] + f_*J([JX_1, X_2] + [X_1, JX_2]) - f_*[JX_1, JX_2]\\ =&\ f_*\left\{[X_1, X_2] + J([JX_1, X_2] + [X_1, JX_2]) - [JX_1, JX_2]\right\}\\ =&\ f_*N_J(X_1, X_2)\\ =&\ f_*N_J(f_*^{-1}Y_1, f_*^{-1}Y_2). \end{align*}
So $f_*Jf_*^{-1}$ is integrable if and only if $J$ is integrable.