Absolutely Continuous Invariant Measures for Piecewise C 2 and Expanding Mappings in Higher Dimensions

In this paper, by using a trace theorem in the theory of functions of 
bounded variation, we prove the existence of absolutely continuous invariant 
measures for a class of piecewise expanding mappings of 
general bounded domains in 
any dimension.


1.
Introduction. Since the pioneering work of Lasota and Yorke [7] in 1973 on the existence of absolutely continuous invariant measures for a class of piecewise C 2 and stretching mappings of an interval, there have been several approaches in generalizing their work to multi-dimensional mappings. A major contribution to the existence result on multi-dimensional absolutely continuous invariant measures was the paper [5] by Góra and Boyarsky in 1989, following the earlier approach of using the modern notion of variation in [3] on rectangular partitions of the domain. In their work the theory of distributions for multi-variable functions was employed to prove the existence of fixed densities for Frobenius-Perron operators associated with piecewise C 2 and expanding mappings in R N with the assumption that where the C 2 pieces of the boundaries meet, the angle subtended by the tangents to these pieces at the point of contact is bounded away from zero. Recently Adl-Zarabi [1] has extended the result of [5] to more general domains with cusps on the boundaries, based on the idea of Keller for two dimensional mappings. To our knowledge, all the previous approaches were by means of investigating the behavior of the function value on the boundary of a domain under various conditions through complicated analysis.
In this paper we want to clarify some points not very clearly stated previously by providing a general variation approach to proving the existence result via directly employing the trace theorem in the theory of functions of bounded variation that was proved in such books as [4] and [9]. While we do not explore function values on the boundary of the partition of the domain for different cases, our purpose here is to give a general condition for the existence of absolutely continuous invariant measures for a domain in any dimension so that the previous work reduces to checking the condition in different cases. However we point out that estimating the constant in the trace theorem for more general cases is a difficult problem that requires delicate analysis from geometric measure theory.
In the next section we briefly review the concept of variation and its basic properties, and in Section 3 we give the unified approach to establishing the existence theorem.
2. Preliminaries. The concept of variation plays an important role in the compactness argument in L 1 spaces (see [4] and [9] for more details). In this section we first introduce the modern notion of variation for functions of several variables and present some results for later use.
Let Ω ⊂ R N be a bounded open set. As standard notation in functions theory, C m (Ω) is the space of functions which have continuous derivatives up to order m in Ω, and C m 0 (Ω) consists of those in C m (Ω) with compact support. C m (Ω; R N ) and C m 0 (Ω; R N ) denote the corresponding vector functions spaces, respectively. Let m denote the Lebesgue measure on R N , and let < ·, · > and | · | denote the usual Euclidean inner product and the Euclidean 2-norm for R N , respectively. The space and norm notations for different functions spaces, such as the Sobolev spaces W 1,1 (Ω) and W 0,1 (Ω) ≡ L 1 (Ω), are used in a standard way as in [4] or [9].
Remark 2.1. The gradient of f ∈ L 1 (Ω) in the sense of distribution will be denoted by Df [4], and so we can write V (f ; Ω) as Ω |Df |. The latter notation for the variation of f comes from the fact that if f ∈ C 1 (Ω), then is the gradient of f in the classic sense, which is deduced from integration by parts, and Ω |Df | is actually the total variation of ω on Ω. More generally A |Df | is the total variation of ω on A ⊂ Ω (see Remark 1.5 of [4]; also [8]). Also the notation A < Df, g > means the integral of the vector function g = (g 1 , . . . , g N ) Now we introduce the definition of the trace of a function in BV (Ω).
is the open ball centered at x with radius r.
In this paper we assume that Ω and any Ω 0 ⊂ Ω involved are open, bounded, and admissible (see [9]), say a domain with piecewise Lipschitz continuous boundary, so that for any f ∈ BV (Ω 0 ) the trace tr Ω 0 f is well defined and the following three results are valid (see [4] or [9]). , tr Ω 0 f < g, n > dH, (2.1) where n is the outward unit normal vector to ∂Ω 0 .
We need the following result which was proved in [1] since it is needed to investigate the Frobenius-Perron operator expression for the class of mappings studied in this paper (see (3.7) in the next section). This result is basically a generalization of the familiar change of variable formula for smooth functions to weakly differentiable functions. For a mapping T , we denote by J T the Jacobian matrix of T and |J T | the absolute value of the determinant of J T . If A is a matrix, then A is defined to be the Euclidean 2-norm of A. Note that A = λ max (A T A). Lemma 2.6. Let Ω 1 , Ω 2 ,Ω 1 ,Ω 2 be bounded open sets in R N , Ω 1 ⊂Ω 1 , Ω 2 ⊂Ω 2 , let S = (S 1 , · · · , S N ) T :Ω 1 →Ω 2 be a diffeomorphism of class C 2 with S(Ω 1 ) = Ω 2 , and let f ∈ BV (Ω 1 ). Then there hold (i) For g = (g 1 , · · · , g N ) T ∈ C 1 (Ω 2 ; R N ), |tr Ω 1 f | dH.

Existence Result.
In this section, we prove the existence result for a class of piecewise C 2 mappings in any dimension. First we prove two useful results.
Proof: Let g ∈ C 1 0 (Ω; R N ). Then from Proposition 2. 3, where n i is the outward unit normal vector to ∂Ω i , i = 1, 2. From the above and using the fact that n 1 = −n 2 along Γ, we have Hence, Taking supremum over all g ∈ C 1 0 (Ω; R N ) such that |g(x)| ≤ 1, x ∈ Ω, we obtain (3.3). Now, from Remark 2.2, we see that |tr Ω 0 f | dH. (3.5) It is time to study the existence problem of absolutely continuous invariant measures for a class of piecewise C 2 and expanding mappings in R N . Specifically this class of mappings is given in the next definition [1]. Definition 3.3. Let S : Ω → Ω and let {Ω 1 , · · · , Ω r } be a partition of Ω. Denote S i = S| Ωi . We say that S is piecewise C 2 and α-expanding if each S i is C 2 and one-to-one on Ω i , can be extended as a C 2 mapping on Ω i (i.e., C 2 on an open neighborhood of Ω i ), and satisfies For the mapping S : Ω → Ω, the corresponding Frobenius-Perron operator P : has the expression From (3.6) it is clear that any fixed density f * of P defines an absolutely continuous invariant measure See [2] and [6] for more details about properties and applications of Frobenius-Perron operators.
A similar result to the following one was presented in [1] for some domain. But we intend to give a complete proof for a very general domain and then combine it with the trace theorem to get Lemma 3.5.
Lemma 3.4. If S : Ω → Ω is piecewise C 2 and α-expanding, then for any f ∈ BV (Ω), where C > 0 is a constant independent of f .
Proof: From Lemma 3.5, the assumption implies that the sequence is uniformly bounded, where 1 is the constant function 1. Hence the sequence {P i 1} i≥1 is precompact in L 1 (Ω) by Proposition 2.5, and it follows from the Kakutani-Yosida theorem [6] that P has a nontrivial fixed point which is the density of an absolutely continuous invariant measure.
Remark 3.1. Under the condition of Theorem 3.6, it can be shown with a standard technique that P : BV (Ω) → BV (Ω) is quasi-compact [2]. Hence many statistical properties of the dynamical system, including the exponential decay of correlation, can be obtained.
Finally, it should be mentioned that since no additional conditions are assumed for Ω, the existence results under different assumptions on Ω obtained before in, e.g., [1,5] can be viewed as consequences of our general framework after giving an upper bound for ρ Ω (S) in different cases.