SMOOTHNESS OF INVARIANT DENSITY FOR EXPANDING TRANSFORMATIONS IN HIGHER DIMENSIONS KOUROSH

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) {Pi}TMi=I be a partition of into a finite number of subsets having piecewise C boundaries of finite (n1)-dimensional measure. Let r: -be piecewise C on z2 where ’i v[ p. is a C2 diffeomorphism onto its image and expanding in the sense that there exists c > 1 such that for any 1, 2,..., rn, II Dr[II E < c1, where D71 is the derivative matrix of 7/1 and II" II 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


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 transforma- tions.Let be a bounded region in Nn and let z) {Pi} TMi=I be a partition of into a finite number of subsets having piecewise C 2 boundaries of finite (n-1)-dimension- al measure.Let r: -be piecewise C 2 on z2 where 'i v[ p. is a C 2 diffeomorph- ism onto its image and expanding in the sense that there exists c > 1 such that for any 1, 2,..., rn, I I Dr[-I I E < c-1, where D7-1 is the derivative matrix of 7/-1 and II" I I 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 properties of interest include" number of ergodic acim [9,2], uniqueness [4, 21], stabi- lity [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-smooth- ness relative to another partition that is obtained from the given one through refine- ment 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 piece- wise 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 transforma- tion is expanding in the sense of the maps considered by Man [16] and of class CM, 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 con- sidered 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].

Definitions and Conditions
We denote by ,k, the Lebesgue measure on n.For a n x n x... x n (k-times) array M k we define its norm by I I Mk I I maxNkl(m)il...ikl, where N k {ili2...ik: 1 <_ ij <_ n for 1 For a real-valued function f:n_ we denote by Df its derivative, and by D(M)f the M-th derivative of f.If f(x)= f(xl,x2,...,xn) then (Df) x is a linear map: Nn--and (Df)x(v I v2, Vn , n i=1 VOxi Let {pi}n= 1 be a partition of f into a finite number of subsets having piece- wise C 2 boundaries of finite (n-1)-dimensional measure.Henceforth we work mainly with the open domain fo-f\Ji 3oOPi" We use the notation vt -k to mean a specific composition of inverses of the form v/--k where each ij G {1, 2,..., m} for 1 _< j _< k, and denote by 1t k the set of subscripts for which vt -k is defined (we use letters t,u,.., as subscript here to distinguish between v v p. and the composition of inverses of such maps as described.)Thus for any open set 4, each rt--k is a C M diffeomorphism of A rI.(Pil onto its image.Of course, r-k(A) U e k vk(A)" Definition 1: For an invariant measure #, absolutely continuous with respect to the invariant density h is given by # f hd.
Man' [16] 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" (a) [J P e 2 0 f (mod 0).
(b) For every k _> 0 and P e k+l,r(P) is a union (mod 0) of atoms of 2k, and rip is injective.
(c) There exists 0 < < 1 and K > 0 such that, -k for every k _> 0, and for every x,y in the domain of r (d) There exists l> 0 such that, for every pair of atoms P,Q E o, we have #(r-I(P)OQ)#O.
(e) 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 (Y) 1 <6"]lr(x)--(y)ll (t is the absolute value of the determinant of the Jacobian matrix of -).Man proved that these conditions are sufficient for the existence of an acim (see Theorem 1, Section 3).
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, -k for every k _> 0, for every (corresponding to the maps v 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 CM, M _> 2. The expanding condition implies that for a > 0, for N large enough and for any u E IN and any x, y in the domain of r-}ven we have" I I N(x) g(y) I I E <<1. (1) Set -u-N_ ((/)1, 2,.Cn)" Thus for a fixed x, if y a.pproaches x in the direction of any of the n coordinate axis, it follows that J(X) ox k < < 1 where 1 _< j, k _< n.
Thus there exists an N such that for all u IN we have IIDrN(x) I I < < 1 for all x in the domain of r N.
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)).

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 probabi- lity 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.
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 CM, M _> 2. Then the invariant density h(x) C M-2 for all x o.

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 2 0 rt inf k(x) <-B()" x fl0 rt
Proof: Starting with Definition 5, the following relations are valid" su S(x x fO x E flO E k rt E k x E fO rt Sk(x <_ B () for every x fo and k >_ 1 (where B () is the bound in inf _k(x) ]k x fO rt Since f aSkd 1, it follows that B@0) < in[ S(x) < sup Sk(x < B ().
The following lemma was proved in where Pi, j,M(D,...,DM) is a polynomial.
Proof: Straightforward proof by induction.

Vi
We now prove that the sequence { I I DM -1Sk(x)II } is uniformly bounded, if " is piecewise CM.This will guarantee that for every 0 < j < M-2, the sequence { I I D3Sk(x)[[ } is also uniformly bounded and equicontinuous.
Proof of Theorem 2: We prove the theorem by induction.First we note that s + ()   s(j ()) ().
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 oD(M)Sk(x)} is uniformly bounded).
Proof of Theorem 3: Using Theorem 2, we conclude that the sequence {D(K)Sk} is uniformly bounded and equicontinuous for 0 _< K _< M-2 on f0" By the Ascoli- Arzela Theorem, there is a subsequence {D(K)Sk,} with a continuous limit, fK(x).
This means f is equal to the invariant density h a.e.Hence, h can be chosen to be M-2 times differentiable and we have f(J) h(j), for 1 Corollary 3.1: Let 7:ff be expanding and vie C M M_ > 2. Then for hJ). 0<_ j <_ M-2, the sequence {D(J)Sk} converges uniformly to Proof: This follows immediately from Theorem 3 and the fact that {D(J)Cokl } is uniformly bounded and equicontinuous.V1

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.