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Lyapunov exponents and rates of mixing for one-dimensional maps
JOSÉ F. ALVES, STEFANO LUZZATTO and VILTON PINHEIRO
Ergodic Theory and Dynamical Systems / Volume 24 / Issue 03 / June 2004, pp 637 - 657
DOI: 10.1017/S0143385703000579, Published online: 04 May 2004
Link to this article: http://journals.cambridge.org/abstract_S0143385703000579
How to cite this article:
JOSÉ F. ALVES, STEFANO LUZZATTO and VILTON PINHEIRO (2004). Lyapunov exponents and rates of mixing for one-
dimensional maps. Ergodic Theory and Dynamical Systems, 24, pp 637-657 doi:10.1017/S0143385703000579
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Ergod. Th. & Dynam. Sys. (2004), 24, 637–657
c
2004 Cambridge University Press
DOI: 10.1017/S0143385703000579 Printed in the United Kingdom
Lyapunov exponents and rates of mixing
for one-dimensional maps
JOS
´
E F. ALVES†, STEFANO LUZZATTO‡ and VILTON PINHEIRO§
† Departamento de Matem
´
atica Pura, Faculdade de Ci
ˆ
encias do Porto,
Rua do Campo Alegre 687, 4169-007 Porto, Portugal
(e-mail: jfalves@fc.up.pt)
‡ Mathematics Department, Imperial College, 180 Queen’s Gate, London SW7, UK
(e-mail: stefano.luzzatto@ic.ac.uk)
§ Departamento de Matem
´
atica, Universidade Federal da Bahia,
Av. Ademar de Barros s/n, 40170-110 Salvador, Brazil
(e-mail: viltonj@ufba.br)
(Received 13 February 2003 and accepted in revised form 2 September 2003)
Abstract. We show that one-dimensional maps f with strictly positive Lyapunov exponents
almost everywhere admit an absolutely continuous invariant measure. If f is topologically
transitive, some power of f is mixing and, in particular, the correlation of H¨older
continuous observables decays to zero. The main objective of this paper is to show that the
rate of decay of correlations is determined, in some situations, by the average rate at which
typical points start to exhibit exponential growth of the derivative.
1. Introduction and statement of results
1.1. Lyapunov exponents. The purpose of this paper is to study the statistical properties
of one-dimensional maps f : I → I with positive Lyapunov exponents, where I may be
the circle S
1
or an interval. Such maps satisfy asymptotic exponential estimates for growth
of the derivative, bu t do not necessarily exhibit exponential estimates for other features of
the dynamics such as the decay of correlations. Our main objective here is to identify some
criterion which distinguishes different degrees of expansivity and which is reflected in the
rate of decay of correlations of the system.
Definition 1 . We say that a map f : I → I has positive Lyapunov exponents almost
everywhere if there exists some λ>0suchthat
lim inf
n→∞
1
n
log |(f
n
)
(x)|≥λ>0(∗)
for Lebesgue almost every point x ∈ I .
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638 J. F. Alves et al
1.2. Decay of correlations. Positive Lyapunov exponents are known to be a cause of
sensitive dependence on initial conditions and other dynamical features which give rise to
adegreeofchaoticity or stochasticity in the dynamics. We can formalize this idea through
the notion of mixing with respect to some invariant measure.
Definition 2 . A probability measure µ defined on the Borel sets of I is said to be
f -invariant if µ(f
−1
(A)) = µ(A) for every Borel set A ⊂ I .
Definition 3 . Amapf is said to be mixing with respect to some f -invariant p robability
measure µ if
|µ(f
−n
(A) ∩ B) −µ(A)µ(B)|→0, when n →∞,
for any measurable sets A, B.
One interpretation of this property is that the conditional probability of B given f
−n
(A),
i.e. the probability that the event A is a consequence of the event B having occurred at
some time in the past, is asymptotically the same as if the two events were completely
independent. This is sometimes referred to as a property of loss of memory, and thus
in some sense of stochasticity, of the system. A natural question of interest both for
application and for intrinsic reasons, therefore, is the speed at which such loss of memory
occurs. Standard counterexamples show that, in general, there is no specific rate: it is
always possible to choose sets A and B for which mixing is arbitrarily slow. However, this
notion can be generalized in the fo llowing way.
Definition 4 . For a map f : I → I preserving a probab ility measure µ and functions
ϕ,ψ ∈ L
1
(µ),wedefinethecorrelation function
C
n
= C
n
(ϕ, ψ) =
(ϕ ◦ f
n
)ψ dµ −
ϕdµ
ψdµ
.
Note that choosing these observables to be characteristic functions of Borel sets g ives
the well-known definition of mixing when C
n
→ 0. By restricting the set of allowed
observables, for example to the class of H¨older continuous functions, it is sometimes
possible to obtain specific upper bounds for the decay of the correlation function C
n
which
depend only on the map f (up to a multip licative constant which is allowed to depend
on ϕ and ψ). Indeed, it is generally possible to allow ψ ∈ L
∞
(µ) and only restrict the
choice of ϕ to the class of H¨older continuous functions. We make several statements below
concerning the decay of correlations for H¨older continuous observables meaning that ϕ is
H¨older continuous and ψ ∈ L
∞
(µ).
Not more than a decade ago, the only examples for which a specific rate of decay
of correlations was known were uniformly expanding maps in one dimension and, more
generally, uniformly hyperbolic systems in higher dimensions [14–16, 36]. In these cases
we always have exponential decay of correlations for H¨older continuous observables, an
estimate which can be expected due to the fact that essentially everything is exponential in
these cases. Relaxing uniform expansion conditions has proved extremely hard and until
now estimates on the decay for correlations for one-dimensional systems satisfying the
asymptotic exponential expansion condition (∗) but strictly not uniformly expanding have
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Lyapunov exponents and rates of mixing 639
only been known in some fairly specific classes o f examples such as maps with indifferent
fixed points [11, 12, 15, 24] or non-flat critical points [5, 6, 22] where the r ate o f d ecay
depends qu ite explicitly on the features of th e neutral fixed point and the critical points,
respectively. See [18] for a more detailed discussion and references.
1.3. Degree of non-uniformity of the expansion. A natural question is what general
characteristics of a system satisfying (∗) determine the rate of decay of correlations?
The purpose of the present paper is to give a partial solution to this question by relating the
rate of decay of correlations to the degree of non-uniformity of the expansivity, i.e. the time
we have to wait for typical points to start behaving as though the system were uniformly
expanding.
Definition 5 . For 0 <λ
<λ,wedefinetheexpansion time function E by
E(x) = min
N :
1
n
log |(f
n
)
(x)|≥λ
, ∀n ≥ N
.
Condition (∗) implies th at E is defined and finite almost everywhere. We think of this as
the waiting time before the exponential derivative growth kicks in. A map is uniformly
expanding if E(x) is uniformly bounded for every x. In general, however, E(x) is defined
only almost everywhere and unbounded, indicating that some points may exhibit no growth
or even arbitr arily large lo ss of growth of the derivative along their orbit for an arbitrarily
long time before starting to exhibit the exponential growth implied by condition (∗).
Our resu lts show that at least in certain situations th e pr operties of the function E (x) are
closely related to the rate of mixing of the map.
1.4. Local diffeomorphisms. We begin with the simplest situation in order to highlight
the main idea of the results.
T
HEOREM 1. Let f : I → I be a C
2
local diffeomorphism with some point having dense
pre-orbit. Suppose that f satisfies condition (∗) and that there exists γ>1 such that for
some 0 <λ
<λwe have
|{E(x) > n}| ≤ O(n
−γ
).
Then there exists an absolutely continuous, f -invariant, probability measure µ on I.
Some finite power of f is mixing with respect to µ and the correlation function C
n
for
H
¨
older continuous observables on I satisfies
C
n
≤ O(n
−γ +1
).
We emphasize here that the asymptotic statements here do not depend on the choice
of λ
. Thus the rate of decay of correlations depends essentially on the average time with
which some given uniform exponential rate of expansion is attained and does not depend
on what turn out to be significantly more subtle characteristics of the system such as the
actual rate o f convergence of the Lyapunov exponents to the limit.
Remark 1. The absolute continuity and ergodicity of µ follow from [2]. Our argument
gives an alternative proof of the absolute continuity of µ and allows us to obtain the
estimates on the rate of decay of correlations which are the main purpose of this p aper.
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640 J. F. Alves et al
Remark 2. The statement about the rate o f decay of the correlations is of interest even if
an absolutely continuous mixing f -invariant, probability measure µ is given to begin with.
Then condition (∗) is just equivalent to the integrability condition
log |f
|dµ > 0,
since Birkhoff’s ergodic theorem then implies that the limit
λ = lim
n→∞
1
n
log |(f
n
)
(x)|= lim
n→∞
1
n
n−1
i=0
log |f
(x)|=
log |f
|dµ > 0
exists for µ-almost every x ∈ I . In particular, the expansion time function E (x) is also
defined and finite almost everywhere and the conclusions of the theorem hold under the
given conditions on the rate of decay of |{E(x) > n}|.
1.5. Multimodal maps. We can generalize our result to C
2
maps with non-flat critical
points if we assume that almost all orbits have slow approximation to the critical set C.Let
dist
δ
(x, C) denote the δ-truncated distance from x to C defined as dist
δ
(x, C) = dist(x, C)
if dist(x, C) ≤ δ and dist
δ
(x, C) = 1otherwise.
Definition 6 . We sa y that a map f : I → I , with a critical set C, satisfies the slow
recurrence condition if given any >0 there exists δ>0 such that for Lebesgue almost
every x ∈ I
lim sup
n→+∞
1
n
n−1
j=0
−log dist
δ
(f
j
(x), C) ≤ . (∗∗)
Condition (∗∗) is an asymptotic statement, just like condition (∗), and we have no a
priori knowledge about how fast this limit is approached or with what degree of uniformity
for different points x. Thus we introduce the analogue of the expansion time function as
follows.
Definition 7 . The recurrence time function is d efined b y
R(x) = min
N ≥ 1 :
1
n
n−1
i=0
−log dist
δ
(f
j
(x), C) ≤ 2, ∀n ≥ N
.
Condition (∗∗) implies that R is well defined and finite almost everywhere in I .
For stating our results we also need the following.
Definition 8 . For each n ≥ 1definethetail of non-uniformity by
n
={x : E (x) > n or R(x) > n}
This is the set of points which at time n have not yet achieved either the uniform
exponential growth or the uniform slow approximation to C given by (∗) and (∗∗).