C 0-CONTINUITY OF THE FRÖBENIUS-PERRON SEMIGROUP

We consider the Fröbenius-Perron semigroup of linear operators associated to a semidynamical system defined in a topological space X endowed with a finite or a σ -finite regular measure. We prove that if there exists a faithful invariant measure for the semidynamical system, then the Fröbenius-Perron semigroup of linear operators is C0-continuous in the space Lμ(X). We also give a geometrical condition which ensures C0-continuity of the Fröbenius-Perron semigroup of linear operators in the space Lμ(X) for 1≤ p <∞, as well as in the space Lloc.


1.
Introduction.An important problem in the study of the dynamics of nonsingular transformations is to know if they admit an absolutely continuous invariant measure (acim).For interval maps, for example, we have a well-known theorem of Lasota and Yorke [5], which roughly states that if the map is smooth by parts (C r , with r ≥ 2) and expanding, then it admits an acim, and with some additional conditions it is exact with respect to this acim (for more details, see [5]); extensions of this result have been obtained for the n-dimensional case (see [3]).When we deal with a continuous semidynamical or a dynamical system (i.e., with a semi-flow or a flow) the problem is more complicated.
A useful technical tool for studying the problem of the existence of an acim is the Fröbenius-Perron operator (see [3,4] for more details).Let X be a topological space and let µ be a regular measure defined on the Borel σ -algebra of X (see Section 2.1); if τ : X → X is a nonsingular transformation, then the Fröbenius-Perron operator associated to τ, denoted by P τ (in fact P τ depend also on µ, and sometimes we use the notation P τ,µ in order to indicate such dependence on the measure), is a linear operator, naturally defined in the space L 1 µ (X).The central point here is that an invariant density, that is, a nonnegative measurable function of unit norm and fixed for the Fröbenius-Perron operator corresponds to a density of an acim for the transformation τ (see Section 2.4).
Let τ t : X → X be a semidynamical system.Denote by P t the Fröbenius-Perron operator associated to the transformation τ t .The family {P t } t≥0 = {P t,µ } t≥0 satisfies P 0 = Id, P t+s = P t • P s , ∀t, s ≥ 0, (1.1) that is, {P t } t≥0 is a semigroup of linear operators on L 1 µ (X).For a semigroup of continuous linear operators defined in the space L 1 µ (X), a central problem is to know if the semigroup is C 0 -continuous, that is, if the following relation holds: If this is the case, we may consider the infinitesimal generator of the semigroup which is defined by for elements f ∈ L 1 µ (X) for which the above limit exists (see [2,6]).It is known that a function f : X → R satisfies P t (f ) = f , for all t ≥ 0, if and only if A is defined for f and the differential equation A(f ) = 0 is satisfied.In this way, the problem of finding or proving the existence of an acim for the semidynamical system is equivalent to the problem of finding or proving the existence of a nontrivial zero for the infinitesimal generator of the Fröbenius-Perron semigroup of linear operators associated, provided that this semigroup is strongly continuous.
Let V be a smooth vector field defined in a smooth manifold, and let {τ t } t∈R be its flow.In this case, the Fröbenius-Perron operator, for f of class C 1 is given by the equation A(f ) = ∇(f V ) (where ∇ denote the divergence operator).We recover in this way a well-known theorem of Liouville which states that a flow preserves the canonical measure in the manifold if and only if the vector field has divergence equal to zero.The operator A defined in (1.3) can be viewed as a generalization of the divergence operator for continuous semi-flows for which the associated Fröbenius-Perron semigroup of linear operators is C 0 -continuous.
In this paper, we study general conditions that ensure this C 0 -continuity.In Section 2, we establish the notation and recall some basic results from the semigroup theory, and the definition of the Fröbenius-Perron operator and some of its properties.
In Section 3, we consider the case in which we know a priori that there exists a faithful acim for the system, that is, an acim with a positive density.In that case, we prove Theorem 3.3 which implies that the problem of finding a faithful acim is equivalent to the problem of finding a zero for the infinitesimal generator.
Since the problem to deal with is exactly the problem of the existence of the acim, we have to search an intrinsic property of the flow that ensures the strong continuity of the semigroup.The condition is: there exists T > 0 such that To understand condition (1.4), we may consider the case of a dynamical system.In that case, each transformation τ t : X → X has an inverse, and the associated Fröbenius-Perron operator is given by where J(τ −t ) is the density of the measure (τ t ) * (µ), where for more details see [4].
Thus, if we have an upper bound and good behaviour for J(τ −t ) near zero, then we use the dominated convergence theorem to prove that condition (1.2) holds for continuous functions.The extension of the result for arbitrary integrable functions is obtained by using the fact that the set of continuous functions is dense in the space of integrable functions.
For a semidynamical system, we do not have an explicit expression like (1.5) for the associated Fröbenius-Perron operator.However, a bound of J(τ −t ) can be interpreted as an estimate of the type that appears in condition (1.4).
In Section 4, we prove that, under general hypotheses, condition (1.4) holds if and only if the Fröbenius-Perron semigroup of linear operators can be defined in the space L p µ (X), and it is C 0 -continuous in that space (see Theorems 4.1 and 4.2).
In Section 5, we prove that condition (1.4) also ensures strong continuity in the space L 1 µ (X) when (X, Ꮽ,µ) is a probability space (Theorem 5.1).The precise statement of the main result is the following (for the concepts involved, see Section 2).Theorem 1.1.Let X be a topological space endowed with a regular probability measure µ.Let {τ t } t≥0 be a continuous proper semidynamical system defined on X.If the semidynamical system satisfies (1.4), then the associated Fröbenius-Perron semigroup of linear operators is Finally, to make this work more complete, we deal with the L 1 µ,loc (X) case in Section 6.The presentation is quite informal since there are some technical difficulties derived from the fact that L 1 µ,loc (X) is only a locally convex space and not a Banach space (we must add some hypotheses in this case in order to make the semigroup approach to the acim problem available).

Basic results.
In this section, we give a survey of definitions, results and notations that are necessary for the rest of the paper.

Measure theory.
Let X be a topological space and let Ꮽ be its Borel σ -algebra.Let µ be a measure defined over Ꮽ.We say that µ is regular if, for all A ∈ Ꮽ, we have We note that if X is a metric space, then a probability measure defined on the Borel σ -algebra is regular.In general, if a measure µ is regular, then the set of continuous functions with compact support is dense in the space L p µ (X), for all 1 ≤ p < ∞.

Semidynamical systems.
Let X be a topological space.A family {τ t } t≥0 of continuous transformations τ t : X → X is a semidynamical system if the following conditions are satisfied: (i) If each transformation τ t has a continuous inverse τ −t , then the family {τ t } t∈R is a continuous flow.However, for general semidynamical systems, the maps τ t may not necessarily have an inverse.
We say that a semidynamical system {τ t } t≥0 is proper if, for each compact set K ⊂ X and for each t > 0, the set ∪ s≤t τ −1 s (K) is compact.

Semigroups in Banach spaces.
Let L be a Banach space with respect to a norm • .A family {T t } t≥0 of continuous linear operators T t : L → L (t ≥ 0) is called a semigroup of linear operators if the following conditions are satisfied: For more details see [6] or [2].We say that a semigroup {T t } t≥0 of linear operators is When a semigroup {T t } t≥0 is C 0 -continuous, there exist constants M ≥ 1 and w ≥ 0 such that, for all f ∈ L we have If {T t } t≥0 is a semigroup defined on L, then the adjoint family {T * t } t≥0 is a semigroup defined on the dual space L * .By a duality theorem, we have that if

2.4.
Fröbenius-Perron operator.Let (X, µ, Ꮽ) be a measure space.We say that a transformation τ : X → X is nonsingular if for all A ∈ Ꮽ such that µ(A) = 0, we have µ(τ −1 (A)) = 0.If a transformation τ : X → X is nonsingular, then associated to it there exists a linear operator P τ = P τ,µ : L 1 µ (X) → L 1 µ (X), called Fröbenius-Perron operator which is characterized by the relation for all f ∈ L 1 µ (X) and all A ∈ Ꮽ.It is well known (see [1,4]) that a probability measure µ on X is τ-invariant (i.e., µ(τ −1 (A)) = µ(A) for all A ∈ Ꮽ) if and only if P τ (1) = 1 (this is also true for σ -finite measure spaces, but in that case there is a problem with the space where the Fröbenius-Perron operator is defined, as we will see).In general, τ preserves a measure dν = f dµ, with f ∈ L 1 µ (X) if and only if P τ (f ) = f .It is also well known that the Fröbenius-Perron operator is a linear contraction in L 1 µ (X) endowed with the L 1 µ -norm, that is, P τ L 1 µ ≤ 1 (see [3,4]).Moreover, for f ∈ L 1 µ (X) and a.e.x ∈ X, we have On the other hand, if we change the measure µ by an absolutely continuous one given by dν = gdµ, then the change in the Fröbenius-Perron operator is given by Another important property of the Fröbenius-Perron operator is given by the equality valid for all f ∈ L 1 µ (X) and all g ∈ L ∞ µ (X).Equation (2.7) permits us define a linear operator The operator K τ is well defined if τ is a nonsingular transformation.This operator is called the Koopman operator.
For more details about these concepts see [1] or [4].
If we have a semidynamical system {τ t } t≥0 such that each transformation τ t is nonsingular, then we denote the family of Fröbenius-Perron operators associated by P t = P τ t .This is a semigroup of continuous linear operators in the space L 1 µ (X) (see [4]).We will also use the notation K t for the Koopman operator K τ t .
We note that the Fröbenius-Perron operator may also be defined and is bounded in other spaces of functions if the transformation τ has a good behaviour, for example, L p µ (X) spaces or BV (X), the space of functions of bounded variation.For example, in Section 4, we consider this operator in L p µ (X) spaces, and we prove that the geometrical condition (1.5) ensures continuity of each operator P t (and also the C 0 -continuity of the semigroup {P t } t≥0 in the space L p µ (X)).We note that if P t is continuous in L p µ (X), then the duality equation (2.7) is valid for all f ∈ L p µ (X) and for all g ∈ L q µ (X), with 1/p + 1/q = 1, that is, K t is the adjoint operator of P t .

Conditional expectation and
Fröbenius-Perron operator.Let (X, Ꮽ,µ) be a probability space.Suppose τ : X → X preserves µ, then we have another way of introducing the Fröbenius-Perron operator associated to τ.This is given for f ∈ L 1 µ (X) by the equality where the expression on the right-hand side denotes the conditional expectation of f with respect to the σ -algebra τ −1 (Ꮽ).In Section 3, we use this approach and the following result of convergence that arises in Martingale theory (see [1, page 81]).
Theorem 2.1.Let {Ꮽ n } n∈N be a collection of σ -algebras such that Ꮽ n ⊂ Ꮽ n+1 , for all n.Then E(f , Ꮽ n ) converges to E(f , Ꮽ ∞ ), in the L 1 sense, where Ꮽ ∞ denotes the σ -algebra generated by all the Ꮽ n .

Strong continuity with an absolutely continuous invariant measure.
Let X be a topological space and let µ be a regular probability measure defined on the Borel σ -algebra on X.In this section, we consider a nonsingular semidynamical system {τ t } t≥0 defined over X.We prove that if there exists a faithful acim (an acim with positive density), then the associated Fröbenius-Perron semigroup is C 0 -continuous in L 1 µ (X).We first prove the following two lemmas.
Lemma 3.1.Suppose that {τ t } t≥0 has an acim ν such that dν = gdµ, with g > 0 (a.e.).Then, Proof.The sequence of σ -algebras {Ꮽ t } t≥0 , where Ꮽ t = τ −1 t (Ꮽ) for all t ≥ 0, is increasing as t goes to zero.By the martingale convergence theorem, we have, for f in L 1 ν (X), where Ꮽ ∞ is the σ -algebra generated by all the σ -algebras Ꮽ t for t > 0. Thus, we must prove that Ꮽ ∞ is equal to Ꮽ.For this, let A be an open set.By continuity of the semidynamical system {τ t } t≥0 , the function |χ A •τ t −χ A | converges pointwise to zero as t goes to zero, and by the dominated convergence theorem, we have converges to zero as t goes to zero.Thus, for each n ∈ N, we may consider a sequence This implies that A ∈ Ꮽ ∞ , and since A is an arbitrary open set, this implies that Ꮽ ∞ is equal to Ꮽ, which completes the proof of Lemma 3.1.
Proof.Let ε be an arbitrary positive number.We take a sequence {f n } n∈N of bounded continuous functions that converges to a function f ∈ L 1 ν (X) and choose n 0 ∈ N such that f n 0 −f ≤ ε/4.By the invariance of ν and since the operator P t,ν is a linear contraction, we have For small t, by Lemma 3.1 we have P t,ν (f n 0 )•τ t −f n 0 L 1 ν ≤ ε/4.Now, by continuity of the semidynamical system and by the dominated convergence theorem, we have lim t→0 f n 0 − f n 0 • τ t L 1 ν = 0. Thus, for t small we have f n 0 − f n 0 • τ t L 1 ν ≤ ε/4.This implies, for t small, that P t,ν (f )−f L 1 ν ≤ ε, which completes the proof of Lemma 3.2.Theorem 3.3.Suppose that the semidynamical system {τ t } t≥0 has a faithful acim.Then, the semigroup Proof.Let ν the faithful acim given by dν = gdµ, with g ∈ L 1 µ (X) and g > 0 (a.e.).Let f ∈ L 1 µ (X).Then, f /g ∈ L 1 ν (X), and we have, by equality (2.6), that By Lemma 3.2, the last quantity converges to zero as t goes to zero, which proves the theorem.

4.
Strong continuity in L p .In this section, we consider a semidynamical system defined over a topological space X provided of a regular measure µ defined on its Borel σ -algebra, and prove that condition (1.4) is equivalent to strong continuity of the associated Fröbenius-Perron semigroup defined over the space L p µ (X).We say that a semidynamical system is strongly nonsingular if it satisfies condition (1.4).It is easy to see that the following conditions for being strongly nonsingular are equivalent: (i) for each t > 0, there exists M t such that for all s ≤ t and all A ∈ Ꮽ; (ii) there exist T > 0 and M = M T > 0 such that for all A ∈ Ꮽ and all s ≤ T .In fact, condition (4.1) implies trivially (4.2), and if condition (4.2) is assumed, then condition (4.1) holds by putting Finally, it is easy to see that every strongly nonsingular semidynamical system is nonsingular.
Theorem 4.1.Let {τ t } t≥0 be a nonsingular semidynamical system such that its associated semigroup of Fröbenius-Perron operators {P t } t≥0 is a C 0 -continuous semigroup of bounded linear operators in the space L p µ (X) for some 1 < p < ∞.Then {τ t } t≥0 is strongly nonsingular.
Proof.By the hypothesis, the semigroup of linear operators {P t } t≥0 is a C 0 -semi group over a reflexive space.Thus, the semigroup of Koopman operators {K t } t≥0 is a C 0 -semigroup in the space L q µ (X), where q = p/(p − 1) is the conjugate of p (see [6, Corollary 10.6, page 41]).Therefore, there exist constants M ≥ 1 and w > 0 such that, for all f ∈ L q µ (X), we have that 1/q and from this it follows that µ(τ −1 t (A)) ≤ (Me wt ) q µ(A), which proves our theorem.
If a measure µ is regular and the semidynamical system {τ t } t≥0 is proper, then we have the converse of the above result.Theorem 4.2.Suppose the measure µ is regular on X.If {τ t } t≥0 is a proper and strongly nonsingular semidynamical system, then, for all 1 < p < ∞, the associated Fröbenius-Perron semigroup {P t } t≥0 is a C 0 -semigroup of linear bounded operators in the space L p µ (X).
The idea for the proof of Theorem 4.2, is to prove that condition (4.1) ensures strong continuity for the dual semigroup {K t } t≥0 in the dual space L q µ (X).For this we first prove the following two lemmas.Lemma 4.3.Under the hypotheses of Theorem 4.2, K s (f ) L q µ ≤ M 1/q t f L q µ for all f ∈ L q µ (X) and all s ≤ t.
Proof.Let f ∈ L q µ (X) be a simple function given by f = n i=1 λ i χ A i , with A i ∩A j = ∅, for i ≠ j, then we have On the other hand, by condition (4.1) we have, for s ≤ t, that and this implies that K s (f ) L q µ ≤ M 1/q t f L q µ .Finally, since the set of simple functions is dense in the space L q µ (X), the lemma follows.Proof.Let f ∈ L q µ (X) and let ε > 0. We take a sequence {f n } n∈N of continuous functions with compact support, say K, such that lim n→∞ f n −f L q µ = 0 and we choose where M = M T is fixed.By continuity of the Koopman semigroup, the function K s (f n 0 ) − f n 0 converges pointwise to zero as s goes to zero.If K is the support of f n 0 , then, since the semidynamical system is proper, the set K = ∪ s≤t τ −1 s (K) is compact.Now, it is clear that supp(K s (f n 0 ) − f n 0 ) ⊂ K for s ≤ t, and by the dominated convergence theorem we have lim s→0 K s (f n 0 ) − f n 0 L q µ = 0. Thus, we may take t 0 > 0 such that, for all s ≤ t 0 , we have Using inequalities (4.5) and (4.6), we have, for s ≤ min{T ,t 0 }, that which proves the lemma.
Proof of Theorem 4.2.It follows directly by the duality theorem (see Sections 2.3 and 2.4).

5.
Strong continuity in L 1 .One of the most important applications of Fröbenius-Perron operator arise in probability spaces by considering its action over the space L 1 µ (X).For these spaces, we have the following result.
Theorem 5.1.Let X be a topological space endowed with a regular probability measure µ and let {τ t } t≥0 be a proper semidynamical system.If the semidynamical system is strongly nonsingular, then the associated Fröbenius-Perron semigroup of operators Proof.Let f ∈ L 1 µ (X) and ε > 0. We fix p > 0 and consider a sequence (5.1) We choose n 0 such that f − f n 0 L 1 µ ≤ ε/3.By Theorem 4.1, there exists t ε > 0 such that P t (f n 0 )−f n 0 L p µ ≤ ε/3 for all t ≤ t ε .Therefore, for t ≤ t ε we have P t (f )−f L 1 µ ≤ ε.Since f and ε are arbitrary, this finishes the proof.Theorem 5.1 is useful for compact metric spaces of finite measure, in that case {τ t } t≥0 is proper.For noncompact or unbounded metric spaces with nonfinite measure we assume that the system is strongly continuous over bounded sets, that is, we assume that for each ball Theorem 5.2.Under the hypotheses of Theorem 5.1, suppose also that condition (5.3) is true and the measure is finite over bounded sets.Then, the Fröbenius-Perron semigroup is C 0 -continuous in L 1 µ (X).
The proof of Theorem 5.2 is based in the reduction of it to Theorem 5.1 in each ball B = B(x, r ).For that, we define a new measure μ in X by putting μ(A) = µ(A ∩ B). (5.4) This new measure is finite, and the semidynamical system {τ t } t≥0 is strongly nonsingular with respect to μ.By Theorem 4.2, we have, for each f ∈ L 1 μ (X), that lim t→0 Pt (f ) = f (5.5) in the L 1 μ sense, where Pt = P t,μ .Using this, we prove the following lemma.(5.9) We claim that µ(C k,t ) converges to zero when t goes to zero, for all k.To prove the claim, we consider the sets ( (5.12) As in the proof of Lemma 3.1, and using the fact that the system is proper, it is not difficult to see that µ(τ −1 t (B)−B) converges to zero when t goes to zero, which implies

Lemma 4 . 4 .
If the hypotheses of Theorem 4.1 are satisfied, then the Koopman semigroup of operators {K t } t≥0 is a C 0 -continuous semigroup of linear bounded operators in the space L q µ (X).

. 2 )
there exists a constant M = M(x, r ) and T = T (x,r ) such thatµ τ −1 t (A) ∩ B ≤ Mµ(A ∩ B) (5.3)for all t ≤ T and A ∈ Ꮽ.Now we have the following theorem.