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Alves-Oliveira-Tahzibi
ON THE CONTINUITY OF ENTROPY FOR
NON-UNIFORMLY EXPANDING MAPS
JOSÉ F. ALVES
Centro de Matemática da Universidade do Porto
Rua do Campo Alegre, 687
4169-007 Porto, Portugal
E-mail: [email protected]
KRERLEY OLIVEIRA
Departamento de Matemática
Universidade Federal de Alagoas
Campus A. C. Simões, s/n
57000-000 Maceió, Brazil
E-mail: [email protected]
ALI TAHZIBI
Departamento de Matemática
ICMC-USP São Carlos
Caixa Postal 668
13560-970 São Carlos, Brazil
E-mail: [email protected]
Let f : M → M of a compact Riemannian manifold M. If an ergodic
absolutely continuous f -invariant probability measure µ exists, then by
Birkhoff’s Ergodic Theorem it has the so-called SRB property: for a positive
Lebesgue measure (Leb) subset of points in x ∈ M
Z
n−1
1X
lim
φ(f i (x)) = φ dµ, for every φ ∈ C 0 (M ).
n→∞ n
j=0
It is a difficult problem to verify the existence of these measures for general Dynamical Systems. For Axiom A diffeomorphisms and flows, and
uniformly expanding endomorphisms classical results by Sinai13 , Ruelle11,8
and Bowen7,8 prove the existence of SRB measures. In this context the
continuous variation of the SRB entropy is known. This follows easily from
the fact that it coincides with the integral of the Jacobian with respect to
1
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the SRB measure, and this measure varies continuously with the map, by
Ruelle12 .
Alves, Bonatti & Viana3 construct SRB measures for a large class of
diffeomorphisms and endomorphisms with a weak form of hyperbolicity.
In this work we are interested in studying the continuous variation of the
SRB entropy of these endomorphisms. We present an abstract model and
give the necessary conditions which imply the continuous variation of the
entropy. Using some recent results we verify that certain classes of nonuniformly expanding endomorphisms satisfy our conditions.
1. Piecewise expanding Markovian induced maps
Let f be a map from a manifold M into itself, and consider the Lebesgue
measure Leb on M . Let F : ∆ → ∆ be an induced map for f : there are
a countable partition P of a full Leb subset of ∆ ⊂ M , and a return time
function τ : P → Z+ such that F |ω = f τ (ω) |ω for each ω ∈ P. Assume that
F : ∆ → ∆ is a piecewise expanding Markovian map:
(c1 ) Markov: F |ω : ω → ∆ is a C 2 diffeomorphism, for each ω ∈ P.
(c2 ) Expansion: there is 0 < σ < 1 such that for ω ∈ P and x ∈ ω
kDF (x)−1 k < σ.
(c3 ) Bounded distortion: there is K > 0 such that for ω ∈ P and x, y ∈ ω
¯
¯
¯ det DF (x)
¯
¯
¯
¯ det DF (y) − 1¯ ≤ K dist(F (x), F (y)).
A piecewise expanding Markovian map F has a unique ergodic absolutely continuous invariant probability measure µF . Moreover, µF is equivalent to Leb | ∆, and its density is bounded from above and from below
by constants. If the return time τ : ∆ → Z+ is integrable with respect to
Leb | ∆, then
µ∗f =
∞
X
f∗j (µF | {τ > j})
(1)
j=0
is an absolutely continuous f -invariant finite measure. We denote by µf
the probability measure which consists in dividing µ∗f by its mass.
Theorem 1.1. If F is an induced Markov map for f , then
hµf (f ) =
1
hµ (F ).
µ∗f (M ) F
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2. Continuity of entropy
Let F be a family of C k maps (k ≥ 2) from M into itself endowed with
the C k topology. Assume that we may associate to each f ∈ F an induced
map Ff : ∆ → ∆. Consider the following uniformity conditions:
(u1 ) τf varies continuously in the L1 norm with f ∈ F.
(u2 ) σ, K associated to Ff in (c2 )-(c3 ) may be chosen uniformly in F .
Alves & Viana6 show that if (u1 )-(u2 ) hold, then the density of the measure
µf varies continuously with f ∈ F in the L1 norm.
Theorem 2.1. If (u1 )-(u2 ) hold in F, then hµf (f ) varies continuously
with f ∈ F.
For the proof of this theorem we need the following consequence of
results by Qian & Zhu10 and Liu9 .
Theorem 2.2. If f : M → M is a C 2 endomorphism with an absolutely
continuous invariant measure µ, then
Z
hµ (f ) =
log | det Df (x)|dµ.
M
A similar result holds for piecewise expanding Markovian induced maps;
see Section 4.
3. Non-uniformly expanding maps
Let f : M → M be a C 2 local diffeomorphism except possibly at a critical
set C ⊂ M near which f behaves like a power of the distance to C; see the
precise definition in the work of Alves, Bonatti & Viana3 . We say that
f : M → M is non-uniformly expanding if there is λ > 0 such that for
Lebesgue almost every x ∈ M
lim sup
n→∞
n−1
1X
−1
log kDf (f i (x)) k < −λ.
n i=0
(2)
We say that orbits have slow recurrence to C if for every ² > 0 there exists
δ > 0 such that for Lebesgue almost every x ∈ M
n−1
1X
lim sup
− log distδ (f j (x), C) ≤ ².
n→+∞ n
j=0
(3)
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Given δ > 0 we define the δ-truncated distance from x to C as distδ (x, C) = 1
if dist(x, C) ≥ δ; and distδ (x, C) = dist(x, C), otherwise. Condition (2)
implies that the expansion time function
(
)
n−1
1X
λ
−1
E(x) = min N ≥ 1 :
log kDff i (x) k ≤ − , for all n ≥ N
n i=0
2
is defined and finite almost everywhere in M . The recurrence time function
)
(
n−1
1X
− log distδ (f j (x), C) ≤ 2ε, for all n ≥ N ,
R(x) = min N ≥ 1 :
n i=0
is also finite almost everywhere in M by (3). We define the tail set
©
ª
Γn = x ∈ M : E(x) > n or R(x) > n .
(4)
The decay on Leb(Γn ) plays a crucial role in the results we present below.
The next result was proved by Alves, Luzzatto & Pinheiro4 .
Theorem 3.1. Let f : M → M be a transitive non-uniformly expanding
map. If there is γ > 0 such that Leb(Γn ) ≤ O(n−γ ), then f has a piecewise
expanding Markovian induced map with Leb{τ > n} ≤ O(n−γ ).
We say that F as above is a uniform family if the constants λ, ε and δ
can be chosen uniformly in F. Alves2 proves the result below for uniform
families of non-uniformly expanding maps.
Theorem 3.2. Let F be a uniform family of C k (k ≥ 2) transitive nonuniformly expanding maps, for which there are C > 0, γ > 1 such that
Leb(Γfn ) ≤ Cn−γ for all n ≥ 1 and f ∈ F . Then (u1 )-(u2 ) hold in F.
As a consequence of these last two theorems and Theorem 2.1 we obtain:
Corollary 3.1. Under the assumptions of the previous theorem hµf (f )
varies continuously with f ∈ F.
3.1. Example: Viana maps
Let a0 ∈ (1, 2) be such that the critical point x = 0 is pre-periodic under
iteration by the quadratic map p(x) = a0 − x2 . Let b : S 1 → R be a Morse
function, for instance, b(t) = sin(2πt). For each α > 0, we define
fα : S 1 × R −→
S1 × R
(θ, x) 7−→ (ĝ(θ), q̂(θ, x))
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where ĝ is the uniformly expanding map ĝ(θ) = dθ (mod 1), d ≥ 2, and
q̂(θ, x) = a(θ) − x2 , with a(θ) = a0 + αb(θ). We take V as a small C 3
neighborhood of fα , with sufficiently small α > 0. Viana14 proves the
following result for the family V:
Theorem 3.3. V is a uniform family of non-uniformly expanding maps
whose orbits have slow recurrence to
the critical set. Moreover, there are
√
f
−γ n
C, γ > 0 such that Leb(Γn ) ≤ Ce
for all n ≥ 1 and f ∈ V.
It follows from the next result, which has been proved by Alves1,6 and
Viana6 , that the maps in V are transitive.
Theorem 3.4. Every f ∈ V is topologically mixing and has a unique ergodic absolutely continuous invariant (SRB) measure. Moreover, the density of the SRB measure varies continuously in the L1 norm with f ∈ V.
These last two theorems together with Corollary 3.1 yield:
Corollary 3.2. The metric entropy (wrt the SRB measure) of f ∈ V varies
continuously with f ∈ V.
4. Main ingredients
In this section we present the main ingredients in the proofs of Theorem 1.1
and Theorem 2.1. Detailed proofs will appear in5 .
Statistical stability. The first ingredient is the continuity of the absolutely continuous invariant measure for the induced map:
(1) If (u2 ) holds, then the measure µF of the induced map Ff varies
continuously in the L1 -norm with f ∈ F.
Entropy formulas. The next ingredients concern an entropy formula for
the induced map, and a formula relating the entropy of the induced map
to the entropy of the original dynamics.
(2) If F is a piecewise expanding Markovian induced map, then
Z
hµF (F ) =
log JF dµF .
∆
(3) If F is a piecewise expanding Markovian induced map for f , then
Z
Z
log JF dµF =
log Jf dµ∗f .
∆
M
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Here Jf are JF the Jacobians of f and F respectively. From (2) and (3)
one has
Z
Z
1
1
∗
log Jf dµf = ∗
log JF dµF .
(5)
hµf (f ) = ∗
µf (M ) M
µf (M ) ∆
The previous equality gives content of Theorem 1.1.
Continuity. Finally we have the continuity of the last two factors in (5).
(4) If (u1 ) holds, then µ∗f (M ) varies continuously with f ∈ F.
Z
log JF dµF varies continuously with f ∈ F.
(5) If (u1 ) holds, then
This is enough for concluding the proof of Theorem 2.1.
References
1. J. F. Alves, SRB measures for non-hyperbolic systems with multidimensional expansion, Ann. Scient. Éc. Norm. Sup., 4e série, 33 (2000), 1-32.
2. ———–, Strong statistical stability of non-uniformly expanding maps,
preprint CMUP 2003.
3. J. F. Alves, C. Bonatti, M. Viana, SRB measures for partially hyperbolic
systems whose central direction is mostly expanding, Invent. Math. 140
(2000), 351-398.
4. J. F. Alves, S. Luzzatto, V. Pinheiro, Markov structures and decay of
correlations for non-uniformly expanding maps, preprint CMUP 2002.
5. J. F. Alves, K. Oliveira, A. Tahzibi, Continuity of entropy via induced
maps, in preparation.
6. J. F. Alves, M. Viana, Statistical stability for robust classes of maps with
non-uniform expansion, Ergod. Th. & Dynam. Sys. 22 (2002), 1-32.
7. R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics 470 (1975), Springer.
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29 (1975), 181-202.
9. P.-D. Liu, Pesin’s entropy formula for endomorphisms, Nagoya Math. J.
150 (1998), 197-209.
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11. D. Ruelle, A measure associated with Axiom A attractors, Amer. Jour.
Math. 98 (1976), 619-654.
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(1972), 21-69.
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85 (1997), 63-96.