  In the literature, there are many APN-like functions that generalize the APN
properties or are similar to APN functions, e.g. locally-APN functions, 0-APN
functions or those with boomerang uniformity 2.
  In this paper, we study the problem of constructing infinite classes of
APN-like but not APN power functions.
  For one thing, we find two infinite classes of locally-APN but not APN power
functions over $\gf_{2^{2m}}$ with $m$ even, i.e.,
$\mathcal{F}_1(x)=x^{j(2^m-1)}$ with $\gcd(j,2^m+1)=1$ and
$\mathcal{F}_2(x)=x^{j(2^m-1)+1}$ with $j = \frac{2^m+2}{3}$. As far as the
authors know, our infinite classes of locally-APN but not APN functions are the
only two discovered in the last eleven years. Moreover, we also prove that this
infinite class $\mathcal{F}_1$ is not only with the optimal boomerang
uniformity $2$, but also has an interesting property that its differential
uniformity is strictly greater than its boomerang uniformity. For another
thing, using the multivariate method, including the above infinite class
$\mathcal{F}_1$, we construct seven new infinite classes of 0-APN but not APN
power functions.
