CORRIGENDUM AND ADDENDUM TO: ON RANDOM
TOPOLOGICAL MARKOV CHAINS WITH BIG IMAGES AND
PREIMAGES
MANUEL STADLBAUER
Abstract. We give a corrected statement and extended proof of theorem 4.1
in [2] and an alternative proof corollary 4.1 which is independent from the
wrong statement in the theorem.
Corrigendum 1. Lines 8, p. 89 – 6, p. 90 should read as follows.
Theorem 4.1. Let (X, T, φ) be a topologically mixing system of divergence type with
(H2), S(1-2) and PG (φ) > −∞. Then there exists a random probability measure
{µω } such that µω is positive on cylinders for a.e. ω ∈ Ω and, for x ∈ Xω ,
ω
dµθω ◦ Tω
(x) = λ(ω)−1 ePG (φ)−φ (x) .
dµω
Proof. By substituting φ with φ − PG (φ), we may assume without loss of generality
that PG (φ) = 0. Let (tk : k ∈ N) be the sequence given by Lemma 4.1. By Lemma
4.2, there exists a subsequence (sn : n ∈ N) of (tk ) and a random probability
measure {µω : ω ∈ Ω} which is the limit of {µω,sn }. For k ∈ N, a ∈ Wωk and
A ⊂ [a]ω such that A = [ab]ω for some b ∈ Wθ∗k ω , it follows as in the proof of
proposition 6.3 in [1] that
µω,s (A) =
+
1
Pω (s)
k
X
sk eφω (x)
e
k∈Jω (Ω),k≤n
k ∩A
x∈Eω
Pθk ω (s)
1
Pω (s) Pθk ω (s)
X
ω
θn ω
sk eφn (τa (x)) eφk−n (x) .
e
n∈Jω (Ω),k>n,
k−n
n (A)
x∈E n ∩Tω
θ ω
Observe that the first summand tends to zero when s → ρ. Hence, for B ⊂ Ωa , we
obtain by integration that
Z
Z
µω (A)dP (ω)
=
B
=
lim µω,sn (A)dP (ω)
Z
Z
ω
Pθk ω (sn )
lim
eφn (τa (x)) dµθk ω,sn (x)dP (ω)
n→∞ B Pω (sn )
Tωk (A)
B n→∞
1
2
MANUEL STADLBAUER
By considering a subset of B, we may assume that Mω Mθω · · · Mθk−1 ω is uniformly
bounded from above on B. Then,
Z
Z
Z
ω
(?) :=
Λk (ω)
eφn ◦τa dµθk ω dP −
µω (A)dP
Tωk (A)
B
Z =
≥
B
Z
ω
Pθk ω (sn )
− Λk (ω)
eφn ◦τa dµθk ω dP
P
(s
)
k
ω n
Tω (A)
B
Z
Z
ω
Pθk ω (sn )
eφn ◦τa d(µθk ω,sn − µθk ω )dP.
+ lim
n→∞ B Pω (sn )
Tωk (A)
Z Z
ω
Pθk ω (sn )
lim
log
Λk (ω)
eφn ◦τa dµθk ω dP = 0.
n→∞ B
Pω (sn )Λk (ω)
Tωk (A)
lim
n→∞
By the same argument, it follows that (?) ≤ 0. Now observe that, since
R X is a subset
of the product space NN × Ω, the measure m defined by m(A) := µθω (Tω (A))dP
for sets A such that T |A is one-to-one can be disintegrated with respect to P .
Hence, by uniqueness of the Radon-Nikodym derivative, it follows that
ω
dµθk ω ◦ Tωk
(x) = e−φk (x) Λk (ω)−1 .
dµω
In order to show that {µω } is positive on cylinders, choose a ∈ W and Ω0 ⊂ Ωa
of positive measure such that µω ([a]ω ) > 0 for all ω ∈ Ω0 . It follows, for a.e.
ω ∈ Ω, n ∈ N and w ∈ Wωn , that there exists k > n such that θk ω ∈ Ω0 and
Tωk ([w]) ⊃ [a]θk ω . In particular, since dµθk ω ◦ Tωk /µω (x) > 0 for all x ∈ [w]ω , we
have µω ([w]) > 0.
n→∞
Remark. The theorem no longer states that Pω (sn )/Pθω (sn ) −−−−→ λ(ω). Since the
statement also follows from a stronger result in the forthcoming paper [3], its proof
will be omitted. Since the convergence was used in the proof of corollary 4.1, we
give an independent and much simpler proof of that result.
Corrigendum 2. Lines 1–22, p. 91 should read as follows.
P
Proof. Since w∈Wωn µω ([w]) = 1, it follows from remark 4.2 and the definition of
n
Aω
n in [2] that, for a.e. ω and n ∈ N with θ ω ∈ Ωbi ,
P
φω
n ([w])
n e
Aω
w∈Wω
n
−1
Bθ n ω ≤
=
≤ Bθn ω Dθ−1
n (ω) .
Λn (ω)enPG (φ)
Λn (ω)enPG (φ)
Corollary 4.1 in [2] is then a consequence of Lemma 3.2 in there.
References
[1] M. Denker, Y. Kifer, and M. Stadlbauer. Thermodynamic formalism for random countable
Markov shifts. Discrete Contin. Dyn. Syst., 22 (2008) 131–164.
[2] M. Stadlbauer. On random topological Markov chains with big images and preimages, Stoch.
Dyn. 10 (2010) 77–95.
[3] M. Stadlbauer. Coupling methods for random topological markov chains. Preprint in arxiv,
2013.
Departamento de Matemática, Universidade Federal da Bahia, Av. Ademar de
Barros s/n, 40170-110 Salvador, BA, Brasil., [email protected]