NON-UNIFORMLY EXPANDING DYNAMICAL SYSTEMS : MULTI-DIMENSION

Dynamical systems on the interval [0, 1], satisfying the Thaler’s condition, have been extensively studied. In this paper we consider invariant density and statistical properties of non-uniformly expanding dynamical systems on Rd (d ≥ 1). We present a critical regular condition that is a supplement and a development of the Thaler’s condition, and it is very closely related to Lamperti’s criterion. Under this new condition, we offer a method for studying the dynamical systems. A continuity description of the invariant density is presented; and a convergence theorem for iterations of Perron-Frobenius operator is set up. Furthermore, we establish a more exact result for one-dimensional systems.

1. Introduction.This paper can be regarded as a follow-up and a supplement of papers [25,26].Recently a lot of attention of mathematicians is focused on the nonuniformly expanding dynamical systems (see e.g.[2,6,7,15,16,17,21,23,29,30]), in particular on interval maps with indifferent fixed points (refer to [8,9,18,22,25,26,27,28,35]).The importance of this kind of maps was addressed by Prellberg and Slawny [22].By refering to the study of the Pomeau-Mannevile map, Young [31] set up an abstract setting, which can be applied to dealing with rates of mixing in terms of the recurrence times.Thaler [27,28] investigated the asymptotic behaviour of the Perron-Frobenius operator.Hu gave a deep study in this topic (see e.g.[9,10,11,12]).It is interesting to note that some other new theories are applied to studying the non-uniformly expanding dynamical systems.For example, the renewal sequences theory and the operator renewal theory are applied to dealing with the correlation of non-uniformly expanding dynamical systems by Sarig [23] first, and then by Gouëzel [7] and Melbourne and Terhesiu [19,20], respectively.
Pomeau-Manneville map (see (1)) is a well-known example in the study of nonuniformly expanding dynamical systems.Given any α > 0, let Roughly speaking, Thaler [25,26,27] and Hu [9] considered the cases α ≥ 1 and 0 < α < 1, respectively.And, the domains of the maps considered in papers 2512 YUAN-LING YE [18,22,23,28,35] are subsets of one-dimensional space R. To my best knowledge, there are few papers concerning invariant density and statistical properties of nonuniformly expanding dynamical systems defined on multi-dimensional space R d with d > 1.In this paper we try to extend the consideration of paper [25] to multidimensional space (R d , | • |) (d ≥ 1), where For any x ∈ R d and ρ > 0, we let We use λ to denote the Lebesgue measure on R d , and denote by 0 = (0, • • • , 0) t ∈ R d .We use E d×d , or E for the simplicity, to denote the d × d identity matrix, which matches identical linear operator T E : R d → R d defined by If there is no confusion, we denote also by 0 = (0) d×d .
Let E 1 and E 2 be Banach spaces, and let C(E 1 , E 2 ) be the set of all bounded linear operators from E 1 to E 2 .Let D be an open subset of E 1 and x 0 ∈ D. We call map A : D → E 2 to be Fréchet differentiable at x 0 [34], if there is a bounded linear operator B ∈ C(E 1 , E 2 ) such that = 0.It is known that the operator B is unique.The operator B is called a Fréchet derivative operator of map A at x 0 , and denoted by A (x 0 ).And we use A (x 0 ) to denote the second derivative of map A at x 0 .We use |A (x 0 )| and |A (x 0 )| to denote the norms of A (x 0 ) and A (x 0 ) as operators, respectively.We say that the map A : D → E 2 is continuously Fréchet differentiable, if A (x) is continuous with respect to x ∈ D. When A(•) is a function defined on set D(⊆ R d ), then A (x) is the gradient of A at x, i.e., A (x) = ∂A(x)  ∂x1 , .
And it matches linear operator S : R d → R d defined as Sy = A (x)y for all y ∈ R d .
We use det A (x) to denote determinant of A (x).And we use A (x) to denote the norm of A (x) as linear operator from R d to R d , i.e., And, we let α • A = (α • a i,j ) m×n for any α ∈ R. Let Let Y 0 be any non-empty subset of X.Let f and g be any two non-negative vectorvalued functions defined on Y 0 .We say that f is increasing, if f (x) ≤ f (y) whenever x ≤ y.
Let x 0 ∈ Y 0 and let lim We write, for brevity, f (x) ≈ g(x) as x → x 0 to mean the existence of the constants c > 0 and δ > 0 such that And, f (x) = O(g(x)), or f (x) g(x), as x → x 0 to mean the existence of the constants c > 0 and δ > 0 such that f (x) ≤ cg(x) for all 0 < |x − x 0 | < δ.
When f and g are positive functions, we write, for brevity, f (x) ∼ g(x) as x → x 0 to mean that lim x→x0 f (x) g(x) = 1.If x 0 = 0, we write, for brevity, f (x) ≈ g(y) as |x| ≈ |y| → 0 to mean the existence of the constants c 1 , c 2 > 0 and δ > 0 such that c −1 1 f (x) ≤ g(y) ≤ c 1 f (x) for all x, y ∈ Y 0 with 0 < c −1 2 |x| ≤ |y| ≤ c 2 |x| < δ.Let S be a linear operator from R m to R n , i.e., S ∈ C(R m , R n ).We denote by S ≥ 0 if S(y) ≥ 0 n×1 for any 0 m×1 ≤ y ∈ R m .
We say that the map τ : Y 0 → Y 0 is expanding, or uniformly expanding for the emphasis here, if there exists a γ > 1 such that |τ (x) − τ (y)| ≥ γ|x − y| for all x, y ∈ Y 0 .
Let τ : X → X and denote t , for all x ∈ X.
Then for any fixed 1 ≤ i ≤ d and any given x ∈ X, τ i (x) is a linear operator from R d to R d .Similar to one-dimension case, we call 0 ∈ X an indifferent fixed point of τ , if lim We say that the indifferent fixed point 0 is a regular source, if for all 1 ≤ i ≤ d, τ i (x) ≥ 0 d×d near 0.
Throughout this paper, without loss of generality, we just consider systems on X := [0, 1] d admitting only one indifferent fixed point, for convenience.For any D ⊆ R d , we use, except for particular remark, D • and D to denote the interior and the closure of D, respectively.We assume that there exist nonempty convex subsets {X j } m j=1 (m ≥ 2) such that For small ρ > 0, we let B(ρ) = {x ∈ X 1 : |x| < ρ}.
Let continuous function ψ : X 1 → [0, +∞) be twice continuously Fréchet differentiable on X • 1 with ψ(x) > ψ(0) = 0 for any 0 = x ∈ X 1 and ψ (x) ≥ 0 for any x ∈ X • 1 .Assume that ψ satisfies the following condition: 1 with β i (x) ≥ 0 near 0. And assume that there exists an α > 0 such that as Define Let τ : X → X be a transformation with In the following we always assume that τ : X → X is twice continuously Fréchet differentiable on each X • j (1 ≤ j ≤ m) and satisfies the following conditions: (C1) for any 1 (ii) 0 is a regular source point of τ, and for small ρ > 0 there exists γ ρ > 1 such that for any x, y We call transformation τ satisfying assumptions (C0)-(C2) a non-uniformly expanding system, and let d denote the class of all such transformations.Let = ∞ d=1 d .In this paper we will study non-uniformly expanding system (X, τ ) with τ ∈ .
A Borel measure µ on X is called invariant measure of the dynamical system (X, τ ), if µ(E) = µ(τ −1 E) for any Borel subset E of X.
Let E 0 = X \ {0}, and let C(E 0 ) denote the set of all continuous functions on E 0 .The Perron-Frobenius operator T : We call 0 h ∈ C(E 0 ) an invariant density of τ if T h = h.In this case, the system (X, τ ) admits an invariant measure µ such that dµ dλ (x) = h(x).We are interested in the statistical properties of dynamical systems (X, τ ) with τ ∈ .For this, we will focus on studying the behaviour of invariant density and the convergence properties of iterations of the Perron-Frobenius operator.And, the following problems arise naturally: (Q1) How do we describe the invariant density of the system (X, τ )? (Q2) Does the functions sequence {T k f } ∞ k=1 converge for any f ∈ C(E 0 )?To state more precisely the necessity of considering the dynamical systems (X, τ ) where τ ∈ , we need to say something about the Lamperti's criterion [14] and Thaler's assumptions [1].Let f be a positive differentiable function defined on (0, 1  2 ].We say that the function f satisfies Lamperti's criterion [14], if there exists a α ∈ R such that lim We will see from (90) and Remark 4 and 7 that (C0) is very closely related to Lamperti's criterion.Let X = [0, 1], i.e., d = 1.We say that τ : X → X satisfies the Adler's condition (see e.g.[1,25,26,35] The system (X, τ ) is said to satisfy the Thaler's assumptions, if the conditions (C1) and (C2), together with the Adler's condition, are satisfied.Schweiger [24] introduced the jump transformation τ * of τ , and showed that τ admits an unbounded invariant density provided that τ * admits a bounded invariant density.And then, under the Thaler's condition, Thaler [25], by making use of Adler's Theorem and Schweiger's result, gave an estimates of the invariant density of τ .Later on, Thaler [26,27,28] continued to study parabolic dynamical systems satisfying the Thaler's assumptions: Paper [26] considered ergodic properties of the system; paper [27] set up an important refinement limit theorem; and then a distributional limit theorem was established in paper [28].Under the Thaler's condition, Zweimüller [35] studied the ergodic properties of the (infinite) invariant measures of the systems.We notice that there is a striking difference between our assumption and the Thaler's assumption.On the one hand, some interesting multi-dimensional nonuniformly expanding dynamical systems are included (see the definition of and Example 4.6).On the other hand, even for one-dimensional non-uniformly expanding dynamical systems, from the following two simple examples, we see that it is a supplement of Thaler's assumption.
(ii) Let Then we can check that τ ∈ 1 .This transformation, however, does not satisfy the Adler's condition (see Example 5.2).
From Remark 3, we will see that (C0)(i) implies (C0)(ii) if d = 1.And by combining (3) with (4), we can see that (i) of (C0) is not a simply replacement of the Adler's condition.In fact, from the Adler's condition, we know that the invariant measure of the system must be infinite (see e.g.[25,26,27,35]).However, there are systems (X, τ ) with τ ∈ 1 do not satisfy the Adler's condition.And their invariant measures maybe finite or infinite (see e.g.Example 5.2 and 5.3).
In this paper we will offer a method to study the systems and the associated Perron-Frobenius operators.For any 0 < ϕ ∈ C(E 0 ), we let

And for any
ϕ (E 0 ) denote the dual space of C ϕ (E 0 ).Our basic result is presented in the following Theorem 1.1.
Theorem 1.1.Let τ ∈ .There exists a locally Lipschitz continuous function Theorem 1.1 will be proved in §4 (see Theorem 4.4).We would like to point out that this theorem deals with multi-dimensional systems.This result is of interest even in one dimension case.As a consequence of Theorem 1.1, we have the following corollaries.They come from Corollary 6 and Proposition 9, respectively.Corollary 1.Let ξ be as in Theorem 1.1.Then, the invariant measure of τ is finite if and only if there exists a constant c > 0 such that ξ = cλ.
We extend the consideration of paper [27] to multi-dimensional systems, and present further, in particular, a supplement of the Thaler's condition set up for one-dimensional systems.We present an explicit description of the invariant density (see Theorem 4.1).It is a critical preparation for the proof of Theorem 4.4.We would like to mention that Hu [9] introduced a concept so-called asymmetrical ball to study the system (X, τ α ), 0 < α < 1, as in (1).In this paper the concept is extended to multi-dimensional systems, and it can be used to study the behaviour of the invariant density (see Theorem 4.1).It is easy to see that τ (x) (x ∈ (0, 1)) is conformal if d = 1.We can take advantage of the conformity of τ (•) to set up Proposition 2. It seems that it is easy to extend the results for one-dimensional systems (see e.g.[27]) to high-dimensional systems.However, the high-dimensional systems are more complicated.Because τ (x) is not conformal in general; and the expansion of τ at boundary point of X, near 0, is complicated.These make it difficult for us to describe the equi-continuity of sequence {T n 1} ∞ n=1 on the boundary of X.And comparing with Proposition 2 established for one-dimensional systems, we set up Proposition 4 for high-dimensional systems.For this, we need to define the asymmetrical ball in another way (see (41)).It is necessary to point out that the Lebesgue measure λ was chosen as an eigen-measure of T * in paper [9].However, the Lebesgue measure maybe not any longer suitable for our consideration.And we need to construct another eigen-measure (see Theorem 4.4 and Proposition 8).Because of the unboundedness of the invariant density, we are unable to define strictly positive "normalized" weight functions on X as in [17].It creates difficulty for us to show the convergence properties (see Theorem 4.4).
We organize the paper as follows.In section 2, we present some elementary facts about non-uniformly expanding dynamical systems on R d (d ≥ 1).In section 3, we discuss the equicontinuous properties of iterations of the Perron-Frobenius operator for systems of one-dimension and multi-dimension, respectively.In section 4, we study the behavior of invariant density, and consider the convergence properties of iteration of the Perron-Frobenius operator.In section 5 we consider, in particular, one-dimensional dynamical systems.And compared with the result appears in §4, more exactly one is set up in this section.

2.
Preliminaries.In the following we always let τ ∈ d (d ≥ 1) be as in the previous section.And let w j : X → X (1 ≤ j ≤ m) be given as in (C1).Let t .For any 1 ≤ i ≤ d, we define And let .
Then, we confirm that We use τ (x) and w j (x) to denote Jacobian matrices of τ and w j at x, respectively.Then from (4), it follows that for any x ∈ X 1 , And, it follows from (8) that for any x ∈ X, Let det τ (x) and det w j (x) denote determinants of τ (x) and w j (x), respectively.From the assumptions on τ , we confirm that each det w j (x) is uniformly continuous on X.
For any A = a i,j d×d , we use A to denote the norms of A as an operator from where ρ(A t A) is the largest absolute eigenvalue of A t A.
For any ω : R d → R and ξ ∈ R d , we let, for the sake of convenience, ∂ω(ξ) ∂xj = x ∈ B ρ (0)} for any small ρ > 0. Define For any a, b ∈ R, we let And generally, for any For small ρ > 0 and any x ∈ B (ρ), we define an asymmetrical ball at x by Proof.For any x ∈ X 1 \ {0} with |x| small, by the convex assumption on X 1 , there exist ζ, ξ ∈ I(0, x) such that Note that 0 is the indifferent fixed point of τ .Then we have lim x→0 τ (x)−E = 0. From this, together with the inverse function theorem, it follows that Hence, we conclude that Remark 1.We would like to point out that the convex assumption is imposed on the partition {X j } m j=1 of X.Since we will find that, for example, the mean value theorem is often applied on X j , see e.g., Lemma 2.3 and 3.11, Proposition 4 and 5.
In the following we always let ψ be the function defined as in §1 satisfying (C0).
From Remark 2, it follows that τ (x) ≥ τ (y) for any x, y near 0 with x ≥ y.
From this, together with (11), we confirm that for any x ∈ X 1 near 0, At the end of this section, we would like to point out that we can deduce from the above arguments that, by noting ( 9) and ( 3), there exist δ 0 > 0 and A 0 > 0 such that for any x ∈ B (δ 0 ) Moreover, from (C2)(i), we conclude that there exists a 0 < r 0 < 1 such that for any 2 3. The equicontinuous properties of iterations of Perron-Frobenius operator.Let E 0 = X \ {0}, and let C(E 0 ) denote the set of all continuous functions on E 0 .Let T be the operator defined by (5).To study the operator T , we need some preparation.It is necessary to remark that det w 1 (x) > 0 near 0. For any multi-index to denote the length of J, and let From this, we can prove inductively that for any integer k, 3.1.One-dimension.In this subsection, we consider one-dimension non-uniformly expanding systems (X, τ ), i.e., X = [0, 1] and τ ∈ 1 .We assume, without loss of generality, that X 1 = [0, 1  2 ], and we let τ (x) = x + xβ(x), for all x ∈ X 1 .
In this case, the assumption (C0) is reduced to that the function ψ is twice continuously differentiable on (0, 1 2 ) and satisfies the condition And ( 3) is reduced to that the function β : [0, 1  2 ] → [0, +∞) satisfies the condition: From ( 17) and ( 18), we conclude that Hence in this subsection we assume, without loss of generality, that The following Proposition 3 is a fundamental result of this subsection.Because the systems under consideration do not have the bounded distortion, to prove this proposition, we need to set up the following Proposition 2 first.Proposition 2. There exist η > 0 and a > 0 such that for any x ∈ B (η), To prove Proposition 2, we need some preparations.From this, together with (11), we conclude the assertion.
Proof.From Lemma 3.2(i), it follows that there exists a constant c ≥ 1 such that for small x From this, we conclude that for any small z 1 and z 2 with From this, we get the assertion.
Remark 3. We would like to point out that (C0)(ii) is reduced to Lemma 3.2(ii) and Corollary 3 if d = 1.And from the proof of Lemma 3.2(ii) and Corollary 3, we see that (C0 From Example 4.6, we will see that, for high-dimensional systems, (C0)(i) does not imply (C0)(ii).Hence, it gives us an explanation why the assumption (C0)(ii) is imposed on the system when d > 1.
Proof.(i) By the Taylor expansion theorem, there exists ζ ∈ I(x, τ (x)) such that This, combined with Lemma 3.2, implies that From this, together with Lemma 3.2(i), it follows that (ii) From ( 4) and (i), we conclude that From this, together with (i) and Lemma 3.2(i), we deduce that Proof of Proposition 2. For small ρ > 0, we let From Lemma 3.2, together with Lemma 3.1, we deduce that there exist η > 0 and a > 0 such that for any x ∈ B (η), Φ(x, ξ) ≤ aβ(x) ∀ξ ∈ B(x).
Note that τ w 1 (x) = x and τ (x) ≥ 1 near 0. It follows that for any ξ ∈ B(x), For any x ∈ B (η) and y ∈ B(x), we conclude, by applying the mean value theorem, that From this, together with (20), it follows that for any x ∈ B (η), In the rest of this subsection, we always assume that f ∈ C(E 0 ).
Lemma 3.4.Let η be given by Proposition 2. Then there exist 0 < ε < η and c 0 > 0 such that for any c ≥ c 0 , provided that for any x ∈ B (ε), Proof.By the assumption that β (x) ≥ 0 near 0, we see that β(x) is increasing near 0. Let η > 0 and a > 0 be given by Proposition 2. From Lemma 3.2(i) and 3.3(i), we deduce that there exists A 2 > A 1 > 0 and 0 < 1 < η such that for any x ∈ B ( 1 ), This implies that Note that for any y ∈ B(x), by the mean value theorem, there exists a ζ ∈ B(x) such that .
From this, together with ( 22), we deduce that there exists a 0 < 2 < 1 such that for any x ∈ B ( 2 ), This, together with (23), implies that for any x ∈ B (ε) and y ∈ B(x), Now let f ∈ C(E 0 ) satisfy the condition (21).Then, for any x ∈ τ B (ε) and y ∈ B(x), we have where From this, we get the assertion.
For such a ε > 0 given by Lemma 3.4, we let Note that w 1 (x) < x.This implies that δ > 0. For any x ∈ B c (ε), we define In the following of this subsection, we always assume the "balls" B(x) is defined by ( 11) if x ∈ B (ε), and defined by ( 26) if x ∈ B c (ε), respectively.It will depend on the location of x.Let Note that τ (X 1 ) = X and τ B (ε) ⊆ X 1 .From the definition of D(ε), it follows that m j=2 X j ⊆ D(ε).Lemma 3.5.Let ε > 0 be the constant given by Lemma 3.4.There exists a constant b 0 > 0 such that for any fixed b ≥ b 0 , Proof.Let ε > 0 and c 0 > 0 be as in Lemma 3.4.For any x ∈ B c (ε), we define d 0, B(x) = inf{y : y ∈ B(x)}, and let We can verify that d 0 > 0 and From this, together with (C2) and ( 16), we deduce that, by making use of the twice continuously differentiability of τ , there exist constants 0 < r 0 ≤ r < 1 and R > 0 such that (a) for any x ∈ B c (ε) and y, z ∈ B(x), Recall that ε > 0 and c 0 > 0 are determined by Lemma 3.4.Let We will prove that this constant b 0 is a desired one.In fact, for any fixed b ≥ b 0 , from (27), together with (a) and (b), we can deduce, by a direct verification, that and for any 2 ≤ j ≤ m, From ( 29) and (30), it follows that for any x ∈ D(ε) and y ∈ B(x) Now for b > 0 and ε > 0, we define We notice that the function ∆(•, •) is determined uniquely by positive constants b and ε.Let This, combined with (30), implies that for any x ∈ τ B (ε) and y ∈ B(x), where Note that B (ε) ⊆ τ B (ε) , and From this, together with (33), we get the following conclusions: From (a) and ( 1), we conclude that for any x ∈ B c (ε) and y ∈ B(x), This, together with (b), implies the assertion.
As a consequence of Proposition 3, we have the following Corollary 4. Proof.Let A 2 be given by (22).From Lemma 3.1, it follows that there exists a constant b 1 ≥ b 0 such that From this, together with the mean value theorem, we conclude that for any x ∈ B (ε) and y ∈ B(x), Let d 0 > 0 be defined by (28).Then there exists a constant b 2 ≥ b 0 such that This implies that for any x ∈ B c (ε), Let b = max{b 1 , b 2 }.From Proposition 3, together with ( 34) and ( 35), we deduce that the set D(ε, b) satisfies the assertion.

Multi-dimension.
In this subsection, we consider, without loss of generality, systems on R 3 , i.e., X = [0, 1] 3 .And we let and ∂(X) = X \X • .We will try to construct, along the way similar to the setting of Proposition 3, a non-empty convex invariant set (see Proposition 6).However, the high-dimensional systems τ ∈ 3 are complicated.Because τ (x) is not conformal in general; and moreover, asymmetrical ball B(x) defined as in (11) is not any relative neighbourhood of x ∈ ∂(X) near 0, because we have
These make it difficult for us to describe the equi-continuity of sequence {T n 1} ∞ n=1 on X \ {0}.Hence, we need first to find a way to define asymmetrical ball "centered at" x so that it is a relative neighbourhood of x ∈ X \ {0}.For this, we need some preparations.
For any c ≥ 1 and any 1 ≤ i ≤ 3, we define To consider relative neighbourhood of interior point of X, where X is regarded as a subset of R d , we let, for any n ∈ N, For any 0 < ρ < 1, we define To consider relative neighbourhood of the boundary point, we let for any 1 ≤ i ≤ 3, To consider relative neighbourhood of point located on P i \ 3 j=1 L j , we let for any n ∈ N and ∈ N {∞}, From this, we see that for any x ∈ P i \ 3 j=1 L j , there exists some n ∈ N such that Then To consider relative neighbourhood of point located on L i , we let for any ∈ N {∞}, Then L i \ {0, (0, 0, 1), (0, 1, 0), (1, 0, 0)} ⊂ L i ( ) Define From the definitions of S n , P n (∞) and L i (∞), we confirm that From this, we see that S n n (∞) and T i (∞), respectively (see (41)).Lemma 3.6.Let ψ be given as in (C0), and let X 1 and α > 0 be given by ( 2) and (3), respectively.Then Proof.Remember that From this, together with ( 3) and ( 7), we can deduce that as k → ∞, From this, we deduce that Since α > 0, it follows that (by (37)) Hence, from the above argument, we conclude that Lemma 3.7.There exists a constant M > 0 such that for any Proof.From ( 3) and ( 7), we deduce that there exists 0 > 0 and a ≥ 1 such that for any ≥ 0 , Note that lim k→∞ ψ w k 1 (z) |w k 1 (z)| α = 0, uniformly on X.Hence, from this, together with Lemma 3.6, we conclude that there exists a constant M > 0 such that x i x j < 1 for all j = i}; (c) for any 1 ≤ i ≤ 3, T i ( 0 ) ⊂ {x = (x 1 , x 2 , x 3 ) t ∈ X : 0 < x i ≤ 1, and x j x i < 1 for all j = i}.
Proof.(i) From the definitions of Y n , Z n ( ) and T i ( ), we can get the assertion.(ii) We claim that there exists an n ≥ 1 such that Indeed, we let And let This implies that for any 1 ≤ i, j ≤ 3, From this, together with Lemma 3.7, we conclude that for any x ∈ Y n , lim sup ≤ c n for some c n > 0.
Note that lim k→∞ sup x∈X |w k 1 (x)| = 0. From this, together with the claim (38), we can deduce And then, the assertion (a) follows.
By making use of Lemma 3.7 and (40), we can deduce assertions of (b) and (c).
, and for small ε > 0 we let Having done the above preparation, we can define asymmetrical ball on F n (ε).In the following we always let 0 be determined by Lemma 3.8.For any n ∈ N and 0 < ε < 1, we define where 1 ≤ i ≤ 3. It is easy to see that B n (x) is a relative neighborhood of x ∈ F n (ε).
And we can confirm, by making use of Lemma 3.8, that We would like to point out that for any x ∈ T i (∞, ε), It is remarkable to note that for one dimensional systems (X, τ ), we have y ∈ B(x) if and only if x ∈ B(y).
However, for high dimensional systems, it does not hold for B n (x) in general, if x ∈ ∂(X).Hence, we adjust (x, y) to be x n (y, z) (see (31) and (61), respectively).For this, we define for any x ∈ F n (ε), Let 0 > 0 be the number determined by Lemma 3.8(ii).It is important to note that there exists a n 0 ∈ N (large enough) such that for any 1 ≤ i, j ≤ 3, From this, it follows that Then we have the following Lemma 3.9.
Lemma 3.9.For any fixed n ∈ N, as G n x → 0 Therefore, from the above argument, we deduce the assertion (i).
The following Proposition 4 is one of fundamental results of this subsection.Its function is similar to the function of Proposition 2. It plays an important role in the following argument, especially in describing the equi-continuity of sequence {T n 1} ∞ n=1 on the boundary of X (see the following Proposition 6).Proposition 4. For any n ∈ N there exists a n > 0 and small η n > 0 such that for any Proof.Let n ∈ N be fixed.For small ρ > 0 and for any x ∈ F n (ρ), we let We claim that there exists θ n > 0 and small n > 0 such that Indeed, From ( 9), together with (3), we deduce that, by making use of Lemma 2.2(i) and Lemma 2.3, as x → 0, (a) It follows from Lemma 3.8(ii)(a) that there exists a ς n > 0 such that for any This, combined with (47), implies that for any From this, together with (C0)(i) and Lemma 2.3, we can deduce that there exist θ n > 0 and n > 0 such that for any x ∈ Y n ( n ) and any ζ ∈ B n (x) where for any 1 ≤ i, j ≤ 3, Then we can confirm that σ(x) ∈ B n (x), and there exists a s n > 0 such that And from (41), we conclude that there exists a ŝn > 0 such that for any x ∈ G n B (ρ), y j ≤ ŝn • σ j (x) for all y = (y 1 , y 2 , y 3 ) t ∈ B n (x).
(50) We know from (C2)(ii) that 0 is a regular source point of τ, i.e., τ i (ζ) ≥ 0 3×3 near 0, for all 1 ≤ i ≤ 3.This implies that for all 1 ≤ i ≤ 3, Note that (3).From this, together with d = 3 > 1, we deduce that It follows that ∂βi(ζ) ∂xj (1 ≤ i, j ≤ 3) are increasing.Then from (3), together with Lemma 2.3 and 3.9, we deduce that, by making use of ( 49) and (50), as G n x → 0, From Lemma 2.3, together with 3.9, we can conclude that as G n x → 0, Hence from this, together with (47) and (51), we deduce that there exist also, as in (a) still denoted by, θ n > 0 and n > 0 such that for any From this, together with (48), we get the claim (46).Now for any and any y = (y 1 , y 2 , y 3 ) t , z = (z 1 , z 2 , z 3 ) t ∈ B n (x), from the mean value theorem, it follows that, by noting (by ( 46)) This implies that for any Note that det w 1 (x) • det τ (w 1 (x)) = 1 for all x ∈ X.From this, together with (52), we deduce that there exist a n > 0 and 0 < η n < n such that for any Proposition 5. Let η n (n ∈ N) be determined by Proposition 4.There exist constants 0 < n < η n and γ n > 0 such that for any b ≥ γ n and any x ∈ F n ( n ), 3), together with Lemma 2.3, we conclude that as x → 0, From ( 7), together with Lemma 2.2 and 2.3, we deduce that, by making use of (3), For any x ∈ F n X 1 and y ∈ B n (x), it follows that, by making use of the mean value theorem, there exists From ( 10), together with (54), we deduce, by applying Lemma 2.3 and 3.9, that there exist 0 < r n < η n and A n > 0 such that for any x ∈ F n (r n ), From this, together with (53), we conclude that there exist 0 < n < r n and Ãn > 0 such that for any x ∈ F n ( n ) and y, z From this, we can get the assertion immediately.
In the following of this subsection, we always assume that f ∈ C(E 0 ).The following Lemma 3.10 and 3.11 are similar to Lemma 3.4 and 3.5, respectively.Lemma 3.10.Let n and γ n > 0 be given by Proposition 5. Then there exists Let f ∈ C(E 0 ) satisfy the condition (55).From (42), it follows that for any n (w 1 (y), w 1 (z))} for all y, z ∈ B n (x).
Hence, we conclude that for any y, z ∈ B n (x), (by Proposition 5) For such a ε n > 0 given by Lemma 3.10, we let Then δ n > 0. For any x ∈ B c (ε n ), we define We always assume the "balls" B n (x) to be defined by ( 41) and (57), respectively.It will depend on the location of x.
We remember that for any n ∈ N, Proof.From (C2), together with ( 16), we deduce that, by making use of ( 10) and the twice continuously Fréchet differentiability of τ , there exist constants 0 < r 0 ≤ r n < 1 and R n > 0 such that (ii) for any 2 ≤ j ≤ m and y, z ∈ X, Then for any fixed b ≥ b n , from (58), together with the above (i) and (ii), we deduce that, by a direct verification, for any x ∈ D n (ε n ) and y, z ∈ B n (x), and for any x ∈ F n and y, z ∈ B n (x), From ( 59) and (60), we deduce that for any In the following we let ε n and b n be determined by Lemma 3.10 and 3.11, respectively.We assume that, without loss of generality, ; Remember that w j : X → X j (1 ≤ j ≤ m) are the uniquely extension of (τ | X • j ) −1 .We assume, without loss of generality, that τ i (x) ≥ τ i (y) for any 1 ≤ i ≤ d and any x, y ∈ X 1 with x ≥ y.Then we have w k+1 1 (X 1 ) ⊂ w k 1 (X 1 ) for any k ∈ N. Hence, for any n > 1 and any 2 If d = 1, we let ε > 0 and b > 0 be determined by Corollary 4, and let δ > 0 be defined by (25).We may assume ε is small enough so that Then we have From this, together with Lemma 3.4 and 3.5 (precisely ( 29) and ( 30)), we deduce that for any multi-index From the fact that τ n w J (z) = z for all z ∈ X, it follows that Hence, we conclude that Note that D(ε) is compact.There exist finite points we assume that, without loss of generality, there exist Z j ∈ D(ε), 0 ≤ j ≤ q, such that It follows from ( 26) that Z j−1 , Z j ∈ B(Q j ).From this, together with (64), we can deduce that Hence, from this, together with (64) , ( 29) and ( 30), we conclude that for any multi-index (65) If d = 3, we let ε n > 0 and b n > 0 be determined by Lemma 3.11, and let δ n be defined by (56).We may assume ε n is small enough so that Then we have From this, together with Lemma 3.10 and 3.11 (precisely (59) and ( 60)), we deduce that for any multi-index Note that Hence, we conclude that Let D ⊆ R d and let H ⊆ C(D).We call H is equicontinuous at x(∈ D) (see e.g.[5]), if for any ε > 0 there exists a δ > 0 such that for any y ∈ D with |y − x| < δ, And, the set H is said to be equicontinuous on D if H is equicontinuous at any x ∈ D. If there is no confusion, we say the set H is equicontinuous.(ii) H 0 is equicontinuous on E 0 .
Proof.(i) It is obvious that H 0 ⊂ C(E 0 ).From Proposition 7, it follows that for any n ∈ N, Similarly, we have Hence, we conclude that This implies that H 0 ⊂ D(M ).
If d = 1, as for any x ∈ E 0 , the set B(x) is a neighborhood of x.And from the definition of ∆(x, y), we conclude that lim y→x ∆(x, y) = 1.
From this, together with the claim (70), we conclude that H 0 is equicontinuous on E 0 .
If d > 1, we know from (41) that for any x ∈ E 0 , there exists a n ∈ N such that B n (x) is a relative neighborhood of x.From the definition (61) of ∆ x n (y, z), it follows that lim y→x ∆ x n (y, x) = 1.
From this, together with (44) and the claim (70), we conclude that H 0 is equicontinuous on E 0 .
Proof of Theorem 4.1.(i) To prove Theorem 4.1, we define for any ∈ N large enough, D = X \ B (1+ ) −1 (0).For any integer n ∈ N, let Then g n ∈ H 0 .From Lemma 4.2(ii), we deduce that, for any ∈ N, the sequence {g n } ∞ n=1 is pointwise bounded and equicontinuous on D , i.e., the sequence n=1 is bounded and equicontinuous on compact set D .We know from the Arzelà-Ascoli theorem that there exists a subsequence {n k=1 converges in the supremum norm on D 1 .Similarly, there exists a subsequence {n k=1 converges in the supremum norm on D 2 .And so on, we can get a subsequence {g n ( ) k=1 , which converges in the supremum norm on D .From Then, the function h is well-defined.And from (69), it follows that From this, together with Proposition 3 and 6, we deduce that ) ∈ H 0 , we conclude that for any fixed x ∈ E 0 , the sequence From this, together with (71), it follows that for any x ∈ E 0 , Hence, T h = h.
(ii) For any Borel set A ⊆ X, we define Then we can confirm that μ is an invariant measure of the system (X, τ ).Let ϕ 0 and g 0 be defined by ( 62) and (67), respectively.From (68), it follows that g 0 ≈ 1 ϕ 0 as x → 0.
And, Theorem 3.1 of paper [32] says that the system (X, τ ) admits a unique invariant measure.From this, together with Radon-Nikodym theorem, we conclude that h = c • g 0 a.e.(λ) for some c > 0. Hence, we conclude that To prove our main result (Theorem 1.1), since the density function h is unbounded, we need to normalize the Perron-Frobenius operator T .For this, we show the following Lemma 4.3 first.Proof.From Theorem 4.1, we conclude that for any x near 0, This, combined with (72), implies the assertion.
Let h > 0 be the density function determined by Theorem 4.1, and let exists .
And for any f ∈ C 0 (E 0 ), we define We can verify that (C 0 (E 0 ), • ) is a Banach space.Let C * 0 (E 0 ) denote the dual space of C 0 (E 0 ).
In the following we restrict the operator T on C 0 (E 0 ).If there is no confusion, we use T to denote T | C0(E0) .We can verify that the operator T is a self-map on C 0 (E 0 ).Hence the operator T can be regarded as to be defined by Hence, we conclude that This means that the operator T is continuous on C 0 (E 0 ).
Let C(X) denote the Banach space of continuous functions on X with the supre- Then, the inverse map π Let C * 0 (E 0 ) be the dual space of C 0 (E 0 ), and let T * denote the dual operator of T .For any ξ ∈ C * 0 (E 0 ), we denote ξ ≥ 0 if ξ(f ) ≥ 0 for all 0 ≤ f ∈ C 0 (E 0 ).
Let 1(x) ≡ 1 for any x ∈ X.We define Let M (X) denote the set of all Borel probability measures on X.Without loss of generality, by noting Riesz representation theorem, we may regards M (X) as a subset of C * (X).And then, M (X) is a convex and weak-star compact subset of C * (X) (see e.g.[5]).

And for any
It is worth to mention that, in the case that τ is expanding, a uniform convergence theorem has been established (see e.g.[2,3,17]); and for non-uniformly expanding systems on the interval [0, 1], readers may refer papers [9,27,28,6,19,20].We would like to point out that Theorem 4.4 deals with multi-dimensional systems and establishes a convergence theorem.In fact, it is a generalization of paper [18] if h ∈ L 1 (λ); and moreover, Theorem 4.4 includes the case that h / ∈ L 1 (λ), where λ is Lebesgue measure, and h is the density function of system (X, τ ) (see e.g.Example 5.1, 5.2 and 5.3).This is a new and interesting result.
Note that G is dense on the set {f ∈ C(X) : f > 0}.From this, together with the equicontinuity of {L n g} ∞ n=1 , we deduce that the sequence {L n f } ∞ n=1 is equicontinuous on E 0 .For any f ∈ C(X), we can find a constant a > 0 such that f + a > 0. Then {L n (f + a)} ∞ n=1 and {L n a} ∞ n=1 are equicontinuous on E 0 .And hence, the assertion follows.
Proof of Theorem 4.4.We claim that for any f ∈ C(X) there exists a constant c f such that For this, let D be defined as in the proof of Theorem 4.1.For any f ∈ C(X), from Lemma 4.5, we see that {L n f } ∞ n=1 is uniformly bounded and equicontinuous on compact set D , ∈ N. By making use of the Arzelà-Ascoli theorem and the diagonal line method showed as in the proof of Theorem 4.1, we conclude that there exists a subsequence {n k } ∞ k=1 such that the sequence {L n k f (x)} ∞ k=1 converges in the supremum norm on all D , ∈ N, i.e., for any ∈ N, there is a continuous function Then, the function f is well-defined.And From (80), (81) and the continuity of L n k f on X, we conclude that f is bounded and continuous on E 0 .Let B c (E 0 ) denote the set of all bounded continuous functions defined on E 0 .For any g ∈ B c (E 0 ), we let By taking limit, we have ς(L f ) ≤ ς( f ) and hence equality holds.And, we get similarly that ς(L n f ) = ς( f ) for all n ∈ N.
Remark 5. From Theorem 1 of paper [25], together with Lemma 3.1 and Lemma 2.3, it follows that ϕ 0 (x) ≈ β(x) as x → 0. And from this, together with Theorem 4.1, we conclude that h(x) ≈ 1 β(x) as x → 0. Corollary 7. Let τ ∈ 1 and let υ be the probability measure determined by Proposition 8.Then, invariant measure of τ is infinite if and only if υ is a point measure at 0.
Let f = f 0 /h.Note that there exists a constant c > 0 such that 1 h(x) ≥ c for any x ∈ K. From this, together with Proposition 8, it follows that From this, together with Theorem 4.4, we conclude that for any x ∈ E 0 , lim n→∞
It is a contradiction.And the claim follows.This, combined with the probability of υ, implies the assertion.
For any Borel subset B ⊆ [0, 1], we let µ(B) = B h(x)dλ(x).Then µ is the invariant measure of τ .The transformation τ is called conservative with respect to λ, if λ(W ) = 0 for any measurable set W ⊂ X such that {τ −n W } n≥0 are pairwise disjoint.Remark 6.We would like to point out that τ (∈ 1 ) is conservative with respect to λ.Indeed, we know from [Lemma,28] that for all x ∈ B • (ε).
From this, together with Theorem 4.1(ii), it follows that there exist constants c 1 , c 2 > 0 such that for any x ∈ E 0 , By applying (1.7) of [26], we conclude that τ is conservative with respect to λ. Find a constant a such that α 0 > a > 1.This implies that there exists a δ > 0 such that log β(x) log x > a for all 0 < x < δ. (ii) can be proved similarly.
From this, together with (89) and (90), we deduce that there exists a α ∈ R such that lim Hence, we conclude from (89) that the function f (x) = τ (x) − x satisfies Lamperti's criterion.
We would like to point out that from [26] and Proposition 9(ii), we see that the Adler's condition is not satisfied provided that α 0 < 0. And from the following examples, we see that the invariant measure may be finite or infinite, if α 0 = 0.
be as in Lemma 3.10.Then there exists a constant b n > 0 such that for any fixed b ≥ b n and x

,
by using the diagonal line, we can get a sequence {g n (k) k } ∞ k=1 .From the above argument, we deduce that the sequence {g n (k) k (x)} ∞ k=1 converges in the supremum norm on all D , ∈ N, i.e., for any ∈ N, there is a h ∈ C(D ) such that lim k→∞ sup x∈D g n (k) k (x) − h (x) = 0. (71) Note that D ⊆ D +1 and E 0

= 1
for any n ∈ N.

Corollary 5 .Proposition 8 .
For any f ∈ C(X),lim n→∞ T n f (x) = ξ(f )h(x) for all x ∈ E 0 .Proof.From Theorem 4.4, we get the assertion immediately.Let h and ξ be determined by Theorem 4.1 and Theorem 4.4, respectively.Then there exists a unique regular Borel probability measure υ on X such that ξ(g) = X g(x) h(x) dυ(x) for all g ∈ C 0 (E 0 ).