aa r X i v : . [ m a t h . M G ] M a y Another point of viewon Kusuoka’s measure
Ugo Bessi*
Abstract
Kusuoka’s measure on fractals is a Gibbs measure of a very special kind, because its potential is discon-tinuous, while the standard theory of Gibbs measures requires continuous (actuallly, H¨older) potentials. Inthis paper, we shall see that for many fractals it is possible to build a class of matrix-valued Gibbs measurescompletely within the scope of the standard theory; there are naturally some minor modifications, but theyare only due to the fact that we are dealing with matrix-valued functions and measures. We shall use thesematrix-valued Gibbs measures to build self-similar Dirichlet forms on fractals. Moreover, we shall see thatKusuoka’s measure can be recovered in a simple way from the matrix-valued Gibbs measure.
Introduction
First of all, let us briefly explain what we mean by a fractal G on R d ; our definition is less general thanthe one in [12]. We consider n contractions ψ , . . . , ψ n ∈ C ,ν ( R d , R d ) (1)with ν ∈ (0 , G ⊂ R d such that G = n [ i =1 ψ i ( G ) . We shall also suppose that the maps ψ i are the ”branches of the inverse” of a Borel map F : G → G . Sincethe maps { ψ i } ni =1 are contractions, their inverse F is expanding; in Dynamical Systems, expanding mapshave been studied extensively (see for instance [13] or [19]); as we shall see, many results on expanding mapscarry over to Dirichlet forms on fractals (see [9] or [17] for an introduction to this theory).Applying ergodic theory to the study of Dirichlet forms on fractals is not new: in section 1.4 of [7] theidea of applying the Ruelle operator to the study of Kusuoka’s measure is attributed to Strichartz. Actually,in this paper we try to understand the results of [7] and [12] by looking at them from a slightly differentperspective. * Dipartimento di Matematica, Universit`a Roma Tre, Largo S. Leonardo Murialdo, 00146 Roma, Italy. email: [email protected]
Work partially supported by the PRIN2009 grant ”CriticalPoint Theory and Perturbative Methods for Nonlinear Differential Equations1n order to explain the connection between expanding maps and Dirichlet forms, we begin recalling thescalar Gibbs measure; though it will play no rˆole in our paper, it will guide us in the construction of thematrix-valued one.For ν ∈ (0 ,
1] we take V ∈ C ν ( G, R ) and define the scalar Ruelle operator as L sc : C ( G, R ) → C ( G, R ) , ( L sc v )( x ) = n X i =1 e V ◦ ψ i ( x ) v ◦ ψ i ( x ) . (2)Using the Perron-Frobenius theorem one can prove ([18]) that there is β > h > L sc h = βh . Since L sc is a continuous operator from the space of continuous functions intoitself, its adjoint L ∗ sc brings the space of Borel measures into itself; it can be shown that there is a measure µ , called the Gibbs measure, such that hµ is probability and L ∗ sc µ = βµ . One of the properties of µ is thefollowing: for all u, v ∈ C ( G, R ) we have that1 β Z G u ( L sc v )d µ = Z G ( u ◦ F ) v d µ. (3)We point out a few consequences of (3). First of all, we can write the adjoint L ∗ sc explicitly, at least formeasures absolutely continuous with respect to µ .1 β L ∗ sc ( uµ ) = ( u ◦ F ) µ. Moreover, (3) tells us that β L sc v is the density of F ♯ ( vµ ), where F ♯ ν denotes the push-forward of the measure ν by F ; the push forward is defined by Z G f d( F ♯ ν ) = Z G f ◦ F d ν for all f ∈ C ( G, R ). Since L sc h = βh , this implies that F ♯ ( hµ ) = hµ , i. e. that hµ is an invariant measure.Moreover, we know how µ [ ψ i ◦ . . . ψ i l ( G )] scales as l → + ∞ ; namely, there is a constant D > i i . . . , such that1 D ≤ µ [ ψ i ◦ . . . ψ i l ( G )] β − l · exp( V ( x ) + V ( F ( x )) + . . . + V ( F l − ( x ))) ≤ D for all x ∈ ψ i ◦ . . . ψ i l ( G ).Kusuoka’s measure κ is defined by a similar scaling property; when the maps ψ i of (1) are affine, i. e. Dψ i is a constant matrix, we have κ [( ψ x ◦ . . . ψ x l )( G )] = 1 β l · tr[ ˆ Q ( Dψ x · . . . · Dψ x l ) Q t ( Dψ x · . . . · Dψ x l )]where Q , ˆ Q are two suitable symmetric d × d matrices, β > t Q denotes the transpose of Q and tr isthe trace. As explained in [7] (see [3] for the proof and further details), κ is a Gibbs measure, but for adiscontinuous potential V , and the standard theory does not apply to it.2et us come to the Dirichlet form. On many fractals, the space L ( S, κ ) admits a ”heat semigroup” P s which is induced, as in R d , by a Brownian motion ([1], [2], [6], [11]). The semigroup P s has a generator AAf = lim h ց P h f − fh which induces a Dirichlet form on L ( G, κ ); this form is defined on a dense subspace D ( E ) ⊂ L ( S, m ) by E ( f, g ) = − Z G ( Af ) · g d m if g ∈ D ( E ) and f ∈ D ( A ) . Conversely ([5]), given a Dirichlet form E it is relatively easy to check whether it is induced by a Brownianmotion; since Dirichlet forms are easier to study than the Brownian motion itself, they immediately attractedattention ([14], [15]; a counterexample to the existence of Dirichlet forms is in [16]).This brings us to the matrix-valued Gibbs measure: under suitable hypotheses on the fractal G ⊂ R d ,the ”natural” Dirichlet form E on L ( G, κ ) can be written in the following way: if u, v ∈ C ( R d , R ), then E ( u, v ) = Z G ( T x ∇ u ( x ) , ∇ v ( x ))d κ ( x )where ( · , · ) denotes the standard inner product in R d , T x is a Borel field of symmetric matrices (in manycases, projections) and κ is Kusuoka’s measure.The matrix-valued measure τ : = T x κ appears in a natural way in the formula above; the aim of thispaper is to show that τ is a Gibbs measure as well.More precisely, we denote by M d the space of symmetric d × d matrices; we can define a Ruelle operator L G : C ( G, M d ) → C ( G, M d )by ( L G A )( x ) = n X i =1 t Dψ i ( x ) A ( ψ i ( x )) Dψ i ( x ) . (4)The dual space of C ( G, M d ) is the space M ( G, M d ) of M d -valued measures on G , and the adjoint L ∗ G of L G brings M ( G, M d ) into itself. As we shall see, L G and L ∗ G have each a positive-definite eigenvector, whichwe shall call Q G and τ G respectively. In lemma 4.8 below we shall prove the following version of (3): if g ∈ C ( G, R ) and A ∈ C ( G, M d ), defining the integral as in section 2 below we have Z G (cid:18) g · (cid:18) β L G A (cid:19) , d τ G (cid:19) HS = Z G ( g ◦ F · A, d τ G ) HS . The formula above implies that the scalar measure ( Q G , τ G ) HS (again, see section 2 for the definition) isinvariant. Theorem 1.
Let the maps ψ , . . . , ψ n and F satisfy hypotheses (F1)-(F4) of section 1 below and (ND)with constant b > at the beginning of section 4. Let G be the fractal associated with ψ , . . . , ψ n and let M d denote the space of d × d symmetric matrices. Let the operator L G be defined as in (4) and let || Dψ i || ν e smaller than a positive constant that depends on b > (which is always true if (ND) holds and the maps ψ i are affine); then the following holds.1) There are Q G ∈ C ( G, M d ) and β > such that ( L G Q G )( x ) = βQ G ( x ) ∀ x ∈ G. The map Q G belongs to C ν ( G, M d ) and is unique up to multiplication by a constant; again up to multipli-cation by a constant, Q G ( x ) is positive-definite for all x ∈ G .2) Let L ∗ G denote the adjoint of L G ; then, there is a Borel measure τ G on G which takes values in M d andsuch that L ∗ G τ G = βτ G . The measure τ G is unique up to multiplication by a constant; again up to multiplication by a constant, τ G takes values in semi-positive definite matrices.3) The measures || τ G || and κ G : = ( Q G , τ G ) HS are mutually absolutely continuous. Moreover, κ G is ergodicfor the map F .4) We define the form E : C ( R d ) × C ( R d ) → R in the following way (the notation for the integral is insection 2 below): E ( f, g ) = Z G ( ∇ f ( x ) , d τ G ( x ) ∇ g ( x )) . Then, E is self-similar, i.e. E ( f, g ) = 1 β n X i =1 E ( f ◦ ψ i , g ◦ ψ i ) for all f, g ∈ C ( R d ) .5) Let the maps ψ i be affine. Then, the measure τ G has the Gibbs property, i. e., for all x , . . . , x l ∈ (1 , . . . , n ) we have that τ G ( ψ x ◦ . . . ◦ ψ x l − ( G )) = 1 β l · ( D ( ψ x ◦ . . . ◦ ψ x l − )) · τ G ( G ) · t ( D ( ψ x ◦ . . . ◦ ψ x l − )) . We haven’t written explicitly the point where we calculate D ( ψ x ◦ . . . ◦ ψ x l ) since these maps are affine andtheir derivative is constant. Note that in point 4) we do not assert that E is closable: actually, we don’t know any criteria forclosability other than the ones in [9] and [12].The paper is organised as follows. In section 1 we recall the notation and the basic facts about thePerron-Frobenius theorem, fractal sets and Dirichlet forms. In section 2 we define the convex cones towhich we are going to apply the Perron-Frobenius theorem. In section 3 we define the Ruelle operator L G on matrices and show that the fixed points of its adjoint L ∗ G induce a self-similar quadratic form E on C ( R d ). In section 4, we apply the Perron-Frobenius theorem to find the maximal eigenvector of L G andthe matrix-valued Gibbs measure τ G . In section 5, we show that τ G has the Gibbs property.4 We follow [19] (see also [4] for the original treatment).Let X be a real vector space; we say that C ⊂ X \ { } is a cone if v ∈ C and t > tv ∈ C . Let
C ⊂ X be a convex cone; we say that w ∈ ¯ C if there are v ∈ C and t n ց w + t n v ∈ C for all n ≥
1. In what follows, we shall suppose that C is a convex cone such that¯ C ∩ ( − ¯ C ) = { } . (1 . v , v ∈ C , we define α ( v , v ) = sup { t > v − tv ∈ C} (1 . β ( v , v ) = sup { t > v − tv ∈ C} (1 . θ ( v , v ) = log β ( v , v ) α ( v , v ) . (1 . θ ( v, λv ) = 0 for all λ >
0, we identify the points of a ray; namely, we say that v ≃ v if v = tv for some t >
0; we shall denote by C≃ the set of equivalence classes.We have that θ ( v , v ) ∈ [0 , + ∞ ] for all v , v ∈ C ; if θ never assumes the value + ∞ , then θ is a distanceon C≃ .The following proposition from [19] allows us to use the contraction principle. Proposition 1.1.
1) Let L : X → X be a linear operator such that L ( C ) ⊂ C and let us define D = sup { θ ( Lv , Lv ) : v , v ∈ C} . Then, if
D < + ∞ , L is a contraction on ( C≃ , θ ) , namely θ ( Lv , Lv ) ≤ (1 − e − D ) θ ( v , v ) ∀ v , v ∈ C .
2) As a consequence of 1), if
D < + ∞ and ( C≃ , θ ) is a complete metric space, there is ( λ, v ) ∈ (0 , + ∞ ) × C ,unique in (0 , + ∞ ) × C≃ , such that Lv = λv. Moreover, if w ∈ C , then θ ( L n w, v ) ≤ θ ( w, v ) (1 − e − D ) n e − D . (1 . Fractal sets.
We make the following hypotheses on the fractal set.5
F1)
There is ν ∈ (0 ,
1] and diffeomorphisms ψ , . . . , ψ n ∈ C ,ν ( R d , R d ) (1 . η : = sup i ∈ (1 ,...,n ) Lip ( ψ i ) < . (1 . G ⊂ R d such that G = n [ i =1 ψ i ( G ) . (1 . R d in such a way thatdiam( G ) ≤ . (1 . F on G can be coded. Indeed, we define Σ as the space of sequencesΣ = { , . . . , n } N = {{ x i } i ≥ : x i ∈ (1 , . . . , n ) , ∀ i ≥ } with the product topology. This is a metric space; for instance, if γ ∈ (0 , d γ ( { x i } i ≥ , { y i } i ≥ ) = γ k where k = inf { i ≥ x i = y i } , with the convention that the inf of the empty set is + ∞ .We define the shift σ as σ : Σ → Σ , σ : { x , x , x , . . . } → { x , x , x , . . . } . If x , . . . , x l ∈ (1 , . . . , n ), we define the cylinder[ x . . . x l ] = {{ y i } i ≥ : y i = x i for i ∈ (1 , . . . , l ) } . We also set ψ x ...x l = ψ x ◦ . . . ◦ ψ x l and [ x . . . x l ] G = ψ x ◦ ψ x ◦ . . . ◦ ψ x l ( G ) . (1 . x = ( x x . . . ) we set ( ix ) = ( ix x . . . ). Now (1.10) implies that ψ i ([ x . . . x l ] G ) = [ ix . . . x l ] G . (1 . ψ i are continuous and G is compact, the sets [ x . . . x l ] G ⊂ G are compact. By (1.8) we havethat ψ i ( G ) ⊂ G for i ∈ (1 , . . . , n ); this implies that, for all { x i } i ≥ ∈ Σ[ x . . . x l − x l ] G ⊂ [ x . . . x l − ] G . From (1.7), (1.9) and (1.10) we get that diam([ x . . . x l ] G ) ≤ η l . (1 . { x i } i ≥ ⊂ Σ; by the last two formulas and the finite intersection property we have that \ l ≥ [ x . . . x l ] G is a single point, which we call Φ( { x i } i ≥ ); formula (1.12) implies in a standard way that the map Φ: Σ → G is continuous. It is not hard to prove, using (1.8), that Φ is surjective. We shall call ˜ d the distance on G induced by the Euclidean distance on R d and, from now on, in our choice of the metric on Σ we take γ ∈ ( η, d γ and (1.12) that Φ is 1-Lipschitz. (F2) If i = j , ψ i ( G ) ∩ ψ j ( G ) is a finite set. We set F : = [ i = j ψ i ( G ) ∩ ψ j ( G ) . (F3) We ask that G is post-critically finite, which means the following: if we set A = Φ − ( F ), then the set ∪ j ≥ σ j ( A ) is finite. (F4) We ask that there are disjoint open sets O , . . . , O n ⊂ R d such that G ∩ O i = ψ i ( G ) \ [ i = j ψ i ( G ) ∩ ψ j ( G ) for i ∈ (1 , . . . , n ) . We define a map F : S ni =1 O i → R d by F ( x ) = ψ − i ( x ) if x ∈ O i . If moreover we ask that O i ⊂ ψ − i ( O i ) (or, equivalently, that ψ i ( O i ) ⊂ O i , since the maps ψ i are diffeos),this implies the first equality below. F ◦ ψ i ( x ) = x ∀ x ∈ O i ⊂ ψ − i ( O i ) and ψ i ◦ F ( x ) = x ∀ x ∈ O i . (1 . a i the unique fixed point of ψ i ; note that, by (1.8), a i ∈ G . If x ∈ F , we define F ( x ) = a j for somearbitrary a j . This defines F as a Borel map on all of G , which satisfies (1.13).We point out a consequence of (F4): the sets O i and ψ j ( G ) do not intersect unless i = j . By thedefinition of the coding this implies that, if z = Φ( { x j } j ≥ ) ∈ O i , then x = i . Since ψ x is a diffeo, also thesets ψ x ( O j ) and ψ x ◦ ψ l ( G ) do not intersect unless l = j . Thus, if in addition z ∈ S nj =1 ψ x ( O j ), z belongs7o a unique ψ x ( ψ x ( G )) and also x is uniquely determined. Going on, we see that z has a unique codingunless it belongs to the countable set [ n ≥ [ i ,...,i n ψ i ...i n ( F ) . By (1.10) and the definition of Φ we easily get that [ x . . . x l ] ⊂ Φ − ([ x . . . x l ] G ); since the set where Φis not injective is countable, we have that ♯ (Φ − ([ x . . . x l ] G ) \ [ x . . . x l ]) ≤ ♯ N . (1 . x = ( x x . . . ), then by the definition of ΦΦ ◦ σ ( x ) = \ l ≥ [ x . . . x l ] G . The definition of F implies the first equality below; if we suppose that Φ( x ) ∈ O x and recall that F = ψ − x on O x we get the middle one while the last equality comes from the formula above. F ◦ Φ( x ) = F \ l ≥ [ x . . . x l ] G = \ l ≥ [ x . . . x l ] G = Φ ◦ σ ( x ) . In other words, the first equality below holds save when Φ( x ) ∈ F . The second equality below follows for all x ∈ G from (1.11). (cid:26) Φ ◦ σ ( x ) = F ◦ Φ( x ) save possibly when Φ( x ) ∈ F ,Φ( i, x ) = ψ i (Φ( x )) ∀ x ∈ Σ , ∀ i ∈ (1 , . . . , n ) . (1 . F . Iterating the first one of (1.15) we get that, for all l ≥ ◦ σ l ( x ) = F l ◦ Φ( x ) save possibly for x ∈ [ j ≥ σ − j (Φ − ( F )) . (1 . − ( F ) is finite by (F3).A particular case we have in mind is the harmonic Sierpinski gasket on R ([8], [10]). We set T = (cid:18) , , (cid:19) , T = , √ √ , ! , T = , − √ − √ , ! ,A = (cid:18) (cid:19) , B = (cid:18) √ (cid:19) , C = (cid:18) − √ (cid:19) and ψ ( x ) = T ( x ) , ψ ( x ) = B + T ( x − B ) , ψ ( x ) = C + T ( x − C ) . Referring to the figure below, ψ brings the triangle ABC into
Abc ; ψ brings ABC into
Bac and ψ brings ABC into
Cba . We take O , O , O as three disjoint open sets which contain, respectively, the triangle Abc minus b, c , Bca minus c, a and
Cba minus a, b . 8e define the map F as F ( x ) = ψ − i ( x ) if x ∈ O i and we extend it as in (F4) on { a, b, c } .It is easy to check that the fractal G generated by ψ , ψ , ψ satisfies hypotheses (F1)-(F4) above; it iseasy to check that it also satisfies (ND) of section 4 below. The invariant Dirichlet form.
Let ( X, ˜ d, ν ) be a metric measure space; we suppose for simplicity that( X, ˜ d ) is compact and ν is probability.A Dirichlet form is a symmetric bilinear form E : D ( E ) × D ( E ) → R defined on a dense set D ( E ) ⊂ L ( X, ν ) such that the two conditions below hold.(D1) D ( E ) is closed under the graph norm; in other words, D ( E ) is a Hilbert space for the norm || u || D ( E ) = || u || L ( S,m ) + E ( u, u ) . (1 . E is Markovian, i. e. E ( η ◦ f, η ◦ f ) ≤ E ( f, f )for all f ∈ D ( E ) and all 1-Lipschitz maps η : R → R with η (0) = 0.We list some additional properties a Dirichlet form can have.(D3) E is regular; this means that D ( E ) ∩ C ( X, R ) is dense in C ( X, R ) for the uniform topology and in D ( E )for the graph norm (1.17).(D4) E is strongly local, i. e. E ( f, g ) = 0whenever f, g ∈ D ( E ) and f is constant in a neighbourhood of the support of g .It can be proven ([5]) that, if E satisfies (D1)-(D4), then it is the Dirichlet form of a Brownian motion.(D5) E is self similar, i. e. there is β > E ( u, v ) = β n X i =1 E ( u ◦ ψ i , v ◦ ψ i )for all u, v ∈ D ( E ).As we stated in the introduction, we shall be able to build a local and self-similar form on C ( R d , R ),but not to prove its closability. § We begin listing three equivalent ways to define the norm of a matrix in M d .9he first one is the sup norm || A || = sup {|| Av || : || v || ≤ } . We denote by t A the adjoint of the matrix A and by tr( A ) its trace; on the space of all matrices we candefine the inner product ( A, B ) HS = tr( t AB )which induces the Hilbert-Schmidt norm || A || HS = tr( t AA ) . If A is symmetric we can set ||| A ||| = sup {| ( Av, v ) | : || v || ≤ } where we have denoted by ( · , · ) the inner product of R d .Clearly, if A is symmetric, ||| A ||| is the modulus of the largest eigenvalue of A , while || A || HS is thequadratic mean of the eigenvalues; it is standard that there is D > D || A || HS ≤ ||| A ||| ≤ D || A || HS (2 . A symmetric.We define M d as the space of d × d symmetric matrices; we recall that A ∈ M d is positive semidefiniteif ( v, Av ) ≥ ∀ v ∈ R d . (2 . B ∈ M d is positive semidefinite if and only if( A, B ) HS ≥ . A ∈ M d satisfying (2.2). We briefly prove this fact. Let B satisfy (2.3); if we let let A vary amongthe one-dimensional projections we easily see that all the eigenvalues of B are positive, and (2.2) follows.For the converse, we note that, since A and B are symmetric and semi positive definite, their square rootsare symmetric and real, which implies the first and third equalities below; for the second one, we use thefact that the trace of a product is invariant under cyclic permutations. tr ( t AB ) = tr ( √ t A √ t A √ B √ B ) = tr ( √ B √ t A √ t A √ B ) = tr ( t ( √ t A √ B )( √ t A √ B )) ≥ . Essentially, (2.3) implies that the angle at the vertex of the cone of positive semidefinite matrices is smallerthan π .An immediate consequence of (2.3) is the following: if A, B, C ∈ M d , if A ≥ B ≤ C , then( A, B ) HS ≤ ( A, C ) HS . (2 . E, ˆ d ) be a compact metric space; in the following, ( E, ˆ d ) will be either one of ( G, ˜ d ) or (Σ , d γ ).We define C ( E, M d ) as the space of continuous functions from E to M d ; for A ∈ C ( E, M d ) we define || A || ∞ = sup x ∈ E || A ( x ) || HS . Let us call M ( E, M d ) the space of the Borel measures on E valued in M d . Putting on M d the Hilbert-Schmidt norm, we can define in the usual way the total variation || τ || of a measure τ ∈ M ( E, M d ); clearly, || τ || is a scalar-valued, non-negative, finite measure on the Borel sets of E .If τ ∈ M ( E, M d ) and B : E → M d belongs to L ( E, || τ || ), we define the real number Z E ( B x , d τ ( x )) HS : = Z E ( B x , T x ) HS d || τ || ( x )where τ = T x || τ || is the polar decomposition of τ ; we recall that || T x || HS = 1 for || τ || -a. e. x ∈ S .Several other products are possible; for instance, if u, v : E → R d are Borel vector fields such that || u ( x ) || · || v ( x ) || ∈ L ( E, || τ || ) , we can define the real number Z E ( u ( x ) , d τ ( x ) v ( x )): = Z E ( u ( x ) , T x v ( x ))d || τ || ( x )where, again, τ = T x || τ || is the polar decomposition of τ . Analogously, if A : E → M d is a Borel field ofmatrices such that || A x || HS ∈ L ( E, || τ || ), we can define the two matrices Z E A x d τ ( x ): = Z E A x T x d || τ || ( x )and Z E d τ ( x ) A x : = Z E T x A x d || τ || ( x ) . If Q ∈ C ( E, M d ) and τ ∈ M ( E, M d ), we define the scalar measure ( Q, τ ) HS in the following way: if B ⊂ E is Borel, then ( Q, τ ) HS ( B ): = Z B ( Q, d τ ) HS . In other words, (
Q, τ ) HS = ( Q x , T x ) HS || τ || .By Riesz’s representation theorem, M ( E, M d ) is the dual space of C ( E, M d ); the duality coupling h· , ·i : C ( E, M d ) × M ( E, M d ) → R is given by h B, τ i = Z E ( B x , d τ ( x )) HS . By Lusin’s theorem we get in the usual way that, if B ⊂ E is a Borel set, then || τ || ( B ) = sup Z B ( A, d τ ) HS (2 . A ∈ C ( E, M d ) such that || A || ∞ ≤ Remark.
In the discussion above, we should have distinguished between M d and its dual ( M d ) ∗ ; strictlyspeaking, the dual of C ( E, M d ) is M ( E, ( M d ) ∗ ). In order to have a simpler notation, we identify M d and( M d ) ∗ thanks to the Riemannian structure on R d . For the same reason, if f ∈ C ( R d , R ), we shall dealwith its gradient ∇ f and not with its differential d f .We shall say that τ ∈ M + ( E, M d ) if τ ∈ M ( E, M d ) and τ ( B ) is a non-negative definite matrix for allBorel sets B ⊂ E . By Lusin’s theorem, this is equivalent to Z E ( v x , d τ ( x ) v x ) ≥ ∀ v ∈ C ( E, R d ) . In turn, by (2.3) this is equivalent to Z E ( A x , d τ ( x )) HS ≥ . A ∈ C ( E, M d ) such that A x is positive semidefinite for all x ∈ E .Let now Q ∈ C ( E, M d ) such that Q x is positive-definite for all x ∈ E ; since E is compact there is D > D Id ≤ Q x ≤ D Id ∀ x ∈ E. (2 . Q satisfying (2.7) we define P Q ( E, M d ) as the set of all τ ∈ M + ( E, M d ) such that Z E ( Q, d τ ) HS = 1 . Lemma 2.1.
Let Q ∈ C ( E, M d ) satisfy (2.7). Then, there is D > (depending on the constant D of(2.7)) such that for all τ ∈ M + ( E, M d ) and all Borel sets B ⊂ E we have || τ || ( B ) ≤ D · ( Q, τ ) HS ( B ) (2 . where || · || denotes total variation. As a consequence, P Q ( E, M d ) is a convex set of M ( E, M d ) , compact forthe weak ∗ topology. Proof.
By the definition of total variation, we must find D > B ⊂ E and all countable Borel partitions { B i } i ≥ of B we have that X i ≥ || τ ( B i ) || HS ≤ D · ( Q, τ ) HS ( B ) . By (2.1), this follows if we show that, for the constant D of (2.7), X i ≥ ||| τ ( B i ) ||| ≤ D · ( Q, τ ) HS ( B ) . (2 . ||| τ ( B i ) ||| before (2.1), we can find unit vectors v i such that ||| τ ( B i ) ||| = ( v i , τ ( B i ) v i ) ∀ i ≥ . Let now v ∈ R d ; the inequality below follows in a standard way from the fact that τ ( B i ) is symmetric andnon-negative-definite; the equality comes from the definition of the Hilbert-Schmidt product.( v, τ ( B i ) v ) ≤ tr( τ ( B i )) || v || = ( τ ( B i ) , Id ) HS || v || . Since v i has unit length, the last two formulas imply the first inequality below; the first equality follows since τ is a measure and { B i } i ≥ is a partition of B . Since τ ∈ M + ( G, M d ), τ ( B ) is positive semidefinite; inparticular, (2.4) holds and together with (2.7) implies the second inequality below. The last equality followsfrom the definition of the measure ( Q, τ ) HS . X i ≥ ||| τ ( B i ) ||| ≤ X i ≥ ( τ ( B i ) , Id ) HS = ( τ ( B ) , Id ) HS ≤ D · Z B ( Q, d τ ) HS = D · ( Q, τ ) HS ( B ) . This is (2.9) and we are done.In order to prove the last assertion, we note that, by (2.6), M + ( E, M d ) is a convex set of M ( E, M d )clesed for the weak ∗ topology; as a consequence, also P Q ( E, M d ) is a closed convex set, while (2.8) impliesthat it is relatively compact for the weak ∗ topology. \\\ Definitions.
Let ( E, ˆ d ) be a compact metric space with diam( E ) ≤
1. We define C + as the set of all the A ∈ C ( E, M d ) such that A x is positive-definite for all x ∈ E ; since E is compact, if A ∈ C + , there is ǫ > A ) such that A x ≥ ǫ || A || ∞ Id ∀ x ∈ E. (2 . a > ν ∈ (0 ,
1] we define C + ( E, a, ν ) as the set of all the A ∈ C + such that A x e − a ˆ d ( x,y ) ν ≤ A y ≤ A x e a ˆ d ( x,y ) ν ∀ x, y ∈ E. (2 . C ν ( E, M d ) as the set of all ν -H¨older maps from E to M d , with the seminorm || A || ν = sup x = y ∈ E || A x − A y || HS ˆ d ( x, y ) ν . As lemma 2.3 below shows, the last two formulas are two different ways to look at the same seminorm, butwe shall need both.
Lemma 2.2.
Let ǫ > and let A, B ∈ M d such that A, B ≥ ǫId. (2 . hen, there is D = D ( ǫ, B ) > such that Be − D || B − A || HS ≤ A ≤ Be D || B − A || HS . (2 . For fixed ǫ , the function D ( ǫ, B ) is bounded when B is bounded.As a converse, there is D > such that the following holds. Let A, B ∈ M d be semi-positive definiteand let us suppose that there is D > such that e − D B ≤ A ≤ e D B. (2 . Then, || B − A || HS ≤ D ( e D − || A || HS . (2 . Proof.
We begin with the direct part. Let C ∈ M d ; it is easy to see (for instance, choosing a base inwhich C is diagonal) that C ≤ || C || HS Id.
Let
A, B ∈ M d ; if we apply the formula above to C = A − B we get that A ≤ B + || B − A || HS Id. (2 . B satisfies (2.12), this implies the first inequality below. A ≤ B (cid:18) ǫ || B − A || HS (cid:19) ≤ Be ǫ || B − A || HS . This yields the inequality on the right of (2.13). We prove the inequality on the left; the first inequalitybelow is (2.16) with the names changed, the second one follows from (2.12). A ≥ B − || B − A || HS Id ≥ B (cid:18) − ǫ || B − A || HS (cid:19) . Again by (2.12) this implies that A ≥ B (cid:18) − ǫ || B − A || HS (cid:19) if || B − A || HS ≤ ǫ ǫId if || B − A || > ǫ . The left hand side of (2.13) now follows from two facts: the first one is that, for D large enough,1 − tǫ ≥ e − D t if 0 ≤ t ≤ ǫ . The second one is the formula below. The first inequality comes taking || B − A || HS ≥ ǫ , the second onetaking γ > γB ≤ Id ; the third one taking D so large that1 γ e − ǫ D ≤ ǫ. e − D || B − A || HS ≤ Be − D ǫ ≤ γ Id · e − D ǫ ≤ ǫId. We prove the converse. By (2.14) we have that − (1 − e − D )( Bx, x ) ≤ (( A − B ) x, x ) ≤ ( e D − Bx, x ) . By the definition of ||| · ||| and the fact that e D − ≥ − e − D this implies that ||| A − B ||| ≤ ( e D − ||| B ||| . Now (2.15) follows from (2.1). \\\
Lemma 2.3.
Let a > and let ν ∈ (0 , ; then, the following holds.1) The sets C + and C + ( E, a, ν ) are convex cones in C ( E, M d ) which satisfy (1.1).2) There is D > such that for all A ∈ C + ( E, a, ν ) we have that || A x − A y || ≤ D || A || ∞ · ˆ d ( x, y ) ν ∀ x, y ∈ G. (2 . Conversely, if A ∈ C + ∩ C ν ( E, M d ) , then A ∈ C + ( E, a, ν ) for some a > (which depends on A ). Proof.
We don’t dwell on the proof of point 1), since it follows immediately from the definitions of C + and C + ( a, ν ).We prove point 2). Let A ∈ C + ( E, a, ν ); this means that, if x, y ∈ E , e − a ˆ d ( x,y ) ν A y ≤ A x ≤ e a ˆ d ( x,y ) ν A y . This is (2.14) for D = a ˆ d ( x, y ) ν ; by lemma 2.2, (2.15) holds, i. e. || A x − A y || HS ≤ D (cid:16) e a ˆ d ( x,y ) ν − (cid:17) || A x || HS ∀ x, y ∈ E. Since we are supposing that the diameter of E is 1 (for E = G this is (1.9)), we get (2.17).Conversely, let A ∈ C + ∩ C ν ( E, M d ); since E is compact, we easily see that there is ǫ > A x ≥ ǫId for all x ∈ E . Thus, setting A = A x and B = A y , we have that A and B satisfy (2.12); by lemma2.2 also (2.13) holds, i. e. A y e − D || A x − A y || HS ≤ A x ≤ A y e D || A x − A y || HS for some D = D ( ǫ, A y ) >
0. Since A ∈ C ν ( E, M d ), we have that || A x − A y || HS ≤ D ˆ d ( x − y ) ν The last two formulas imply that A y e − D D ( ǫ,A y ) ˆ d ( x,y ) ν ≤ A x ≤ A y e D D ( ǫ,A y ) ˆ d ( x,y ) ν . A ∈ C + ( E, a, ν ) if a ≥ D sup { D ( ǫ, A y ) : y ∈ E } . Note that the term on the right is finite since, by lemma 2.2, D ( ǫ, · ) is bounded on bounded sets and || A || ∞ is finite. \\\§ From now on, we suppose that (F1)-(F4) hold; we let ( G, ˜ d ) be the fractal defined by (1.8) with thedistance ˜ d induced by the immersion in R d .We define the Ruelle operator L G on G as L G : C ( G, M d ) → C ( G, M d )( L G A )( x ) = n X i =1 t Dψ i ( x ) A ψ i ( x ) Dψ i ( x ) . (3 . L Σ on C (Σ , M k ): first, if x = ( x x . . . ) ∈ Σ, we set ( ix ) = ( ix x . . . ).Then, we define L Σ : C (Σ , M k ) → C (Σ , M k )( L Σ A )( x ) = n X i =1 t Dψ i | Φ( x ) A ( ix ) Dψ i | Φ( x ) . (3 . x → Dψ i | Φ( x ) is ν -H¨older, since Dψ i is ν -H¨older by (F1) and we saw in section 1 that Φis Lipschitz; if A = ˜ A ◦ Φ for some ˜ A ∈ C ( G, M d ), we get by the second formula of (1.15) that L Σ A ( x ) = n X i =1 t Dψ i | Φ( x ) ˜ A ψ i ◦ Φ( x ) Dψ i | Φ( x ) . The next lemma shows that the fixed points of the adjoint of L G , which we call L ∗ G , induce a self-similarform on C ( R d ). Lemma 3.1.
Let G be a fractal satisfying (F1)-(F4). Let ( β, τ ) ∈ (0 , + ∞ ) × M + ( G, M d ) be such that L ∗ G τ = βτ. Let f, g ∈ C ( R d , R ) and let us define, with the notation of section 2 for the integral, E τ ( f, g ) = Z G ( ∇ f ( x ) , d τ ∇ g ( x )) . Then, n X i =1 E τ ( f ◦ ψ i , g ◦ ψ i ) = β E τ ( f, g ) . (3 . roof. By the polarisation identity, it suffices to show (3.3) when f = g ; we shall take advantage of thefact that ∇ f ⊗ ∇ f ∈ C ( G, M d ), i. e. is in the domain of L G . We recall that, if a ∈ R d and A ∈ M d , then( a, Aa ) = ( a ⊗ a, A ) HS where by a ⊗ a we denote the tensor product of the column vector a with itself: a ⊗ a = a · t a. As we shall see in the formula below, this explains the position of the transpose sign in (3.1).The definition of E τ and the formula above imply the first equality below; the second one comes fromthe chain rule (recall that ∇ ( f ◦ ψ i ) = t Dψ i · ∇ f ) and the definition of L G ; this third one follows since L ∗ G τ = βτ and the last one is again the definition of E τ . n X i =1 E τ ( f ◦ ψ i , f ◦ ψ i ) = n X i =1 Z G ( ∇ ( f ◦ ψ i )( x ) ⊗ ∇ ( f ◦ ψ i )( x ) , d τ ( x )) HS = Z G ( L G ( ∇ f ⊗ ∇ f )( x ) , d τ ( x )) HS = β Z G ( ∇ f ⊗ ∇ f, d τ ) HS = β E τ ( f, g ) . \\\ Remark.
We can read lemma 3.1 as a statement about the push-forward of the measure τ , the positiveeigenvector of L ∗ . Indeed, let f, g ∈ C ( R d ); by (F3), f ◦ F and g ◦ F are not defined at the points of F , which are a finite set. As we shall see in lemma 5.3 below, the measure τ G of point 2) of theorem 1 isnon-atomic; in particular, the points where f ◦ F and g ◦ F are not defined have measure zero. Togetherwith (1.13), this implies the first equality below; the second one follows from lemma 3.1; the last one followsfrom the chain rule. Z G ( ∇ f, d τ ∇ g ) = 1 n n X i =1 Z G ( ∇ ( f ◦ F ◦ ψ i ) , d τ ∇ ( g ◦ F ◦ ψ i )) = βn Z G ( ∇ ( f ◦ F ) , d τ ∇ ( g ◦ F )) = βn Z G ( t DF ( x ) ∇ f | F ( x ) ⊗ t DF ( x ) ∇ g | F ( x ) , d τ ( x )) HS . (3 . F ∗ ω the pull-back by F of the two-tensor ω , we have that τ = βn ( F ∗ ) ♯ τ where the ”push-forward” ( F ♯ ) ♯ τ is defined as in the last term on the right in the formula above. Thoughthis is not the standard push-forward operator, it is natural if we regard τ not as a measure, but as a linearoperator on 2-tensors. § ixed points of the Ruelle operator We shall suppose that the maps { ψ i } ni =1 satisfy the following nondegeneracy condition; it is strongerthan the one in [12], but it allows us to use the Perron-Frobenius theorem without modifications. (ND) We suppose that, for all v ∈ R d \ { } and all x ∈ G , the set { t Dψ i ( x ) v } ni =1 generates R d . Actually, weask for a quantitative version of this, i. e. that there is b > v, v ∈ R d and let x ∈ G ; then, there is ¯ i ∈ (1 , . . . , n ), depending on x , v and v , such that( Dψ ¯ i ( x ) v, v ) ≥ b || v || · || v || . (4 . P the orthogonal projection on v , the formula above implies the inequality below. || P · Dψ ¯ i ( x ) v || ≥ b || v || . (4 . L A . Lemma 4.1.
Let the maps { ψ i } ni =1 satisfy (F1)-(F4) and (ND). Let ( E, L ) denote either one of ( G, L G ) or (Σ , L Σ ) .Then, for all a > there is D = D ( a, b ) > such that the following happens. Let ν ∈ (0 , ν ] and let A ∈ C + ( E, a, ν ) ; then, for all x ∈ E , D ( a, b ) || A || ∞ · Id ≤ ( L A ) x ≤ D ( a, b ) || A || ∞ · Id. (4 . Proof.
We prove the left hand side of (4.3); for the right hand side it suffices to note that L is continuousfrom the || · || ∞ topology to itself.By compactness, there is x max ∈ E such that || A x max || HS = || A || ∞ . (4 . ||| A x max ||| we can find v max ∈ R d with || v max || = 1 such that ||| A x max ||| = ( A x max v max , v max ) . (4 . E ) = 1 we have that e − a ( A x max v, v ) ≤ ( A x v, v ) ≤ e a ( A x max v, v ) ∀ x ∈ E, ∀ v ∈ R d . Let v ∈ R d and let ¯ i ∈ (1 , . . . , n ); the formula above implies the first inequality below. Since v max is aneigenvector of the symmetric matrix A x max , we have that A x max preserves the space generated by v max andits orthogonal complement. Thus, if we denote by P the orthogonal projection on v max , we get the second18nequality below. Next, we choose ¯ i in such a way that (4.1) holds with v = v max ; the choice of ¯ i dependson x and v . By (4.2) we get the third inequality below; the last equality comes from (4.5) and the lastinequality from (2.1) and (4.4). ( A ψ ¯ i ( x ) Dψ ¯ i ( x ) v, Dψ ¯ i ( x ) v ) ≥ e − a ( A x max Dψ ¯ i ( x ) v, Dψ ¯ i ( x ) v ) ≥ e − a ( A x max P Dψ ¯ i ( x ) v, P Dψ ¯ i ( x ) v ) ≥ e − a b ( A x max v max , v max ) · || v || = e − a b ||| A x max ||| · || v || ≥ D · e − a b · || A || ∞ · || v || . We choose ¯ i as above; the definition of L implies the first inequality below; the second one comes from theformula above. (( L A ) x v, v ) ≥ ( A ψ ¯ i ( x ) Dψ ¯ i ( x ) v, Dψ ¯ i ( x ) v ) ≥ D · e − a b · || A || ∞ · || v || ∀ x ∈ E. \\\ Lemma 4.2.
Let the maps { ψ i } ni =1 satisfy (F1)-(F4) and (ND) for some b > . Then, for all a > thereis ω ( a, b ) > such that, if A ∈ C + ( G, a, ν ) and || Dψ i || ν ≤ ω ( a, b ) ∀ i ∈ (1 , . . . , n ) , (4 . then for all x, y ∈ G we have that e −|| x − y || ν n X i =1 t Dψ i ( x ) A ψ i ( x ) Dψ i ( x ) ≤ n X i =1 t Dψ i ( y ) A ψ i ( x ) Dψ i ( y ) ≤ e || x − y || ν n X i =1 t Dψ i ( x ) A ψ i ( x ) Dψ i ( x ) . (4 . G Note that, if the maps ψ i are affine and satisfy (ND), (4.6) is always verified.Analogously, possibly reducing ω ( a, b ) in (4.6), for all A ∈ C + (Σ , a, ν ) and all x, y ∈ Σ we have that e − d γ ( x,y ) ν n X i =1 t Dψ i | Φ( x ) A ( ix ) Dψ i | Φ( x ) ≤ n X i =1 t Dψ i | Φ( y ) A ( ix ) Dψ i | Φ( y ) ≤ e d γ ( x,y ) ν n X i =1 t Dψ i | Φ( x ) A ( ix ) Dψ i | Φ( x ) . (4 . Σ roof. We shall prove (4 . G , since (4 . Σ is analogous. We begin recalling an inequality on matrices. Let B ∈ M d be positive semidefinite and let C, C ′ two invertible matrices; we suppose that || C || HS , || C ′ || HS ≤ D (4 . D > B is symmetric,( t C ′ BCx, x ) = ( t CBC ′ x, x ) . Together with a simple calculation, this implies that( t C ′ BC ′ x, x ) = ( BC ′ x, C ′ x ) =( BCx, Cx ) + 2( B ( C ′ − C ) x, Cx ) + ( B ( C ′ − C ) x, ( C ′ − C ) x ) . Since || C − C ′ || HS ≤ D by (4.8), this implies that, for some D > D , but not on C, C ′ and B , t CBC − D || B || HS · || C − C ′ || HS · Id ≤ t C ′ BC ′ ≤ t CBC + D || B || HS · || C − C ′ || HS · Id.
We set B = A ψ i ( x ) , C ′ = Dψ i ( y ) and C = Dψ i ( x )which immediately implies that || C − C ′ || HS ≤ || Dψ i || ν · || x − y || ν . Recalling that by (F1) (4.8) holds with D = η , we get from the last three formulas that t Dψ i ( x ) A ψ i ( x ) Dψ i ( x ) − D || A ψ i ( x ) || HS · || Dψ i || ν · || x − y || ν Id ≤ t Dψ i ( y ) A ψ i ( x ) Dψ i ( y ) ≤ t Dψ i ( x ) A ψ i ( x ) Dψ i ( x ) + D || A ψ i ( x ) || HS · || Dψ i || ν · || x − y || ν Id.
Summing over i ∈ (1 , . . . , n ) and setting || Dψ || ν : = sup i ∈ (1 ,...,n ) || Dψ i || ν we get n X i =1 t Dψ i ( x ) A ψ i ( x ) Dψ i ( x ) − D || Dψ || ν · n X i =1 || A ψ i ( x ) || HS · || x − y || ν Id ≤ n X i =1 t Dψ i ( y ) A ψ i ( x ) Dψ i ( y ) ≤ X i =1 t Dψ i ( x ) A ψ i ( x ) Dψ ( x ) + || Dψ || ν · D n X i =1 || A ψ i ( x ) || HS · || x − y || ν Id.
Since A ∈ C + ( a, ν ) we can apply lemma 4.1 and get that there is D = D ( a, b ) such that[1 − D ( a, b ) || Dψ || ν · || x − y || ν ] n X i =1 t Dψ i ( x ) A ψ i ( x ) Dψ i ( x ) ≤ n X i =1 t Dψ i ( y ) A ψ i ( x ) Dψ i ( y ) ≤ [1 + D ( a, b ) || Dψ || ν · || x − y || ν ] n X i =1 t Dψ i ( x ) A ψ i ( x ) Dψ i ( x ) . We take ω ( a, b ) = 14 D ( a, b )and we recall that, if x ∈ [0 , − x ≥ e − x and 1 + 14 x ≤ e x . From the last three formulas, (4.6) and the fact that diam( E ) = 1 we get that e −|| x − y || ν n X i =1 t Dψ i ( x ) A ψ i ( x ) Dψ i ( x ) ≤ n X i =1 t Dψ i ( y ) A ψ i ( x ) Dψ i ( y ) ≤ e || x − y || ν n X i =1 t Dψ i ( x ) A ψ i ( x ) Dψ i ( x )which is (4 . G . \\\ Lemma 4.3.
Let ( E, L ) be either one of ( G, L G ) or (Σ , L Σ ) ; let (F1)-(F4) and (ND) hold. Then, thereis a > such that, for a > a and || Dψ i || ν ≤ ω ( a, b ) , L ( C + ( E, a, ν )) ⊂ C + ( E, a − , ν ) . (4 . Proof.
We follow [19]. It is immediate from the definition of L that L ( C + ) ⊂ C + . Thus, it suffices toshow that, if A ∈ C + satisfies (2.11) for a and ν , then L A satisfies (2.11) for a − ν , provided a islarge enough.We shall prove the lemma on G , since the proof on Σ is analogous. Let x, y ∈ G ; the first equality below isthe definition of L G in (3.1); the first inequality is the left hand side of (4 . G and holds if || Dψ i || ν ≤ ω ( a, b ).The second one follows from two facts: the map : A → t BAB is order-preserving and A ∈ C + ( a, ν ), i. e.21t satisfies (2.11). The third inequality comes from the fact that η is the common Lipschitz constant of themaps ψ i , i. e. formula (1.7). ( L G A )( y ) = n X i =1 t Dψ i ( y ) A ψ i ( y ) Dψ i ( y ) ≤ e || x − y || ν n X i =1 t Dψ i ( x ) A ψ i ( y ) Dψ i ( x ) ≤ e || x − y || ν + a || ψ i ( x ) − ψ i ( y ) || ν n X i =1 t Dψ i ( x ) A ψ i ( x ) Dψ i ( x ) ≤ e (1+ aη ν ) || x − y || ν n X i =1 t Dψ i ( x ) A ψ i ( x ) Dψ i ( x ) . Since η ∈ (0 ,
1) we can choose a so large that, for a ≥ a ,1 + aη ν ≤ a − . From the last two formulas we get that( L G A )( y ) ≤ e ( a − || x − y || ν ( L G A )( x ) . The opposite inequality follows similarly, implying (4.9). \\\
Definitions.
Let λ , ǫ >
0; we denote by C ǫ + ( E, λ a, ν ) the subset of the A ∈ C + ( E, λ a, ν ) such that, forall x ∈ E , A x ≥ ǫ || A || ∞ Id. (4 . θ ( a,ν ) the hyperbolic distance on C + ( E, a, ν ) and θ + the hyperbolic distanceon C + ; we recall that the hyperbolic distance on a cone has been defined at the beginning of section 1. Lemma 4.4.
Let E be either one of G or Σ ; let a > . Then, the following holds.1) ( C + ( E,a,ν ) ≃ , θ ( a,ν ) ) is a complete metric space.2) Let C ǫ + ( E, λ a, ν ) be defined as in (4.10). If λ ∈ (0 , , then diam θ ( a,ν C ǫ + ( E, λ a, ν ) < + ∞ . Proof.
Again we follow closely [19]; we begin with point 1). We consider a Cauchy sequence { A n } n ≥ in( C + ( E,a,ν ) ≃ , θ ( a,ν ) ); we choose the representatives which satisfy || A n || ∞ = 1 ∀ n ≥ . (4 . Step 1.
We begin to show that { A n } n ≥ converges uniformly to A ∈ C ( E, M d ).22ince C + ( E, a, ν ) ⊂ C + , the definition of the hyperbolic distance implies that θ + ≤ θ ( a,ν ) ; thus, { A n } n ≥ is Cauchy also for the θ + distance; the definition of θ + in (1.4) implies that β + ( A m , A n ) α + ( A m , A n ) → n, m → + ∞ . (4 . α + and β + in (1.2) and (1.3) respectively, we get that, for all x ∈ E , α + ( A m , A n ) A m ( x ) ≤ A n ( x ) ≤ β + ( A m , A n ) A m ( x ) . Let δ >
0; by (4.12) we have that, for n and m large enough and all x ∈ E ,(1 − δ ) β + ( A m , A n ) A m ( x ) ≤ A n ( x ) ≤ β + ( A m , A n ) A m ( x ) . By the converse part of lemma 2.2 this implies that, for n and m large, || A n − β + ( A m , A n ) A m || ∞ ≤ D · δ || A n || ∞ . By (4.11) and the triangle inequality this implies that β + ( A m , A n ) →
1; again by the formula above, β + ( A m , A n ) → { A n } n ≥ is Cauchy for || · || ∞ ; thus, there is A ∈ C ( G, M d ) such that A n → A uniformly. Step 2.
We show that θ ( a,ν ) ( A n , A ) → n → + ∞ . It is easy to see that θ ( a,ν ) is lower semicontinuousunder uniform convergence; together with step 1, this yields the first inequality below. Since { A n } n ≥ isCauchy for θ ( a,ν ) , there is δ n → θ ( a,ν ) ( A, A n ) ≤ lim inf m → + ∞ θ ( a,ν ) ( A m , A n ) ≤ δ n . End of the proof of point 1.
It only remains to prove that A ∈ C + ( E, a, ν ). First of all, A satisfies (2.11),since this condition is closed under uniform convergence. We have to show that A x is positive-definite for all x ∈ E . We recall that A n,x is positive-definite for all x ∈ E , since A n ∈ C + ( E, a, ν ); since θ + ( A n , A ) < + ∞ ,we have that α + ( A n , A ) >
0; since by (1.2) A ≥ α + ( A n , A ) A n and A n satisfies (2.10) for some ǫ > A n ), we get that A x is positive-definite for all x ∈ E . Proof of point 2).
The proof is in two steps: first, we show thatdiam θ + C ǫ + ( E, λ a, ν ) < + ∞ (4 . θ ( a,ν C ǫ + ( E, λ a, ν ) ≤ diam θ + C ǫ + ( E, λ a, ν ) + D ( λ ) . (4 . Step 3.
We prove (4.13); this follows if we show that C ǫ + ( E, λ a, ν ) is compact in ( C + , θ + ). Thus, let { A n } n ≥ ⊂ C ǫ + ( E, λ a, ν ); we can suppose that { A n } n ≥ is normalised, i. e. that (4.11) holds. Now,point 2) of lemma 2.3 implies that the H¨older seminorm of A n is bounded. Thus, by Ascoli-Arzel`a thereis a subsequence { A n h } h ≥ which converges uniformly to A ∈ C ( E, M d ); we see as in point 1) that A satisfies (2.11); it also satisfies (4.10) because this formula is stable under uniform convergence. Thus,23 ∈ C ǫ + ( E, λ a, ν ). Since A n h and A satisfy (4.10) and A n h → A uniformly, we can apply the direct partof lemma 2.2 and get that θ + ( A n h , A ) →
0, ending the proof of compactness.
Step 4.
We prove (4.14) on the general space E with distance ˆ d . For starters, let us see how α + and α ( a,ν ) are related. Let A , A ∈ C + ( E, a, ν ); by the definition of α ( a,ν ) ( A , A ) we have that A − α ( a,ν ) ( A , A ) A ∈ C + ( E, a, ν )which by (2.11) implies that, for all x, y ∈ E , e − a ˆ d ( x,y ) ν [ A ( x ) − α ( a,ν ) ( A , A ) A ( x )] ≤ A ( y ) − α ( a,ν ) ( A , A ) A ( y ) ≤ e a ˆ d ( x,y ) ν [ A ( x ) − α ( a,ν ) ( A , A ) A ( x )] . Rearranging the terms of the inequality on the left, we get that e − a ˆ d ( x,y ) ν A ( x ) − A ( y ) ≤ [ e − a ˆ d ( x,y ) ν A ( x ) − A ( y )] α ( a,ν ) ( A , A )for all x, y ∈ E . Since A , A satisfy (2.11) for λ a , the formula above implies that A ( x ) ≤ e − a ˆ d ( x,y ) ν − e − λ ad ( x,y ) ν e − a ˆ d ( x,y ) ν − e λ ad ( x,y ) ν α ( a,ν ) ( A , A ) A ( x ) . (4 . D = sup (cid:26) z − z λ z − z − λ : z ∈ (0 , (cid:27) and by a function study we see that D ∈ (0 , A ( x ) ≤ D α ( a,ν ) ( A , A ) A ( x )which by the definition of α + implies that D α ( α,ν ) ( A , A ) ≥ α + ( A , A ) . (4 . D = inf (cid:26) z − z − λ z − z λ : z > (cid:27) . A function study shows that D > β ( a,ν ) ( A , A ) ≤ D β + ( A , A ) . Using this, (4.16) and the definition of θ + in (1.4) we get that θ ( a,ν ) ( A , A ) ≤ θ + ( A , A ) + log D − log D which ends the proof of (2.18). \\\ roposition 4.5. Let ( E, L ) be either one of ( G, L G ) or (Σ , L Σ ) . Let (F1)-(F4) and (ND) with constant b > hold. Then, if sup i ∈ (1 ,...,n ) || Dψ i || ν is small enough, the following holds.1) There is a simple, positive eigenvalue β of L : C ( E, M d ) → C ( E, M d ) . Denoting by Q the eigenfunction of β , we have that Q ∈ C + ( a, ν ) . In particular, Q x is positive-definite forall x ∈ E . If the maps ψ i are affine, then there is ¯ Q ∈ M d such that Q x = ¯ Q for all x ∈ E .2) Recall that after formula (2.7) we defined P Q ( E, M d ) ; we assert that there is τ ∈ P Q ( E, M d ) such that L ∗ τ = βτ .3) If B ∈ C ( E, M d ) , we have β l L l B → Q Z E ( B, d τ ) HS (4 . uniformly on E . Note that this implies that the measure τ of the previous point is unique and that β issimple eigenvalue of L . Moreover, if B ∈ C + ( E, a, ν ) with ν ∈ (0 , ν ] , the convergence above is exponentiallyfast. Proof. Step 1.
Let sup i ∈ (1 ,...,n ) || Dψ i || ν be so small that lemma 4.3 hold. Since L : C ( E, M ) → C ( E, M )is continuous, the left hand side of (4.3) implies that, possibly increasing the constant D ( a, b ),( L A ) x ≥ D ( a, b ) ||L A || ∞ · Id ∀ x ∈ E. Together with lemma 4.3 this implies that L ( C + ( E, a, ν )) ⊂ C ǫ + ( E, λ a, ν )for some λ ∈ (0 ,
1) and ǫ equal to the constant D ( a,b ) of the formula above; by point 2) of lemma 4.4 thisimplies that diam θ ( a,ν ) L ( C + ( E, a, ν )) < + ∞ . By point 1) of proposition 1.1, we get that L is a contraction of C + ( E, a, ν ) into itself; since (cid:18) C + ( E, a, ν )) ≃ , θ ( a,ν ) (cid:19) is complete by point 1) of lemma 4.4, we get that L has a unique fixed point in C + ( E,a,ν )) ≃ . In other words,there are1) Q ∈ C + ( E, a, ν ), unique up to multiplication by a scalar, and2) a unique β ∈ (0 , + ∞ ) such that L Q = βQ. (4 . Q is unique up to multiplication by a scalar, we can normalise it in such a way that || Q || ∞ = 1.25 tep 2. If the maps Dψ i are constant, we see that, if A is a constant matrix, then L A is constant too.Applying the Perron-Frobenius theorem to the positive cone of M d , we can find a constant, positive-definitematrix ¯ Q and β ′ > L ¯ Q = β ′ ¯ Q . By the uniqueness of step 1, we have that Q ≡ ¯ Q . Step 3.
We prove point 2). We saw in lemma 2.1 that P Q ( E, M d ) is a convex, compact set of M ( E, M d );thus, by Schauder’s fixed point theorem, it suffices to show that β L ∗ brings P Q ( E, M d ) into itself. Let˜ τ ∈ P Q ( E, M d ); we skip the proof that β L ∗ ˜ τ is non-negative definite (it follows easily by (2.3) and thedefinition of the adjoint), but we show that its integral against Q is 1. The first equality below is thedefinition of the adjoint, the second one is point 1) and the last follows since ˜ τ ∈ P Q ( E, M d ). Z E (cid:18) Q, d (cid:18) β L ∗ (cid:19) ˜ τ (cid:19) HS = Z E (cid:18) β L Q, d˜ τ (cid:19) HS = Z E ( Q, d˜ τ ) HS = 1 . Step 4.
We prove point 3). Since L is linear and B = B + − B − with B + , B − ≥
0, it suffices to prove (4.17)when B ∈ C ( E, M d ) and B ≥
0; in other words, when B ∈ ¯ C + .We begin to show that, if B ∈ C + , then θ + ( L l B, Q ) → . (4 . ǫ > a > B ∈ C + ( E, a , ν ) such that θ + ( B, ˜ B ) < ǫ . This follows, for instance,since H¨older functions are dense for the || · || ∞ topology. We can also require that ˜ B ∈ C + ( E, a, ν ) for afixed a ≥ a .b) If a is large enough, θ ( a,ν ) ( L l ˜ B, Q ) →
0. This follows since ˜ B ∈ C + ( E, a, ν ) and L is a contractionon C + ( E, a, ν ) by step 2. By (1.5), this convergence is exponentially fast. If we apply this argument toˆ B ∈ C + ( E, a, ν ) we get the last assertion of the thesis.c) Since C + ( E, a, ν ) ⊂ C + , the definition of hyperbolic distance in section 1 immediately implies that θ + ≤ θ ( a,ν ) ; by the triangle inequality, this implies the first inequality below; the second one comes fromthe fact that, since L ( C + ) ⊂ C + , then Lip θ + ( L ) ≤ θ + ( L l B, Q ) ≤ θ + ( L l B, L l ˜ B ) + θ ( a,ν ) ( L l ˜ B, Q ) ≤ θ + ( B, ˜ B ) + θ ( a,ν ) ( L l ˜ B, Q ) . Now the first term on the right is arbitrarily small by point a) and the second one tends to zero by point b).We show how (4.19) implies (4.17) when B ∈ C + . We begin to note that, since L ∗ τ = βτ by point 2)of the thesis, we have for all l ≥ Z E (cid:18) β L (cid:19) l B, d τ ! HS = Z E ( B, d τ ) HS . (4 . || (cid:16) β L (cid:17) n B || ∞ ; in turn, by lemma 4.1 this implies thatthe matrices (cid:16) β L (cid:17) n B are uniformly positive-definite. Together with (4.19) and lemma 2.2 this implies thatthere is α l > || α l (cid:18) β L (cid:19) l B − Q || ∞ → . (4 . α l Z E ( B, d τ ) HS → Z E ( Q, d τ ) HS . The right hand side in the formula above is 1 since τ ∈ P Q ( G, M d ); this implies that α l → α : = R E ( Q, d τ ) HS R E ( B, d τ ) HS . Note that the numerator is 1 since τ ∈ P Q ( G, M d ); the denominator is different from zero since B ∈ C + ,(2.4) holds and τ ∈ P Q ( E, M d ).Recall that (4.21) and the last formula imply that (cid:18) β L (cid:19) l B → α Q uniformly. Now (4.17) follows from the last two formulas.The last case is when B ∈ ¯ C + \ { } . In this case, we consider B + δId for δ >
0; since B + δId ∈ C + we have just shown that (cid:18) β L (cid:19) l ( B + δId ) → Q Z G ( B + δId, d τ ) HS (4 . n → + ∞ . We saw above that, since Id ∈ C + , (cid:18) β L (cid:19) l Id → Q Z E ( Id, d τ ) HS uniformly. Now the thesis follows subtracting the last two formulas. \\\ Definition.
By point 1) of proposition 4.5, the operator L G on C ( G, M k ) has a couple eigenvalue-eigenvectorwhich we call ( β G , Q G ); the operator L Σ on C (Σ , M k ) has a couple eigenvalue-eigenvector which we call( β Σ , Q Σ ). By point 2) of proposition 4.5 there is a Gibbs measure on G , which we call τ G , and one on Σ,which we call τ Σ . We shall say that κ G : = ( Q G , τ G ) HS is Kusuoka’s measure on G and that κ Σ : = ( Q Σ , τ Σ ) HS is Kusuoka’s measure on Σ. Since τ G ∈ P Q G and τ Σ ∈ P Q Σ , κ G and κ Σ are both probability measures.The next lemma shows that there is a natural relationship between these objects. Lemma 4.6.
We have that β G = β Σ ; we shall call β their common value. Up to multiplying one of themby a positive constant, we have that Q Σ = Q G ◦ Φ . Moreover, τ G = Φ ♯ τ Σ and κ G = Φ ♯ κ Σ . Proof.
The first equality below comes from the formula after (3.2) and the second one from the definitionof L G in (3.1). L Σ ( A ◦ Φ)( x ) = n X i =1 t Dψ i | Φ( x ) A ψ i ◦ Φ( x ) Dψ i | Φ( x ) = L G ( A ) ◦ Φ( x ) for all A ∈ C ( G, M k ) . (4 . Q G is a fixed point of L G on C + ( G,a,ν ) ≃ , the formula above implies that Q G ◦ Φ is a fixed point of L Σ on C + (Σ ,a,ν ) ≃ . By the uniqueness of proposition 4.5 we get that, up to multiplying one of them by a positiveconstant, Q Σ = Q G ◦ Φ. Since β Σ Q Σ = L Σ Q Σ = L Σ ( Q G ◦ Φ) = L G ( Q G ) ◦ Φ = β G Q G ◦ Φwe get that β Σ = β G .We prove the relation between the Gibbs measures. Let A ∈ C ( G, M k ); the first equality below is thedefinition of the adjoint, the second one follows from the definition of push-forward; the third one comesfrom (4.23) while the fourth one comes from the fact that τ Σ is an eigenvector of L ∗ Σ ; the last one comesagain by the definition of push-forward. h A, L ∗ G (Φ ♯ τ Σ ) i = hL G A, Φ ♯ τ Σ i = h ( L G A ) ◦ Φ , τ Σ i = hL Σ ( A ◦ Φ) , τ Σ i = β h A ◦ Φ , τ Σ i = β h A, Φ ♯ τ Σ i . Moreover, the fact that Q Σ = Q G ◦ Φ easily implies that Φ ♯ τ Σ ∈ P Q G ; since lemma 4.5 implies the uniquenessof the eigenvector of L ∗ G in P Q G , the last formula implies that τ G = Φ ♯ τ Σ .We leave to the reader the easy verification that κ G = Φ ♯ κ Σ . \\\ In order to prove that Kusuoka’s measure (
Q, τ ) HS is ergodic, we need a lemma. Lemma 4.7.
Let β > and let τ ∈ M + ( E, M d ) be as in point 2) of proposition 4.5. Let A : E → M d bea bounded Borel function. Then, Z E ( L A, d τ ) HS = β Z E ( A, d τ ) HS . Proof.
Let us define the measure t as t : = || τ || + n X i =1 ( ψ i ) ♯ ( || τ || )if we are on G ; on Σ we set t : = || τ || + n X i =1 ( a i ) ♯ ( || τ || )where a i : ( x x . . . ) → ( ix x . . . ).By Lusin’s theorem there is a sequence A k ∈ C ( G, M d ) such that A k → A t -a. e. on G ; moreover, || A k || ∞ is bounded. By dominated convergence, this implies that Z E ( A k , d τ ) HS → Z E ( A, d τ ) HS Z E ( L A k , d τ ) HS → Z E ( L A, d τ ) HS . Since A k is continuous, point 2) of proposition 4.5 implies that Z E ( L A k , d τ ) HS = β Z E ( A k , d τ ) HS . The thesis follows from the last three formulas. \\\
The next lemma recalls some properties of Kusuoka’s measure.
Lemma 4.8.
Let E = G or E = Σ ; in the first case we set S = F , in the second one we set S = σ . Then,the following holds.1) Let Q and τ be as in proposition 3.2, let g ∈ C ( E, R ) and let A ∈ C ( E, M k ) . Then we have that Z E ( g ◦ S l · A, d τ ) HS → Z E ( gQ, d τ ) HS · Z E ( A, d τ ) HS . (4 .
2) The scalar measures κ G and κ Σ defined above are ergodic. Proof.
We begin with point 1) on Σ; we follow [PP].First of all, we note that, if h : Σ → R is a bounded Borel function, then h ◦ σ ( ix ) = h ( x ) for all i ∈ (1 , . . . , n ) and all x ∈ Σ . (4 . A ∈ C (Σ , M d ),[ L Σ ( h ◦ σ · A )]( x ) = h ( x )( L Σ A )( x ) ∀ x ∈ G. Integrating against τ Σ , we get the first equality below; the second equality comes from lemma 4.7. Z Σ (cid:18) h · (cid:18) β L Σ A (cid:19) , d τ Σ (cid:19) HS = Z Σ (cid:18) β L Σ ( h ◦ σ · A ) , d τ Σ (cid:19) HS = Z Σ ( h ◦ σ · A, d τ Σ ) HS . (4 . A = Q Σ where Q Σ is the eigenfuction of proposition 3.2, we have that Z Σ h d( Q Σ , τ Σ ) HS = Z Σ h ◦ σ d( Q Σ , τ Σ ) HS . (4 . h = g ∈ C (Σ , R ) we get that Z Σ g (cid:18) β L Σ (cid:19) k A, d τ Σ ! HS = Z Σ ( g ◦ σ k · A, d τ Σ ) HS . Now (4.24) for (
E, S ) = (Σ , σ ) follows from point 3) of proposition 4.5.29ext, we show (4.24) when E = G . Anticipating on lemma 5.3 below, the measure τ Σ is non-atomic.In particular, the countable set N ⊂ Σ on which Φ is not injective is a null set for κ Σ . Let g ∈ C ( G, R )and let A ∈ C ( G, M k ); the first equality below comes from lemma 4.6; the second one is the definition ofpush-forward; the third one comes from (1.16), which holds save on a null-set; the limit is (4.24) on Σ, whichwe have just proven. The last equality follows again from lemma 4.6. Z G ( g ◦ F l · A, d τ G ) HS = Z G ( g ◦ F l ( x ) · A ( x ) , d(Φ ♯ τ Σ )( x )) HS = Z Σ ( g ◦ F l ◦ Φ( y ) · A ◦ Φ( y ) , d τ Σ ( y )) HS = Z Σ ( g ◦ Φ ◦ σ l ( y ) · A ◦ Φ( y ) , d τ Σ ( y )) HS → Z Σ ( g ◦ Φ( y ) · Q Σ ( y ) , d τ Σ ( y )) HS · Z Σ ( A ◦ Φ( y ) , d τ Σ ( y )) HS = Z G ( g ( x ) Q G ( x ) , d τ G ( x )) HS · Z G ( A ( x ) , d τ G ( x )) HS . This is (4.24) for G , ending the proof of point 1).We prove point 2). First of all, since (4.27) holds for all h ∈ C (Σ , R ) we get that σ ♯ κ Σ = κ Σ , i. e. that κ Σ is σ -invariant. With the same argument we used for the formula above this implies that Z G h d( Q G , τ G ) HS = Z G h ◦ F d( Q G , τ G ) HS i. e. that κ G is F -invariant.Now we work on E , with E = G or E = Σ. Setting A = f Q for a continuous function f , (4.24) impliesthat Z E g ◦ F l · f d( Q, τ ) HS → Z E g d( Q, τ ) HS · Z E f d( Q, τ ) HS which implies that ( Q, τ ) HS is strongly mixing; in particular, it is ergodic. \\\ At this stage it is natural to ask whether, when the maps ψ i are affine, Kusuoka’s measure κ G coincideswith ( Q G , τ G ) HS ; in the remark at the end of section 5 we shall prove that this is the case. § In section 1 we defined the cylinder [ x . . . x l ] ⊂ Σ and the cell [ x . . . x l ] G ⊂ G .From now on, we shall suppose that the maps ψ i are affine and we set ψ x ...x l = ψ x ◦ . . . ◦ ψ x l . efinition. Let M + ( G, M d ). Following [18], we shall say that µ ∈ M + ( G, M d ) is a Gibbs measure if thereis there are constants C, D > l ≥ x ∈ G \ ˜ N , e − Cl − D · ( Dψ x ...x l − ) · µ ( G ) · t ( Dψ x ...x l − ) ≤ µ ([ x . . . x l − ] G ) ≤ e − Cl + D · ( Dψ x ...x l − ) · µ ( G ) · t ( Dψ x ...x l − ) . (5 . G We say that µ ∈ M + (Σ , M d ) is a Gibbs measure if there is there are constants C, D > l ≥ x ∈ Σ, e − Cl − D · ( Dψ x ...x l − ) · µ (Σ) · t ( Dψ x ...x l − ) ≤ µ ([ x . . . x l − ]) ≤ e − Cl + D · ( Dψ x ...x l − ) · µ (Σ) · t ( Dψ x ...x l − ) . (5 . Σ In the formula above, we have not specified at which point we calculate Dψ x ...x l − , since ψ x ...x l − isaffine.Let τ Σ be the positive eigenvector of L ∗ as in lemma 4.5; we briefly prove that τ Σ (Σ) = 0. Let Q Σ beas in proposition 4.5; the inequality below comes from (2.4) and the fact that, for some ǫ > ǫQ Σ ( G ) ≤ Id for all x ∈ G by compactness; the equality comes from the fact that τ Σ ∈ P Q (Σ , M d ).( Id, τ Σ (Σ)) HS ≥ ǫ Z Σ ( Q Σ ( x ) , d τ Σ ( x )) HS = ǫ. We have the following analogue of proposition 3.2 of [18].
Lemma 5.1.
Let (F1)-(F4) and (ND) hold; let the maps ψ i be affine. Let ( β, τ Σ ) be as in proposition4.5. Then for all l ≥ and all x = ( x , x , . . . ) ∈ Σ we have τ Σ ([ x . . . x l ]) = 1 β ( Dψ x ) · τ Σ ([ x . . . x l ]) · t ( Dψ x ) . (5 . If l = 0 , (5.2) holds with τ Σ (Σ) instead of τ Σ ([ x . . . x l ]) on the right. Proof.
Let x = ( x , x , . . . ) ∈ Σ be fixed; clearly, we have that1 [ x x ...x n ] ( iz ) = ( [ x ...x n ] ( z ) if i = x A ∈ M d be a fixed, positive semidefinite matrix. The formula above implies the second equalitybelow, while the first one comes from the fact that A is constant; the third one follows by multiplying anddividing and recalling that 1 [ x ...x l ] ( iz ) = 0 if i = x ; the fourth one from the definition of L Σ in (3.2); thelast one follows by lemma 4.7.( A, τ Σ ([ x . . . x l ])) HS = Z Σ ( A [ x ...x l ] ( z ) , d τ Σ ( z )) HS =31 Σ n X i =1 A [ x ...x l ] ( iz )) , d τ Σ ( z ) ! HS = Z Σ n X i =1 t Dψ i · t ( Dψ x ) − · A [ x ...x n ] ( iz ) · ( Dψ x ) − · Dψ i , d τ Σ ( z ) ! HS = Z Σ (cid:0) L Σ (cid:0) t ( Dψ x ) − · A [ x ...x l ] · ( Dψ x ) − (cid:1) ( z ) , d τ Σ ( z ) (cid:1) HS = β Z [ x ...x l ] (cid:0) t ( Dψ x ) − · A · ( Dψ x ) − , d τ Σ (cid:1) HS . (5 . β ( A, Dψ − x · τ Σ ([ x . . . x l ]) · t Dψ − x ) HS = ( A, τ Σ ([ x , . . . x l ])) HS . Letting A vary among the one-dimensional projections we get that β · Dψ − x · τ Σ ([ x . . . x l ]) · t Dψ − x = τ Σ [ x . . . x l ] . To get (5.2) it suffices to multiply the formula above by1 β · ( Dψ x ) on the left and by t ( Dψ x ) on the right. \\\ Corollary 5.2.
Let (F1)-(F4) and (ND) hold; let us suppose that the maps ψ i are affine and let ( β, τ Σ ) be as in proposition 4.5. then, τ Σ is a Gibbs measure for the constant C = log β . Proof.
Iterating the right hand side of (5.2) and using the chain rule we get the following. τ Σ ([ x . . . x l ]) = 1 β · ( Dψ x ) τ Σ ([ x . . . x l ]) t ( Dψ x ) =1 β · ( Dψ x x ) τ Σ ([ x . . . x l ]) t ( Dψ x x ) = . . . =1 β l · ( Dψ x x ...x l − ) τ Σ ([ x l ]) t ( Dψ x x ...x l − ) =1 β l +1 · ( Dψ x ...x l ) τ Σ (Σ) t ( Dψ x ...x l ) . \\\ Lemma 5.3.
Let the maps ψ i satisfy (F1)-(F4) and let (ND) hold. Then, we have the following.1) The measure τ Σ is positive on open sets.2) The measures τ Σ and τ G are non-atomic. roof. We begin with point 1) for τ Σ . It suffices to show that, for all cylinders [ x . . . x l ] ⊂ Σ, the matrix τ Σ [ x . . . x l ] is not zero. We get from (5.3) that( Id, τ Σ [ x . . . x l ]) HS = β Z [ x ...x l ] ( t ( Dψ x | Φ( x ) ) − · ( Dψ x | Φ( x ) ) − , d τ Σ ( x )) HS . This easily implies that, if τ Σ [ x . . . x l ] is not zero, then also τ Σ [ x . . . x l ] is not zero. Iterating, we see that τ Σ [ x . . . x l ] is not zero if τ Σ (Σ) is not zero, a fact we showed before stating this lemma.As for point 2), we begin to recall the standard proof that κ Σ is non-atomic. By point 2) of lemma4.8, κ Σ is ergodic; let us suppose by contradiction that it has an atom { ¯ x } . We are going to show that κ Σ ( { ¯ x } ) = 1 and, consequently, κ Σ ( { ¯ x } c ) = 0. This will be the contradiction, since by point 1) τ Σ is positiveon open sets.First of all, let us suppose that ¯ x is a periodic orbit of period q and let us set A = [ l ≥ σ − lq ( { ¯ x } ) . Clearly, A is σ q -invariant, i. e. σ − q ( A ) ⊂ A . Since σ preserves κ Σ , we have that κ Σ ( σ − lq { ¯ x } ) = κ Σ ( { ¯ x } );since σ q fixes ¯ x , we see that ¯ x ∈ σ − q ( { ¯ x } ). This implies that τ Σ on σ − q ( { ¯ x } ) concentrates on { ¯ x } ; iterating,we get that τ Σ on σ − lq ( { ¯ x } ) concentrates on { ¯ x } . This and the definition of A easily imply that κ Σ ( A ) = κ Σ ( { ¯ x } ) and that κ Σ ( A \ σ − q ( A )) = 0. By ergodicity, this implies that κ Σ ( { ¯ x } ) = 1.The second case is when ¯ x has an antiperiod, say of length l . We consider ˜ x = σ l (¯ x ), which is periodic.Since ¯ x ∈ σ − l (˜ x ), invariance implies that κ Σ (˜ x ) >
0; now the same argument as above applies.The last case is when ¯ x is not periodic; then, it is easy to see that the sets σ − l ( { ¯ x } ) are all disjoint.Since they have the same measure, we get that κ Σ ( { ¯ x } ) = 0, i. e. that { ¯ x } is not an atom.In order to show that κ G is non-atomic, it suffices to recall three facts: that τ Σ is non-atomic, that τ G = Φ ♯ τ Σ by lemma 4.6 and that Φ is finite-to-one. \\\ End of the proof of theorem 1.
Points 1) and 2) come from proposition 4.5. The self-similarity of point4) comes from lemma 3.1. The ergodicity of point 3) is point 2) of lemma 4.8; mutual absolute continuityin one direction follows from (2.8), in the other one is trivial. For point 5) we begin to note that, by thedefinition of Φ, [ x . . . x l ] ⊂ Φ − ([ x . . . x l ] G ); the points of Φ − ([ x . . . x l ] G \ [ x . . . x l ] are those with multiplecodings, which we have seen in section 1 to be a countable set. Since τ Σ is non-atomic, we get that τ G (Φ − ([ x . . . x l ] G \ [ x . . . x l ]) = 0 . Since τ G = Φ ♯ τ Σ by lemma 3.4, the last formula implies that τ Σ ([ x . . . x l ]) = τ G ([ x . . . x l ] G ) . Since τ Σ has the Gibbs property by corollary 5.2, we are done.33 \\ Remark.
In corollary (4.2) we have supposed that the maps Dψ i are constant, which is the case of Kusuoka’spaper [12]. We prove that, up to multiplication by a positive constant, ( Q G , τ G ) HS coincides with Kusuoka’smeasure κ .When Dψ i is constant, point 1) of proposition 4.5 implies that Q G is constant too and solves Q G = 1 β n X i =1 t Dψ i QDψ i . (5 . β E τ ( f, g ) = n X i =1 E τ ( f ◦ ψ i , g ◦ ψ i ) = n X i =1 Z G ( t Dψ i · ∇ f | ψ i ( x ) , d τ t Dψ i · ∇ g | ψ i ( x ) ) . If we choose as f and g two linear functions and we recall that Dψ i is a constant matrix, we see that thelast formula implies that βτ G ( G ) = n X i =1 Dψ i · τ G ( G ) · t Dψ i . (5 . κ is defined by the following formula: if x = ( x x . . . ), then κ ([ x . . . x l ] G ) = 1 β l ( Q, t ( Dψ x ...x l ) ˆ Q t ( Dψ x ...x l )) HS where Q solves (5.4) and ˆ Q solves (5.5). Since the solution to both equations is unique by Perron-Frobenius(up to multiplication by a constant, of course), the last formula and corollary (5.2) imply that κ G : = ( Q G , τ G )coincide with κ . References [1] M. T. Barlow, R. F. Bass, The construction of Brownian motion on the Sierpinski carpet, Ann. IHP, ,225-257, 1989.[2] M. T. Barlow, E. A. Perkins, Brownian motion on the Sierpiski gasket, Probab. Th. Rel. Fields, ,543-623, 1988.[3] R. Bell, C. W. Ho, R. S. Strichartz, Energy measures of harmonic functions on the Sierpinski gasket,Indiana Univ. Math. J. , 831-868, 2014.[4] G. Birkhoff, Lattice theory, Third edition, AMS Colloquium Publ., Vol. XXV, AMS, Providence, R. I.,1967.[5] M. Fukushima, Y. Oshima, M. Takeda, Dirichlet forms and symmetric Markov processes, De Gruyter,G¨ottingen, 2011. 346] S. Goldstein, Random walks and diffusions on fractals, in H. Kesten (ed), Percolation theory and ergodictheory of infinite particle systems, IMA vol. Math. Appl., , Springer, New York, 121-129, 1987.[7] A. Johansson, A. ¨Oberg, M. Pollicott, Ergodic theory of Kusuoka’s measures, J. Fractal Geom., , 185-214,2017.[8] N. Kajino, Analysis and geometry of the measurable Riemannian structure on the Sierpinski gasket,Contemporary Math. , Amer. Math. Soc., Providence, RI, 2013.[9] J. Kigami, Analysis on fractals, Cambridge tracts in Math., , Cambridge Univ. Press, Cambridge,2001.[10] P. Koskela, Y. Zhou, Geometry and Analysis of Dirichlet forms, Adv. Math., , 2755-2801, 2012.[11] S. Kusuoka, A diffusion process on a fractal, in: K. Ito and N. Ikeda (eds.), Probabilistic methods inMathematical Physics, Academic Press, Boston, MA, 251-274, 1987.[12] S. Kusuoka, Dirichlet forms on fractals and products of random matrices, Publ. Res. Inst. Math. Sci., , 659-680, 1989.[13] R. Ma˜n´e, Ergodic theory and differentiable dynamics, Berlin, 1983.[14] U. Mosco, Composite media and asymptotic Dirichlet forms, J. Functional Analysis, , 368-421, 1994.[15] U. Mosco, Variational fractals, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), , 683-712, 1997.[16] R. Peirone, Existence of self-similar energies on finitely ramified fractals, J. Anal. Math., , 35-94,2014.[17] R. Peirone, Convergence of Dirichlet forms on fractals, mimeographed notes.[18] W. Perry, M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Aster-isque,187-188