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Invariant Measures for Set-Valued Dynamical Systems TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 351, Number 3, March 1999, Pages 1203{1225 S 0002-9947(99)02424-1 INVARIANT MEASURES FOR SET-VALUED DYNAMICAL SYSTEMS WALTER MILLER AND ETHAN AKIN Abstract. A continuous map on a compact metric space, regarded as a dy- namical system by iteration, admits invariant measures. For a closed relation on such a space, or, equivalently, an upper semicontinuous set-valued map, there are several concepts which extend this idea of invariance for a measure. We prove that four such are equivalent. In particular, such relation invari- ant measures arise as projections from shift invariant measures on the space of sample paths. There is a similarly close relationship between the ideas of chain recurrence for the set-valued system and for the shift on the sample path space. 1. Introduction All our spaces will be nonempty, compact, metric spaces. For such a space X let P (X) denote the space of Borel probability measures on X with δ : X P (X) → the embedding associating to x X the point measure δx. The support µ of a measure µ in P (X) is the smallest∈ closed subset of measure 1. If f : X | |X is 1 → 2 Borel measurable then the induced map f : P (X1) P (X2) associates to µ the measure f (µ) defined by ∗ → ∗ 1 (1.1) f (µ)(B)=µ(f− (B)) ∗ for all B Borel in X2. We regard a continuous map f on X as a dynamical system by iterating. A measure µ P (X) is called an invariant measure when it is a fixed point for the map f : P (∈X) P (X), i.e. f (µ)=µ. ∗ → ∗ A relation F : X1 X2 is a subset of X1 X2. We associate to x X1 the subset of X given→ by F (x)= y:(x; y) ×F . The relation is a function∈ 2 { ∈ } when each F (x) is a singleton. For A X1,theimage F (A)= F(x):x ⊂ 1 { ∈ A = y:y F(x)forsomex A.Theinverse relation F − : X2 X1 is } { ∈ ∈ } 1 S → (y;x):(x; y) F .SoforB X2,F− (B)= x:F(x) B= . In particular, { ∈ } ⊂ { 1 ∩ 6 ∅} the domain of F , denoted Dom(F ), is given by F − (X2)= x X1 :F(x)= . F is a closed (or Borel) relation when it is a closed (resp. Borel){ ∈ subset of X 6 X∅}. 1 × 2 A function is a closed relation iff it is a continuous function. If F : X1 X2 and G : X X are relations then the composed relation G F : X →X is 2 → 3 ◦ 1 → 3 (x; z):z G(y)forsomey F(x) , i.e. the projection to X1 X3 of the subset ({F X ) ∈(X G) X X∈ X }. The composition of closed× relations is closed. × 3 ∩ 1 × ⊂ 1 × 2 × 3 Received by the editors June 14, 1996. 1991 Mathematics Subject Classification. Primary 54H20, 58F10, 34C35. Key words and phrases. Set-valued dynamical system, dynamics of a relation, sample path spaces, invariant measure, basic set, chain recurrence. c 1999 American Mathematical Society 1203 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1204 WALTER MILLER AND ETHAN AKIN For the elementary facts about relations, and the notation we will use, see Akin (1993), Chapter 1. Both for its own sake and for application to the usual function case, the dynamics of a relation F on X, i.e. F : X X, have recently received some attention, e.g. Aubin and Frankowska (1990), McGehee→ (1992) and Akin (1993). As in the function case, the dynamics are obtained by iterating. By convention, F 0 is the identity map 1 2 n 1X , F = F , F = F F and F is the n-fold composition for any positive integer n ◦ 1 n n; and we write F − for (F − ) . For a relation there are two concepts of invariant measure in the literature which extend the continuous function notion. We introduce two more and will prove in Section 3 that for a closed relation F on X all four definitions are equivalent. (1) Aubin, Frankowska and Lasota (1991) call a measure µ P (X) invariant for F if ∈ 1 (1.2) µ(B) µ(F − (B)) ≤ for all Borel subsets B of X. We will then call µ an invariant1 measure.IfF=f 1 1 is a function then f − (A)andf− (X A) are disjoint, and so, applying (1.2) to B = A and B = X A, we obtain equality.\ By (1.1) this condition says µ = f µ, and so µ is an invariant\ measure for the map. ∗ (2) A Markov kernel κ from X1 to X2 is a function from X1 to P (X2), associating to each x X1 ameasureκx on X2 such that for each Borel set A, κx(A)isa Borel measurable∈ real valued function on X . For example, if f : X X is a 1 1 → 2 Borel measurable function, then δf = δ f : X P (X ) is the Markov kernel ◦ 1 → 2 associated with f, associating to x the point measure δf(x). For a Markov kernel κ : X1 P (X2) there is an induced mapping on measures κ : P (X1) P (X2) defined→ by ∗ → (1.3) κ (µ)(B)= κx(B)µ(dx) ∗ ZX1 for B a Borel subset of X2. Notice that κ (µ) depends only on the measures κx ∗ for x µ. Clearly, for κ = δf , κ = f .IfF:X1 X2we say that the Markov kernel∈|κ : |X P (X )issupported∗ by ∗F when → 1 → 2 (1.4) κx F(x) for all x X : | |⊂ ∈ 1 For example, let f : X X be a Borel measurable selection function for F ,a 1 → 2 function satisfying f(x) F (x) for all x X1.Thenδf :X1 P(X2) is a Markov kernel supported by F . ∈ ∈ → 1.1. Lemma. If F : X1 X2 is a closed relation such that Dom(F )=X1, i.e. 1 → F − (X2)=X1, then there exists a Borel measurable selection function for F . Proof. Let X [0; 1] denote the Cantor set. By Hocking and Young (1961), 3 ⊂ Theorem 3.28, there exists a continuous map h from X3 onto the compact metric 1 ~ 1 space X2. h− F is a closed relation from X1 to X3.Letf(x)=suph− (F(x)). 1 ◦ ~ 1 Since each h− (F (x)) is nonempty and closed, f is a selection function for h− F . ~ ◦~ f : X1 R is upper semicontinuous and so is Borel measurable. f = h f : X X→is therefore a Borel measurable selection map for F . ◦ 1 → 2 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use INVARIANT MEASURES 1205 For a closed relation F on X, Miller (1995) calls µ invariant for F if there exists a Markov kernel κ : X P (X) such that → (1.5) κx F(x) for all x µ and µ = κ (µ): | |⊂ ∈| | ∗ We will then call µ an invariant2 measure. If F = f is a continuous function on X,thenδf is the unique Markov kernel supported by f and (δf ) (µ)=f(µ). So condition (1.5) specializes to the usual notion of invariant measure∗ when ∗f is a map. (3) If µ is a measure on the product space X X , then the coordinate 12 1 × 2 projections πi : X1 X2 Xi (i =1;2) induce the marginal measures µi = πi∗ (µ12) (i =1;2). × → For a closed relation F on X X we will call µ an invariant3 measure if there exists µ P (X X) such that× 12 ∈ × (1.6) µ F; π (µ )=µ=π (µ ): | 12|⊂ 1∗ 12 2∗ 12 If F = f is a continuous function then the restriction of π1 to the subset f is a homeomorphism with inverse given by grphf : X X X: grphf (x)=(x; f(x)). The unique measure µ on X X with support→ in f and× satisfying π (µ )=µ 12 × 1∗ 12 is (grphf ) (µ). Since π2 grphf = f, π2∗ (grphf ) (µ)=f(µ). So again (1.6) specializes∗ to the proper notion.◦ ∗ ∗ (4) Let XZ denote the set of bi-infinite sequences in X regarded as functions Z of Z to X. With the product topology it is compact. Let π0 : X X be the projection π (ξ)=ξ .Theshift homeomorphism s : X Z X Z is defined→ by 0 0 → (1.7) s(ξ)i = ξi ;iZ: +1 ∈ For a closed relation F on X we denote by XF the sample path space for F , Z ξ : ξi F (ξi) for all i Z , a closed, s invariant subset of X .Sosrestricts { +1 ∈ ∈ } to a homeomorphism on XF , which we denote sF . The measures on XF are called Z sample path measures.Soν P(X) is a sample path measure when ν XF. Such a measure is called an invariant∈ sample path measure when it is s| invariant|⊂ or, equivalently, when it is sF invariant regarded as a measure on XF . We will call µ P (X)aninvariant4 measure for F if there exists an invariant ∈ Z sample path measure which projects to µ via π0 : X X, π0(ξ)=ξ0.Thatis, there exists ν P (XZ) such that → ∈ ν XF;s(ν)=ν | |⊂ ∗ (1.8) and π0∗ (ν)=µ. If F = f is a homeomorphism on X then π0 restricts to a homeomorphism of Xf to X inducing a conjugacy of sf with f.Soµis f invariant on X exactly when it is the image of an sf invariant measure under the conjugacy π0.
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