p. 9In Section 1.3, the sketch of proof of Green’s theorem (Theorem 1.3.7) has a serious gap; the problem is that the quotiented orbit of $(g^{n+h} x, g^n x)$ may possibly have a constant image with respect to a horizontal character. It seems that to use this type of argument to obtain the full strength of Theorem 1.3.7 (and not just some special cases) one needs the more complicated quantitative van der Corput argument from [GrTa2009c].
p. 21In the first display after (1.19), $\sum_{j=1}^\infty \frac{1}{n_j} X_{\leq n_j}$ should be $\sum_{j=1}^\infty \frac{1}{n_j} |X_{\leq n_j}|^2$.
p. 23In (1.23), $1 + \mu dt$ should simply be $\mu dt$.
p. 24After (1.30), $\Theta_\infty(x_\infty)-1/x_\infty$ should be $(\Theta_\infty(x_\infty)-1)/x_\infty$.
p. 25In (1.32), (1.33), (1.34), $k_\infty$ should be $k_\infty^\times$.
p. 27In (1.37), $|t|$ should be $|t|_p$.
p. 64In the two long displays the symbol P is missing just before the right bracket ] on most of the lines of the displays.
p. 69In the final paragraph, “$(n-1)$-chain” should be “$(n+1)$-chain”.
p. 70In the final display, $\rho(g,x)$ should be $\rho(h,x)$. In the final paragraph, “$n$-cocycles and $n$-cochains” should be “$n$-cocycles and $n$-boundaries”.
p. 76In Example 2.1.3, delete the parenthetical reference to Example 2.1.1.
p. 87In Exercise 2.2.4, the last sentence should be phrased as a question, i.e. “Does there exist analogous claims in the categories of dynamical systems and measure-preserving systems?”.
p. 90In the proof of Lemma 2.3.3, add “By passing to a subsequence and relabeling, we may assume $T^{n_m} x$ converges to $y$” before “But then one verifies…”.
p. 95Remark 2.3.7 is inaccurate regarding the left-continuity of $\beta S$ (see this paper for the subtle issues arising here) and should be deleted.
p. ?In Corollary 2.3.9, $x,y$ should lie in $X$, not ${\Bbb Z}$.
p. ?In Exercise 2.13.4, ${\Bbb E}(f|Y)(y,z)$ should be ${\Bbb E}(f|Y)(y)$.
p. 99Exercise 2.4.5 is not relevant at this juncture and should be deleted.
p. ?In the paragraph before Exercise 2.4.10, $T^n V$ should be $T^n U$.
p. 102in the proof of Proposition 2.4.11, $p_*$ should lie in $*{\Bbb N}\backslash {\Bbb N}$ rather than $*{\Bbb Z}\backslash {\Bbb Z}$. In particular, the parenthetical remark about setting $p_*$ equal to $p$ should be deleted.
p. 104In the proof of Lemma 2.4.13, $V$ should be $U$.
p. 110A similar ultrafilter proof also appears in Section 3 of N. Hindman’s paper “Problems and new results in the algebra of Beta S and Ramsey Theory” in “Unsolved problems on mathematics for the 21st century”, J. Abe and S. Tanaka eds., IOS Press, Amsterdam (2001), 295-305.
p. 113Lemma 2.5.14 should be called the Ellis-Numakura lemma rather than the Ellis-Namakura lemma. (Similarly for the index entry for this lemma.)
p. 127In Definition 2.6.16, it should be stressed that the fibre metrics $d_y$ are compatible with (i.e. generate) the topology on the fibres inherited from the full space. (More generally, in this text, when we refer to a metric on a topological space, it should be understood that that metric generates the topology of that space unless otherwise specified.)
p. 128In the first paragraph, $f(y_0)$ should be $f(x_0)$.
p. 132In Lemma 2.6.30, $Y \times_\sigma ({\bf R}/{\bf Z})$ should be $Y \times_\sigma ({\bf R}/{\bf Z})^d$.
p. 134In Example 2.7.2, $(0,1/2n)$ should be $(1/2n,0)$, and $(\alpha, \frac{n(n-1)}{2} \alpha + \frac{1}{2})$ should be $(n\alpha, \frac{n(n-1)}{2} \alpha + \frac{1}{2})$.
p. 135In Exercise 2.7.2(5), it should be explicitly stated that X is assumed to be distal.
p. ?In Exercise 2.7.6(3), $F(y,z)$ should be $d(y,z)$.
p. 137After Exercise 2.7.8, the reference to Exercise 2.7.5 should be to Exercise 2.7.3 instead.
p. 139After (2.54), W should be K.
p. 141Exercise 2.7.14 is the same as 2.9.13 and should be deleted.
p. 143The last sentence of the proof of Theorem 2.8.2 is redundant and should be deleted. In Exercise 2.8.3, $\mu(X)$ should read $\mu(E)$ (two occurrences), and “any smaller” should be “any larger”.
p. 144The first proof of von Neumann’s ergodic theorem is due to F. Riesz, rather than von Neumann, and the text should be edited accordingly. After (2.63), “uniformly in $n$” should be “uniformly in $N$“. Also $H^U + \overline{W}$ should be $H^U + W$.
p. 146After (2.67), $\frac{\lambda^N-\lambda}{\lambda-1}$ should be $\frac{\lambda^N-1}{\lambda-1}$.
p. 149In Exercise 2.8.6(1), “${\mathcal X}$-valued” should be “${\mathcal X}$-measurable”.
p. 150In Exercise 2.8.9, Corollary 2.8.12 should be Corollary 2.8.16.
p. 152In Theorem 2.9.1, in the definition of $Mf$, the summation should be from $0$ to $N-1$, rather than from $1$ to $N$.
p. 153In the first display, the equality sign should be a $\leq$ sign instead.
p. 157In Exercise 2.9.6, the probability space should be assumed to be standard Borel (in order to define the countable product space properly).
p. 158In the first paragraph of Section 2.9.4, “Borel $\sigma$-algebra of $T$” should be “Borel $\sigma$-algebra of ${\mathcal F}$“. In Exercise 2.9.10, “measure on $T$” should be measure on $X$“. For Exercise 2.9.9, “if and only if” should just be “only if”, and the additional hypothesis that $U$ has a boundary of measure zero should be added.
p. 159In Exercise 2.9.13, one needs to add the additional hypothesis that the support of the invariant measure $\mu$ is equal to the whole space $X$.
p. 160In Example 2.9.17, “from $Y$ to $X$ and from $X$ to $Z$” should be “from $Y$ to $Z$ and from $X$ to $Y$“. Also, all integrals here should be over Y rather than over X.
p. 162In the right-hand side of (2.96), the factor $g(y)$ should be moved outside the inner integral (for clarity). In Exercise 2.9.14, $\nu_y$ should be $\mu_y$.
p. 163In the final parenthetical of Exercise 2.9.15, add “, but I do not know of a way to prove Proposition 2.9.22 in full generality just from Choquet.”
p. 167After (2.100), the range $1 \leq a \leq N$ should be replaced with $1 \leq a \leq N/N_0$.
p. 189194: In Exercise 2.12.15, and also in the first paragraph of Section 2.12.4, Corollary 2.12.8 should be Corollary 2.12.13. After Remark 2.12.24, Proposition 2.12.15 should be Proposition 2.12.14.
p. 190In footnote 44 in Theorem 2.12.14, “always has full measure” should be “always has full (outer) measure”.
p. 191Replace the first sentence in the paragraph preceding 2.12.17 by “Given a Hilbert space H, define its complex conjugate $\overline{H}$ to be the set of formal conjugates $\overline{H} = \{ \overline{v}: v \in H \}$ of elements of $H$, with the addition structure $\overline{v} + \overline{w} := \overline{v + w}$, the conjugated scalar multiplication structure $\overline{z \overline{v}} := \overline{\overline{z} v}$ and the conjugated inner product $\langle \overline{z},\overline{w} \rangle_{\overline H} := \overline{\langle z, w \rangle_H} = \langle w, z \rangle_H$.”. In equation (2.130), $v \otimes v'$ should then be $\overline{v} \otimes v'$, and the second inner product should be subscripted by $\overline{H} \otimes H'$.
p. 195In Exercise 2.12.22, $\int_X (f\ d\mu)^2$ should be $(\int_X f\ d\mu)^2$.
p. 203In (2.159), the $O(\varepsilon)$ term is unnecessary, and (2.151) and”and relative Cauchy Schwarz again” may be deleted from the preceding line. After (2.160), the parenthetical remark can be deleted, and after (2.161), “again” may be deleted.
p. 207In (2.168), the second $f_1$ should be $f_d$.
p. 208the integration $\int_X\ d\mu$ is missing from the summand.
p. ?In Remark 2.14.4, the hypothesis that $Y$ is ergodic needs to be added.
p. 210In Proposition 2.14.11, the “weak operator topology” should be clarified to “the weak operator topology of $L^2(X)$“, and it should also be parenthetically noted that the $S_{f,N}$ are uniformly bounded in the Hilbert space $L^2(X)$. In the statement of this proposition, “technology” should be “topology”. In Definition 2.14.13, $\langle \dots \rangle_{X,Y}$ should be $\langle \dots \rangle_{X|Y}$.
p. 216After (2.188), “on a set of measure $O_c(\varepsilon^2)$” should be “outside of a set of measure $O_c(\varepsilon^2)$“.
p. 218In Exercise 2.16.1(7), “H/[H,K] and K/[H,K] become abelian” should be “the images of H and K become groups that commute with each other”.
p. 221In Example 2.16.9, $[0,y+x \hbox{ mod } 1]$ should just be $[0,y]$.
p. 222In Example 2.16.13, the group element g should have a coefficient of -1 instead of 1 in the third column, second row position.
p. 223In (2.203), $n+1$ should be $n-1$.
p. ?The statement and proof of Corollary 2.16.21 and Corollary 2.16.22 need to be modified, because the character $\chi$ in the proof of the former need not be primitive. In Corollary 2.16.21(2), one needs to partition the orbit $(T^n x)_{n \in {\bf Z}}$ into finitely many suborbits $(T^n x)_{n \in P}$ for various congruence classes $P$ before the claim holds separately for each suborbit, and similarly for Corollary 2.16.22.
p. 231The proof of Lemma 2.17.5 is incomplete, because U and D do not fully generate $SL_2({\bf R})$. To finish the argument, observe that $d^t w^\varepsilon d^{-t}$ converges to the identity as $t \to +\infty$, and thus $\langle \rho(d^t w^\varepsilon d^{-t}) v, v \rangle \to \langle v, v \rangle$. Using the D-invariance we conclude that $\rho(w^\varepsilon) v, v \rangle = \langle v, v \rangle$, and thus as before v is also invariant with respect to the group U’ generated by the $w^\varepsilon$. Since U and U’ (and D, if desired) generate $SL_2({\Bbb R})$, the claim follows.
p. 232-233The proof of Lemma 2.17.9 requires some changes. In the penultimate paragraph, “any g in L” should be “any g in L with $gx_0$ sufficiently close to $x_0$“. The final paragraph needs to be changed to the following: “Suppose that $Lx_0$ is not closed; then one can find a sequence $g_n x_0$ in $Lx_0$ that converges to $x_0$ but with the $g_m g_n^{-1}$ staying bounded away from the identity for $m \neq n$. For a sufficiently small compact neighbourhood $K$ of the identity in $L$, the sets $K g_n x_0$ then are disjoint and all have the same measure for $n$ large enough; since $\mu(Lx_0)=1$, this forces these sets to be null. But then the invariant measure $m$ annihilates $K$ and is thus null as well, a contradiction.”
p. 235In Proposition 2.17.12, $x_n,y_n$ should lie in $G/\Gamma$ rather than $G$. In the proof of that proposition, $g_*$ should be $g^*$.
Volume II
34 corrections · 9 awaiting a page number
p. ?In Footnote 36, $U^n$ should be $(U+E)^n$.
p. 40After (1.17), “multiply $c_k$ by a scalar” should be “multiply $v_k$ by a scalar”. Two pages previously, the display for U+E has an extraneous space. In (1.18), all appearances of $w_j$ and $w_k$ should be $w_j^*$ and $w_k^*$ instead.
p. ?In example (8) of Section 1.4, $x^2+y^2+z^2$ should be $x^2+y^2+z^2=1$.
p. 67Before (2.11), $\Gamma(M)$ should be $\Gamma(TM)$.
p. ?After (2.19), $\gamma \circ \phi$ should be $\phi \circ\gamma$.
p. 71In (2.29), $\nabla_\beta \nabla_\alpha \nabla_\delta$ should be $\nabla_\beta \nabla_\delta \nabla_\alpha$.
p. 62After (2.32), “(2,0) tensor” should be “(0,2) tensor”.
p. 74In Definition 2.1.14 II, $X_\alpha Y_\beta$ should be $X^\alpha Y^\beta$.
p. 77Before (2.38): the heat equation $\partial_t u + \Delta u = F$ should be $\partial_t u - \overline{\Delta} u = F$.
p. 78In the second line of (2.45), the first negative sign should be positive, and the positive sign should be negative. In (2.48), the last two minus signs should be plus signs, and in (2.49), $\hbox{Riem}^\delta_{\alpha \gamma \beta}$ should be $g^{\sigma \gamma} \hbox{Riem}^\delta_{\sigma \alpha \beta}$.
p. ?In the discussion before (2.53), the manifold should be complete in addition to smooth and connected.
p. 81After (2.60), $\phi^*(t) \dot \phi(t)$ should be $\phi_*(t) \dot \phi(t)$.
p. ?In (2.67), $Riem^\delta_{\alpha \gamma \beta}$ should be $Riem^\delta_{\gamma \alpha \beta}$.
p. 115In (2.121), $K_M$ should be $K_\Sigma$.
p. 127In (2.143), the factor of $1/2$ should be deleted.
p. 132In (2.162), $\nabla_T \nabla_T k$ should be $\nabla_T k$.
p. 136After (2.170), “slows down the flow of time by $1/\lambda$” should be “slows down the flow of time by $1/\lambda^2$“.
p. 152On the last line, (2.72) should be (2.73).
p. ?In Exercise 2.9.4, $-4\Delta+4R$ should be $-4\Delta+R$.
p. ?In (2.247), $\frac{1}{\tau}$ should be $\tau$.
p. ?In (2.264), $\frac{t}{2N}$ should be $\frac{2t}{N}$; in (2.266), $\frac{r^2}{4N^2}$ should be $\frac{r^2}{N^2}$. In (2.274), $r_0^{N/2}$ should be $r_0^N$.
p. 160Strictly speaking, the derivation given of the log-Sobolev inequality is only valid for those $\phi$ for which $\phi^2$ is the backwards time evolution of a non-negative test function $u_0$ by the backwards heat equation for time $\tau$. However, if one runs the argument with $u$ the backwards evolution of $\phi^2$ starting from time $-\tau$, rather than starting from time 0, one obtains the log-Sobolev inequality for all test functions $\phi$. On the penultimate line, (2.239) should be (2.246).
p. 175In the second line, ${\mathcal M}$ should be $\tilde {\mathcal M}$.
p. ?In equations (2.303)-(2.305), $\sqrt{2N}$ should be $\frac{1}{\sqrt{2N}}$.
p. 191In (2.328), the two $\leq$ signs should both be $\geq$.
p. 205“manifild” should be “manifold”.
p. 213In (2.416) and immediately afterwards, $\nabla_f f$ should be $\nabla_{\nabla f} f$.
p. 217In the sentence after (2.437), “every $v$” should be “every $X$“.
p. 229The formulation of the Hamilton compactness theorem given here needs an additional hypothesis, namely a uniform lower bound on the Ricci curvature. More precisely, for any compact interval J there exists a K such that for every radius r one has $\hbox{Ric} \geq -K$ on $J \times B_{g_n(t_0)}(p_n,r)$ for all sufficiently large n. This is needed to prevent the length of a geodesic going off to infinity from collapsing to a finite length, causing incompleteness. (It was recently shown by Topping that the formulation of the compactness theorem give in the text can fail without such a hypothesis. However, in the applications to the Poincare conjecture one has the uniform lower bound on curvature, so this is ultimately not a major issue.)
p. 234In Corollary 2.16.11, $\frac{d}{d\tau}$ should be $\frac{d}{dt}$.
p. 257Before Proposition 2.18.15, the final “oriented” should be “unoriented”.
p. 263In the sketch of proof of Proposition 2.19.9, $d_{\tilde g_n(0)}(x_n,y_n)$ should be $d_{\tilde g_n(0)}(x_n,\tilde y_n)$.
p. 270“width of the necks goes to infinity” should be “width of the necks goes to zero”.
p. 290The reference [Zhang2007] should be “Zhang, Qi S., Strong noncollapsing and uniform Sobolev inequalities for Ricci flow with surgeries. Pacific J. Math. 239 (2009), no. 1, 179–200”.
Contributors
Thanks to all those who have contributed corrections. Corrections received on or before 2026-07-09 were reported by the readers listed below over the years; individual per-erratum attributions for these legacy entries were not preserved in migration.