p 177: "For any absolute value" probably $v\in M_{k'}$, where $k'$ is the splitting field of $\mathbb{Q}(x)$, since one applies it to all $x_i$ simultaneously.

p 180: Remark B.2.7 has parts (i), (ii), (2).

p 181: $\prod_{v\in M_k} \varepsilon_v(r)_v^n = \prod_{v\in M_k^\infty} r_v^n = r^{[k:\mathbb Q]}\qquad$ should be $\qquad\prod_{v\in M_k}\varepsilon_v(r)^{n_v} = \prod_{v\in M_k^\infty} r^{n_v}= r^{[k:\mathbb Q]}$

p 184: In Theorem B.3.2(b) need to assume $\phi(V) \not\subset D$.

p 190: Top display should read $h_{V,D} = h_{V,\phi^*H} = h_{{\mathbb P}^n,H}\circ\phi+O(1) = h\circ\phi+O(1).$

p 195: In Theorem B.4.1, $V$ should probably be projective.

p 198: The proof of B.4.2(b) is missing. (By (a) the preperiodic points satisfy $\hat h_{V,\phi,D}(P) = 0$ hence $h_{V,D}(P)=O(1)$. By the finiteness property the set of rational preperiodic points is finite.)

p 200: Statement (b) follows from (d), so it is natural to prove in the order (a),(c),(d),(b),(e).

p 321: the last two displays should be $\partial_{i_1,\ldots,i_m} P$.

p 328 bottom inequality is reversed

p 329: The first equation in D.6 in display mode should say: for all $i/r < I$

p 330: The first equation in display mode should say: for all $i/r < I$
In the sentence starting: "On the other hand, we know from above …" it should say $i < r I$ and $j+1 < r I$
The displayed equation that comes after the sentence "Eliminating $\beta_2$ from these …" should say "for all $i < r I, j < r I +1$".
Right after that, it should say "it follows that $\partial_k W(\beta_1) =0$ for all $k < r I - 1$".

p 331: It says "Now let $\phi_1, \phi_2, …, \phi_k\in K(X)$ be rational functions...", but it should say "Now let $\phi_1, \phi_2, …, \phi_k \in K(X_1, X_2, …, X_m)$ be rational functions…".

p 336: definition of $W$, second line the sum term of $\phi_r$ - the parentheses are not balanced.

P 376 line 1 and page 378 line 2 from bottom the constant $s$ should be $M$

P 377 first display second line should have $\delta_2$, as in $=\delta_1h(\phi_{NA}(z)) + \delta_2h(\phi_{NA}(w))$

P 377 first display second line should have $\delta_2$, as in $=\delta_1h(\phi_{NA}(z)) + \delta_2h(\phi_{NA}(w))$

P 394 paragraph "The fact that": the property that the coefficient of the monomial $x_j^N$ is nonzero follows from the assumption in E.2.2(i) that $C$ does not meet codimension-2 strata.

P 423 display two lines below (96): $=\frac{2|z|^2}{|w|^2}$ should be an inequality $\leq\frac{2|z|^2}{|w|^2}$.

P 423 bottom display and P 424 top display and following line: $4N$ should probably be $12N$

p 405, inequalities (43),(45),(46) - it seems the inequality should be still an equality, as it seems that the estimation starts on the next page only.

p 420, Equation (87), the $\xi_1$ and $\xi_2$ should probably be in the subscript: $\deg_{\xi_1}(Q) \leq Nd_1$ etc.

Still p 420, on the next display, the statement in B.7.2 doesn't seem to imply precisely what is asked for.