Journal
of Applied Mathematics and Stochastic Analysis,
13:1
(2000),
33-40.
SMOOTHNESS OF INVARIANT DENSITY FOR
EXPANDING TRANSFORMATIONS
IN HIGHER DIMENSIONS
KOUROSH ADL-ZARABI
Dawson College, Department of Mathematics
300 Sherbrooke St. West, Montreal, Canada H3Z 1A
Concordia University, Department of Mathematics and Statistics
7141 Sherbrooke St. West, Montreal, Canada It4B
E-mail: kourosh@cicma.concordia.ca
HARALD PROPPE
Concordia University, Department of Mathematics and Statistics
7141 Sherbrooke St. West, Montreal, Canada H4B 1R5
E-mail: proppe@alcor.concordia.ca
_
(Received August, 1998; Revised September, 1999)
n
Let be a bounded region in
and let 9- {Pi}im= 1 be a partition of
into a finite number of closed subsets having piecewise C 2 boundaries of
finite (n-1)-dimensional measure. Let 7:Q-Q be an expanding transformation on where, v
v] Pi and 7 E C rn 2. We show that the
v-invariant density h C M -2.
Key words: Absolutely Continuous Invariant Measure, Ergodic, Expanding Transformation, Perron-Frobenius Operator.
AMS subject classifications: 28D05, 58Fll.
M,
1. Introduction
There has been a recent surge of interest in the study of existence and properties of
absolutely continuous invariant measures (acim) of higher dimensional transformations. Let
be a bounded region in Nn and let z) {P i} TMi=I be a partition of into
a finite number of subsets having piecewise C 2 boundaries of finite (n- 1)-dimensional measure. Let r:
be piecewise C 2 on z2 where ’i v[ p. is a C 2 diffeomorphism onto its image and expanding in the sense that there exists c > 1 such that for
any
1, 2,..., rn, Dr[- E < c- 1, where D7- 1 is the derivative matrix of 7/- 1
and
E is the Euclidean matrix norm. Then, under general conditions [1], it has
been shown that r has an acim, which is a generalization of the results proved in [11,
12, 5] and [8]. We are then interested in properties associated with the acim. The
II" I
Printed in the U.S.A.
I
I
(C)2000 by North
Atlantic Science Publishing Company
33
KOUROSH ADL-ZARABI and HARALD PROPPE
34
properties of interest include" number of ergodic acim [9, 2], uniqueness [4, 21], stability [13, 6] and the smoothness of their invariant densities.
It is important to know which properties the invariant density, if one exists,
inherits from its underlying transformation. For Lasota-Yorke maps [15] on [0,1],
Szewc [22] proved that the densities of invariant measures for Lasota-Yorke maps of
class C M are of class C M- 1. It should be noted, however, that these smoothness
properties are assumed to hold only piecewise that is, relative to a partition of
The smoothness of the invariant density in Szewc’s result is actually piecewise-smoothness relative to another partition that is obtained from the given one through refinement with all of its forward images. Thus in most cases (with the exception of some
simple classes of maps, such as Markov maps) the underlying partition for the piecewise smoothness of the invariant density has (possibly infinitely) many more elements
than the original one.
In this paper, we investigate the smoothness of invariant densities of acim in
higher dimensions for a subset of Lasota-Yorke maps. We prove that if a transformation is expanding in the sense of the maps considered by Man [16] and of class C M,
then its invariant density is of class C M- 2. It should be noted, however, that this
somewhat weaker smoothness result is valid on the original partition. We conjecture
that this is the sharpest result possible for the given partition. In dimension one, the
smoothness of the invariant density of an acim for transformations on an interval considered by Rnyi [18] was established by Halfant [10]. An alternate proof of this has
been presented in [17]. Some applications of expanding maps in a number theoretical
context can be found in [20].
.
2. Definitions and Conditions
We denote by ,k, the Lebesgue measure
M k we define its norm by I Mk I
on
n.
For a n x n x... x n (k-times) array
maxNkl(m)il...ikl, where
Nk
{ili2...ik: 1 <_ ij <_ n for
1
D(M)f
For a real-valued function f:n_ we denote by Df its derivative, and by
the M-th derivative of f. If f(x)= f(xl,x2,...,xn) then (Df) x is a linear map:
,
n
Nn-- and (Df)x(v I v2, Vn
i=1
VOxi
Let
{pi}n= 1 be a partition of f into a finite number of subsets having piecewise C 2 boundaries of finite (n-1)-dimensional measure. Henceforth we work
We use the notation vt-k to
mainly with the open domain fo-f\Ji
3oOPi"
mean a specific composition of inverses of the form
ij G {1, 2,..., m}
(we
1
1
v/--k o vik_l- o... o v/-1
where
_< j _< k,
and denote by 1t k the set of subscripts for which
as subscript here to distinguish between
v
v p. and the composition of inverses of such maps as described.) Thus for any
onto its image. Of
is a C M diffeomorphism of A
open set 4, each
course, r- k(A) U e
k(A)"
each
for 1
-k is defined
vt
rt--k
use letters
t,u,..,
rI.(Pil
k v-
Definition 1: For an invariant measure #, absolutely continuous with respect to
the invariant density h is given by #
hd.
f
Smoothness
Man’ [16]
of Invariant
Density
for Expanding Transformations
35
defined a class of almost Markov expanding maps satisfying five condi-
tions.
Definition 2: Let (f, %, #) be a probability space, where f is a separable metric
space and % its Borel a-algebra. We say a map r: f+f is Man-expanding if there
exists a sequence of partitions (i)i > 0 of open sets such that"
[J P e 20
(a)
(b)
f
(mod 0).
(c)
For every k _> 0 and P e k+l,r(P) is a union
and rip is injective.
There exists 0 < < 1 and K > 0 such that,
(d)
for every k _> 0, and for every x,y in the domain of r -k
There exists l> 0 such that, for every pair of atoms P,Q E
(mod 0)
of atoms of
#(r-I(P)OQ)#O.
o,
2k,
we have
There exists C > 0 such that, for every k _> 0 and 0 < (_< 1 whenever x,y
are contained in the same atom of k we have
(e)
(Y)
1
<6"]lr(x)--(y)ll
(t is the absolute value of the determinant of the Jacobian matrix of -).
conditions are sufficient for the existence of an acim (see
Man proved that these
Theorem 1, Section 3).
.
Remarks: Such maps need not be Markov, but all Markov maps satisfy condition
considered by Man is, by condition
(b) (see1 [16, pp. 170-171]). The class of maps
(e), C + This class contains cM; M >_ 2.
For our considerations (assuming that
an adm
exists) only
one of these conditions
is needed:
Expanding condition: There exists 0 <
< 1 and K > 0 such that,
for every k _> 0, for every (corresponding to the maps v -k as described above) and
for every x, y in the domain of r t- k.
We will also need Man’ ’s condition (e), but, as remarked above, this condition is
satisfied whenever r is C M, M _> 2. The expanding condition implies that for a
we
> 0, for N large enough and for any u E and any x, y in the domain of
have"
r-}ven
IN
I
Set
-u-N_ ((/)1, 2,.
Cn)"
N(x)
g(y) I E <<1.
Thus for a fixed x, if y a.pproaches x in the direction
of any of the n coordinate axis, it follows that
Thus there exists an N such that for all u
IIDrN(x) I
(1)
< < 1 for all
J(X) < < 1
ox k
where 1
IN we have
x in the domain of
r N.
_< j,
k
_< n.
KOUROSH ADL-ZARABI and HARALD PROPPE
36
For most of the results that follow we need only expanding in the sense of (2). In
other words, if we assume that an acim exists, the results of this paper follow from
the weaker expanding condition (2) and the distortion condition (see Lemma 1).
Definition 3: We define the measures A k by Ak(A
A(T-k(A)).
Definition 4: The Perron-Frobenius operator
1()1()is
Prf(x)
-.
Pr:
f(vE .1() and
I’r (7"/-" 1 (X))’ f
Definition 5: We define iterated densities
sum
Sk(x
pkrl(x
]kfrt_
f AdAk, and therefore S k
Sk(x
m
#().
for almost all x E
k(k). We note that,
by the finite
f A(Pkrl)dA f r_ k(A)ldA
dA k
dA"
3. Main Results
The following result was proved in [16, Chapter III, Theorem 1.3].
Theorem 1: (Man). If v is Man-expanding then it admits an invariant probability measure #, absolutely continuous with respect to Lebesgue measure, and
limkAk(A #(A) for all A in %, where is the Borel (r-algebra of
The next result does not require the existence of an acim.
Theorem 2: Let v:---. satisfy the expanding condition and v C M, M >_ 2.
Then the sequence {SUPx e
[[ D(hT]l } is uniformly bounded.
Our final result, Theorem 3, states that if an acim exists, and if v is of class C M
and satisfies our expanding condition, then the density function must be of class
C M- 2. In this context, the existence of the acim is a separate problem, and, as
stated in the introduction, there are many existence proofs in the literature. Of
course, they all place some additional condition(s) on v. For example, v could satisfy
all of Man ’s conditions, and then Theorem 1 applies.
Theorem 3: Let v:-- admit an absolutely continuous invariant measure and
satisfy the expanding condition, and let v C M, M _> 2. Then the invariant density
h(x) C M-2 for all x
.
1)Sk(x)
o.
4. Proofs and Laminas
The following lemma establishes what is historically called the distortion condition.
Lamina 1: There exists a constant B (0) such that for any k >_ 1 and all t k, we
have
sup
x
20
rt
inf
x
fl0
Proof: We adapt the proof of
have
[16]
rt
k(x) <-
B()"
for dimension 1. From the definition of
j
0v----- l(x))
fir we
of Invariant
Smoothness
Density
for Expanding Transformations
37
k-1
<_
I ( (x)
(+
(
() I ),
j=O
where v E
rk
:]j + 1 (respectively
j)
wE
corresponds to v k (j + 1) o rt-- k (respectively
J o r[- k). Thus
=
This converges if and only if the series
sup
x
E
flO
inf
x
Lemma 2:
Lemma 1).
Sk(x <_ B () for
E gt 0
0r/j converges.
Thus since r/< 1,
r (.)
rt
r r k(x) <- B()"
every x
fo
and k
>_
1
(where B ()
is the bound in
Proof: Starting with Definition 5, the following relations are valid"
E
su
su S(x)
xFt 0
inf
E
()
()
sup
tk xEflO vt
tk vt
<
inf Sk(x )Y ff _k(x)
E x inf ff _k(x)
x fO
x E flO
E k rt
E k E fO rt
xEft 0
inf
_k(x)
inf ff
_k(x)
tE k XO vt
E
]k
Since
f aSkd
x
fO
(byLemmal)-B ().
rt
< in[ S(x) < sup Sk(x
B@0)
in dimension 1.
proved in
1, it follows that
< B ().
u
The following lemma was
[7]
Lemma 3: Let F(x) f((x))(x) for all x efl. Then
M-1
DM F DM f ()(D)MJ’ -+- E
DJf()
3--0
E DiPi,
j,M
(D’’’’’DM)
i=0
where Pi, j,M(D,...,DM) is a polynomial.
Vi
Proof: Straightforward proof by induction.
We now prove that the sequence { DM -1Sk(x)II } is uniformly bounded, if is
piecewise C M. This will guarantee that for every 0 < j < M-2, the sequence
{ D3Sk(x)[[ } is also uniformly bounded and equicontinuous.
Proof of Theorem 2: We prove the theorem by induction. First we note that
I
I
"
KOUROSH ADL-ZARABI and HARALD PROPPE
38
s + ()
Let M- 2. By differentiation
s(j ())
N
u
We take N
E
N
so that
().
we obtain
{DS(: ())m: ()V
DS + ()
ru
(2)is satisfied (for esup
xEfo
uE N
ru
() + S(: ())Or
B--0))"
I Dr-N(x)I]
<
ru
()).
We put
1
B--- < 1.
Since r is piecewise C 2 on f, we also have
)
sup
xfl0
u
JN
I D ru g()II
<
and B k sup
xfl 0
BKTB()+ B()
Using Lemma 2 and Definition 5, we obtain B K + N
implies that the sequence {B K + iN}- 1, is uniformly bounded by some nmber B K
and hence the entire sequence {Bk} isounded by B (1) max0 < < N I{Bi}" Thus
{suPx e 0 DSk(x)II } is uniformly bounded by B (1).
Now we assume the theorem is true for M and prove it for M + 1 (i.e., we assume
thatr C M + 1 and prove that {supx e
is uniformly bounded).
By Lemma 3 we have
oD(M)Sk(x)}
D(M)SK + N(x)
u
D(M)SK(V N(x))(Dv: N(x))Mru N(x)
e Nk
Now by induction we have. for j
for the sequences
{supx E
foD(J)Sk(x)} respectively.
Since r is piecewise C M +
(
i=0
Let
BM)
supx e
0, 1,...,M- 1, constants B (j) which are bounds
on
f, for each j there exists
a constant
Di -N(X)Pi, j,M(Dr N(x),...,DMT
Vu
ao I D(M)sk I () < ,
?.M) such that
N(x))) < .
Then
M-1
?--0
Since
7MB()< 1,
the sequence
{BI:iN}iC= 1
is uniformly bounded by some
Smoothness
number
v/).
of Invariant
Density
for Expanding Transformations
Thus the entire sequence
1 {!M)}"
{BM)}
is bounded by
39
B(M)=
V!
max0 < < N
Lemma 4: Let f. Nn__N and
{Sk} be
Sk- f. If DSk=g (uniformly) then Df
a sequence
of functions Sk:Nn---N
such that
g.
Proof: This is a standard result, e.g., Theorem 7.17 of
[19].
{D(K)Sk}
Proof of Theorem 3: Using Theorem 2, we conclude that the sequence
is
uniformly bounded and equicontinuous for 0 _< K _< M- 2 on f0" By the Ascoli-
{D(K)Sk,}
Arzela Theorem, there is a subsequence
with a continuous limit, fK(x).
In particular, we have limloSkl(x --fo(X)= f(x) Vx E fo" From Lemma 4, it
follows that fK(X)--D(K)f(x)for K-0,1,2,...,M-2 and for all x Ef 0. To
complete the proof, we note that since limlo f
lim__.,k- f fd,, and
cokldl-
since we know
f Afd,
{’k}
converges
(Theorem 1)it
#(A).
means f
follows that
This
is equal to the invariant density h a.e.
M- 2 times differentiable and we have
h (j), for 1
f(J)
Hence, h
(limk,k)(A)
can be chosen to be
Corollary 3.1: Let 7:ff be expanding and vie C M M_
> 2.
0 <_ j <_ M- 2, the sequence
converges uniformly to hJ).
Proof: This follows immediately from Theorem 3 and the fact that
uniformly bounded and equicontinuous.
{D(J)Sk}
Then
for
{D(J)Cokl } isV1
5. Conclusions
Our results could be improved in two directions. One problem to consider is to
establish the smoothness property of invariant densities for Lasota-Yorke maps in
higher dimensions. Another problem would be to increase the degree of smoothness
from C M-2 to C M- 1, (but as noted in the introduction this would in general
require a finer partition). In one dimension, this was proved by [22]. Furthermore, ti
seems possible to establish the existence of an acim for random maps (see [3] and
[14]) composed of expanding transformations, and to derive smoothness properties of
invariant densities based on the technique used in this paper.
Acknowledgements
We
are grateful to Dr. P. Gdra, the associate editor, and the referees who made many
helpful suggestions. We also thank A. Osman and P.K. Kamthan for their interest in
this work. The research of KAZ was supported by NSERC and ISM.
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KOUROSH ADL-ZARABI and HARALD PROPPE
40
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