Abstract
This note corrects an error in our paper Anosov flows, growth rates on covers and group extensions of subshifts, Invent. Math. 223:445–483, 2021. This leaves our main results, Theorem 1.1, Corollary 1.2, Theorem 1.3 and Theorem 5.1, unchanged. We also fill a gap in the arguments presented in Sect. 9; this requires a small modification to the results in this section.
Avoid common mistakes on your manuscript.
1 Correction to the proof of Theorem 5.1
In [1], the authors mistakenly claim in Lemma 5.2.(i) that
There exists \(C>0\) so that for any \((\eta ,g)\) and \((\xi ,h)\) with \(T^{k}_{\psi}(\eta ,g)=(\xi ,h)\) we have
This is in error as the proof actually shows the opposite inequality. Most significantly, the inequality was then used in the proof Lemma 5.3 (which is crucial to the main result Theorem 5.1). Less significantly, the proof of Theorem 5.1 is written with \(\xi \), \(\eta \) not restricted to the same cylinder, and the inequality in this case was not justified. Both of these errors are corrected by using transitivity of the system. In the beginning of the paper the authors set up the notation for a skew product with \(\psi \) but then proceed to write all the proofs (and the definition of the transfer operator) with respect to a skew product with \(\psi ^{-1}\) – this easily seen to not change the validity of any of the statements. We give a revision of Lemma 5.2 to take into account the case where \(\xi \), \(\eta \) belong to different cylinders. We give a revision of Lemma 5.3. that does not depend on the mistaken claim. In addition show that the original statement claimed in Lemma 5.2(i) does in fact hold under transitivity (although it is no longer required for the proof of Theorem 5.1).
We remind the reader that we define the skew product \(T_{\psi}\) with right multiplication (i.e. \(T_{\psi}(\eta ,g) = (\sigma \eta ,g\psi (\eta ))\)), and so for the skew product with \(\psi \) we have \(T^{k}_{\psi}(\eta ,g)=(\xi ,h)\) if and only if \(\sigma ^{k} \eta = \xi \) and \(g\psi (\eta )\psi (\sigma \eta )\cdots \psi (\sigma ^{k}\eta )=h\). In this way for any letters \(j_{1}\), \(j_{2}\) we have \(T_{\psi}^{p} ([j_{1}]\times \left \{ e\right \}) \cap [j_{2}]\times \left \{ h\right \} \ne \varnothing \) if and only if \(T_{\psi}^{p} ([j_{1}]\times \left \{ g\right \}) \cap [j_{2}]\times \left \{ gh\right \} \ne \varnothing \). Or more precisely \(T^{k}_{\psi}(\eta ,e)=(\xi ,h)\) if and only if \(T^{k}_{\psi}(\eta ,g)=(\xi ,gh)\).
Lemma 5.2.
Corrected version
We have the following:
-
(i)
There exists \(C>0\) so that for any \((\eta ,g)\) and \((\xi ,h)\) with \(T^{k}_{\psi}(\eta ,g)=(\xi ,h)\) we have
$$ \nu _{\eta ,g}(v) \le C^{k} \nu _{\xi ,h}(v). $$ -
(ii)
For any \(\xi \), \(\eta \) in the same length 1 cylinder we have
$$ \nu _{\eta ,g}(v) \le C_{f} \nu _{\xi ,g}(v). $$
Assume in addition that \(T_{\psi}\) is transitive. Then we also have the following:
-
(iii)
There exists \(D>0\) so that for any \(\xi \), \(\eta \) we have
$$ \nu _{\xi ,g}(v) \le D \nu _{\eta ,g}(v). $$
Proof
Parts (i) and (ii) are the statements proved in [1] but with corrected inequality in the statement of (i).
We show (iii). Using transitivity of \(T_{\psi}\), there is \(r\in \mathbb{N}\) so that for any letters \(j_{1}\), \(j_{2}\) there is \(p\le r\) with \(T_{\psi}^{p} ([j_{1}]\times \left \{ e\right \}) \cap [j_{2}]\times \left \{ e\right \} \ne \varnothing \), whence for any \(g\in G\) we have \(T_{\psi}^{p} ([j_{1}]\times \left \{ g\right \}) \cap [j_{2}]\times \left \{ g\right \} \ne \varnothing \).
Let \(\eta \in [a]\) and \(\xi \in [b]\). Let \((\zeta ,e)\in [a]\times \left \{ g\right \} \) with \(T_{\psi}^{p} (\zeta ,g)\in [b]\times \left \{ g\right \}\) and set \(T^{p}_{\psi }(\zeta ,g) = (\zeta ^{\prime}, g)\). Then \(\zeta \), \(\eta \) are in the same length 1 cylinder and \(\zeta ^{\prime }\), \(\xi \) are in the same length 1 cylinder. Part (i) tells us that \(\nu _{\zeta ,g}(v) \ge C^{p} \nu _{\zeta ^{\prime },g}(v)\); and then part (ii) gives
Noting that \(p\) is bounded by \(r\), which is independent of \(g\), gives the result. □
We now give a correct proof of Lemma 5.3. and include the mistaken claim as a consequence.
Lemma 5.3.
Corrected version
Assume \(T_{\psi}\) is transitive. We have the following:
-
(i)
For any \(a\in G\) there is a constant \(M_{a}<\infty \) so that
$$ \sup _{g\in G} \frac{\nu _{o,ga}(v)}{\nu _{o,g}(v)} = M_{a}. $$ -
(ii)
There exists \(L>0\) so that for any \((\eta ,g)\) and \((\xi ,h)\) with \(T^{k}_{\psi}(\eta ,g)=(\xi ,h)\) we have
$$ \nu _{\eta ,g}(v) \ge L^{k} \nu _{\xi ,h}(v). $$
Proof
We begin with part (i). Let \(a\in G\) and let \(o\in \Sigma ^{+}\). Denote \(b\) the first letter of \(o\). So \(o\in [b]\). Since \(T_{\psi}\) is transitive there is \((\eta ,a)\in [b]\times \left \{ a\right \}\) and \(k\) with \(T^{k}_{\psi}(\eta ,a)\in [b]\times \left \{ e\right \}\). Then for any \(g\in G\) we have \((\eta ,ga)\in [b]\times \left \{ ga\right \}\) and \(T^{k}_{\psi}(\eta ,ga)\in [b]\times \left \{ g\right \}\). Set \((\xi ,g)=T^{k}_{\psi}(\eta ,ga)\). We use Lemma 5.2.(i) to say that
and we use Lemma 5.2.(ii) to say that
We have deduced part (i).
We now show part (ii). For brevity we will consider \(\psi \) as being defined on letters.
Let \((\eta ,g)\) and \((\xi ,h)\) with \(T^{k}_{\psi}(\eta ,g)=(\xi ,h)\). Let \(b_{0}\cdots b_{k}\) be the initial \(k\) letters of \(\eta \). Since \(\eta \in [b_{0}\cdots b_{k}]\) we have
and by hypothesis
Therefore \(\xi \in [b_{k}]\) and \(g \psi (b_{0})\psi (b_{1})\cdots \psi (b_{k-1}) = h\). Upon setting \(s_{i} = \psi (b_{i})\) we may write \(g=h s_{0}\cdots s_{k-1}\). We use Lemma 5.2(iii) to say that \(\nu _{\xi ,h}(v) \le D\nu _{\eta ,h}(v)\) and then use Lemma 5.2.(ii) to change to some fixed \(o\) belonging to the same cylinder as \(\eta \), giving
Hence it remains to find a lower bound for \(\frac{\nu _{o,g}(v)}{\nu _{o,h}(v)}\). Now we have
As \(s_{i}\) belong to the bounded set of generators \(S=\left \{\psi (B) : |B|=1 \right \}\) the result follows by setting
and taking the minimum over the finitely many choices of \(o\) (\(M_{s}\) depends on \(o\)). □
2 Corrections for Sect. 9
Section 9 of [1] gives large deviations and and equidistribution theorems for amenable covers. In these results, we consider periodic \(\phi \)-orbits \(\gamma \) satisfying \(T < l(\gamma ) \le T+\epsilon \) for arbitrary \(\epsilon >0\). This needs to be modified so that, in general, we only consider sufficiently large \(\epsilon \), since for small \(\epsilon \) we cannot verify the existence of such \(\gamma \) (despite knowing existence for the compact weak-mixing base). More precisely, Definition 9.1 should require that there exists \(\epsilon _{0}>0\) such that the convergence holds for all \(\epsilon \ge \epsilon _{0}\) and Theorem 9.6 should also be formulated for such \(\epsilon \). Secondly, equation (9.1) of [1] asserts (but does not prove) that (when \(G\) is amenable)
The existence of the limit in (9.1) is crucial for the proof of Theorem 9.6. It is the goal of the remainder of this section to show that the limit exists.
To show the limit exists, rather that continuing directly, we proceed via symbolic dynamics. Let \(\sigma : \Sigma ^{+} \to \Sigma ^{+}\) be a mixing one-sided subshift of finite type with alphabet \(S\) and let \(\psi : \Sigma ^{+} \to G\) satisfy \(\psi (x)=\psi (x_{0},x_{1})\), where \(x = (x_{i})_{i=0}^{\infty}\). Define \(T_{\psi }: \Sigma ^{+} \times G \to \Sigma ^{+} \times G\) by \(T_{\psi}(x,g) = (\sigma x,g\psi (x)) \); we assume this skew product is transitive. Let \(r : \Sigma ^{+} \to \mathbb{R}^{>0}\) be Hölder continuous and define \(r : \Sigma ^{+} \times G \to \mathbb{R}^{>0}\) by \(r(x,g)=r(x)\). Let
where the equivalence relation ∼ is defined by \((x,g, r(x,g)) \sim (T_{\psi}(x,g),0)\). Let \(\upsilon ^{t}\) be the suspension semiflow on \(\Upsilon \), i.e. \(\upsilon ^{t}(x,g,s) = (x,g,s+t)\) mod ∼. For \(a \in S\), set
Lemma 1
The limit
exists.
This lemma follows from arguments in the proof of Theorem 1.1 of [3], using the following lemma, which appears as Lemma 4.3.1 in [2]. (A similar result for sequences was used by Sarig [4].) In [3], the base system is assumed to be mixing but this is not essential.
Lemma 2
Let \(\alpha : \mathbb{R}^{+} \to \mathbb{R}\) be such that there are constants \(c_{1},c_{2} \in \mathbb{R}\) with
-
(1)
\(\alpha (s+t+c_{2}) + c_{1} \ge \alpha (s)+\alpha (t) \), for all \(s,t \ge 0\);
-
(2)
for each \(T>0\), \(\alpha (t)\) is bounded above on the finite interval \([0,T]\).
Then \(\lim _{t \to \infty} \alpha (t)/t\) exists (in \((-\infty ,+\infty ]\)).
Hence Lemma 1 follows from Lemma 2 with \(\alpha (t) = \log N_{a}(t,\epsilon )\). Note that taking \(\epsilon \ge \epsilon _{0}\) ensures that \(N_{a}(t,\epsilon ) >0\) and thus that \(\alpha (t)\) is well-defined, at least for large values of \(t\). (Kempton does not state the second condition but it is needed in the proof, and it is clearly satisfied in our application.) Standard arguments then give the existence of the limit
where \(W\) is the image of \([a] \times \{e\} \times (-\eta ,\eta )\) in the symbolic coding of \(\phi _{X}\). It is also straightforward to compare with orbits in
We conclude that for \(\epsilon \ge \epsilon _{0}\),
exists.
References
Dougall, R., Sharp, R.: Anosov flows, growth rates on covers and group extensions of subshifts. Invent. Math. 223, 445–483 (2021)
Kempton, T.: Thermodynamic formalism for symbolic dynamical systems. PhD thesis, University of Warwick (2011)
Kempton, T.: Thermodynamic formalism for suspension flows over countable Markov shifts. Nonlinearity 24, 2763–2775 (2011)
Sarig, O.: Thermodynamic formalism for countable Markov shifts. Ergod. Theory Dyn. Syst. 19, 1565–1593 (1999)
Acknowledgement
The authors are grateful to Zijie Lin for pointing out the errors discussed in Sect. 1.
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Dougall, R., Sharp, R. Correction to “Anosov flows, growth rates on covers and group extensions of subshifts”. Invent. math. 236, 1505–1509 (2024). https://doi.org/10.1007/s00222-024-01251-7
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DOI: https://doi.org/10.1007/s00222-024-01251-7