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Algebraic Polymorphisms– ERRATUM

KLAUS SCHMIDT and ANATOLY VERSHIK

Ergodic Theory and Dynamical Systems / Volume 29 / Issue 04 / August 2009, pp 1369 - 1370DOI: 10.1017/S0143385709000352, Published online: 26 June 2009

Link to this article: http://journals.cambridge.org/abstract_S0143385709000352

How to cite this article:KLAUS SCHMIDT and ANATOLY VERSHIK (2009). Algebraic Polymorphisms– ERRATUM.Ergodic Theory and Dynamical Systems, 29, pp 1369-1370 doi:10.1017/S0143385709000352

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Ergod. Th. & Dynam. Sys. (2009), 29, 1369–1370 c© 2009 Cambridge University Pressdoi:10.1017/S0143385709000352 Printed in the United Kingdom

ERRATUM

Algebraic Polymorphisms – ERRATUM

KLAUS SCHMIDT and ANATOLY VERSHIK

doi:10.1017/S0143385707001022, Published by Cambridge University Press,24 March 2009.

We give three corrections to the paper [K. Schmidt and A. M. Vershik, Algebraicpolymorphisms, Ergod. Th. & Dynam. Sys. 28 (2008), 633–642].

(1) The statement in the penultimate paragraph on [1, p. 634] has to be corrected asfollows: if π1 is an injection then P is (the graph of) an endomorphism, and if π2 is aninjection then P is (the graph of) an exomorphism.

In other words, the symbols π1 and π2 should be interchanged.(2) [1, Corollary 1.7] is incorrect as stated. The correct statement should be the

following.

COROLLARY. Let P⊂ G × G be a correspondence and let H ⊂ G be a closed normalsubgroup. We denote by K (i)

Pn , i = 1, 2, the closed normal subgroups of G associated with

the correspondence Pn, n ≥ 2, in (1.4) by (1.9). The sequences of subgroups (K (i)Pn , n ≥ 1)

are non-decreasing, and we write H (i)0 =

⋃n≥1 K (i)

Pn for the closure of⋃

n≥1 K (i)Pn . Then

the following holds.(1) H (2)

0 is smallest invariant subgroup of 5P.

(2) H (1)0 is the smallest co-invariant subgroup of 5P.

Proof. By definition, ηP(K(1)P K (2)

Pn )= K (2)Pn+1 for all n ≥ 1.

If a closed normal subgroup H ⊂ G is invariant under 5P then [1, Theorem 1.6(1)]shows that K (2)

P2 = ηP(K(1)P K (2)

P )⊂ ηP(K(1)P H)⊂ H . Hence

K (2)P3 = ηP(K

(1)P K (2)

P2 )⊂ ηP(K(1)P H)⊂ H

and, by induction, K (2)Pn ⊂ H (2)

0 ⊂ H for every n ≥ 1.

In order to verify that H (2)0 is invariant under 5P we note that

K (2)Pn+1 = ηP(K

(1)P K (2)

Pn )⊂ H (2)0 for every n ≥ 1,

http://journals.cambridge.org Downloaded: 01 Aug 2014 IP address: 134.153.184.170

1370 K. Schmidt and A. Vershik

and by letting n→∞we see that ηP(K(1)P H (2)

0 )⊂ H (2)0 . According to [1, Theorem 1.6(1)]

this proves that H (2)0 is invariant.

The proof of the second assertion is analogous. 2

(3) The third correction concerns the semigroup P f (Tm) of all finite-to-onecorrespondences of Tm . Denote by L the semigroup of all finite index subgroups of Zm

with intersection as composition (and not, as stated wrongly in [1, p. 637], with addition).We consider the semigroup

M= {(Q, 3) | Q ∈ GL(m,Q), 3 ∈ L, 3⊂3Q := Zm∩ QZm

}, (1)

with composition(Q, 3) · (Q′, 3′)= (Q Q′, 3 ∩ Q3′), (2)

where we again have replaced addition by intersection.This correction does not affect any of the results or proofs in that section.

Acknowledgement. The authors are grateful to Alexei Levin for pointing out these errors.

REFERENCES

[1] K. Schmidt and A. M. Vershik. Algebraic polymorphisms. Ergod. Th. & Dynam. Sys. 28 (2008), 633–642.