Pre-image Entropy. Wen-Chiao Cheng and Sheldon Newhouse. February 11, 2004

Transcription

1 Pre-image Entropy Wen-Chiao Cheng and Sheldon Newhouse February, 2004 Abstract We define and study new invariants called pre-image entropies which are similar to the standard notions of topological and measuretheoretic entropies. These new invariants are only non-zero for noninvertible maps, and they give a quantitative measurement of how far a given map is from being invertible. We obtain analogs of many known results for topological and measure-theoretic entropies. In particular, we obtain product rules, power rules, analogs of the Shannon- Breiman-McMillan Theorem, the ergodic decomposition of entropy, and a variational principle. Introduction. The notions of entropy are fundamental to our current understanding of dynamical systems. The two main notions are, of course, the topological entropy and the measure-theoretic (or metric) entropy. The former measures the maximal exponential growth rate of orbits for an arbitrary topological dynamical system, and the latter measures the maximal loss of information of the iteration of finite partitions in a measure preserving transformation. It is well-known that these two invariants are related by the so-called Variational Principle which gives that the topological entropy is the supremum of the metric entropies for all invariant probability measures of a given topological dynamical system. There are many useful properties shared by these two invariants. Using the word entropy to denote either of these concepts, we have:

2 February, entropy is an invariant in the appropriate category: topologically conjugate systems have the same topological entropy and measure theoretically conjugate systems have the same metric entropy, 2. the entropy of a direct product is the sum of the entropies of the factors, 3. the entropy of f n equals n times the entropy of f, 4. if g is a factor of f, then the entropy of g is no larger than the entropy of f, 5. entropy is preserved under the passage to natural extensions. The last property above shows a weakness of both of these invariants. Namely, they provide no information for non-invertible systems which cannot be gleaned from their associated (invertible) natural extensions. In several recent papers [7], [6], [4], [8], and [3] some new topological invariants of dynamical systems have been defined and studied which only give non-trivial information when the associated systems are non-invertible. In a certain sense, these new invariants give a quantitative estimate of how non-invertible a system is. For simplicity, if we have a numerical quantity h(f) defined for dynamical systems which in preserved under topological conjugacy and which is zero on invertible systems, we call the quantity a non-invertible invariant. Let us be more precise and recall some non-invertible invariants defined by M. Hurley. Let f : X X be a continuous self-map of the compact metric space (X, d). Let n > 0 be a positive integer. Define the d f,n metric in X by d f,n (x, y) = max 0 j<n d(f j x, f j y). A set E is an (n, ɛ) separated set if, for any x y in E, one has d f,n (x, y) > ɛ. Given a subset K X, we define the quantity r(n, ɛ, K) to be the maximal cardinality of (n, ɛ) separated subsets of K. Uniform continuity of f j for 0 j < n guarantees that r(n, ɛ, K) is finite for each n, ɛ > 0. Following Hurley [4], we define the quantities h p (f), h m (f) by h p (f) = sup lim lim sup x X ɛ 0 n n log r(n, ɛ, f n x),

3 February, h m (f) = lim lim sup ɛ 0 n n log sup r(n, ɛ, f n x). x X Of course, the topological entropy, h top (f) is given by h top (f) = lim lim sup log r(n, ɛ, X). ɛ 0 n n It is evident that, in general, one has the inequalities h p (f) h m (f) h top (f). In [8], it is proved that for α = p or α = m, and any positive integer k we have hα(f k ) = khα(f), () and hα(f g) hα(f) + hα(g), (2) the quantity hα(f) is preserved under topological conjugacy. (3) Notice that if f is invertible, then r(n, ɛ, f n x) = for all x, n, ɛ so that h p (f) = h m (f) = 0. Thus, h p and h m are non-invertible invariants. It is not known if the inequality in (2) can be strict. In a recent paper of D. Fiebig, U. Fiebig, and Z. Nitecki,[3], it is proved that h p = h m = h top for forward expansive maps (in particular for subshifts on finitely many symbols), and examples are constructed in which h p (f) < h m (f). It follows from work of Langevin and Przytycki [6], and Nitecki and Przytycki [8] that h m = h top for holomorphic endomorphisms of the Riemann Sphere and for continuous endomorphisms of a real interval. Using the sub-additivity of the product above (2), one sees that if T = f g with f an arbitrary endomorphism and g an automorphism with positive topological entropy, then, h m (T ) < h top (T ). We started this research by considering whether there is a variational principle for h p (f) or h m (f). This is still unknown. However, we discovered that one can define a new non-invertible invariant h pre (f) which is between

4 February, h m (f) and h top (f) for which a variational principle does indeed hold. This required a new kind of metric entropy to be defined which takes into account the past behavior of f. The appropriate quantity is simply the metric entropy of f conditioned on the infinite past σ algebra n f n B where B is the Borel σ algebra. We call this the metric pre-image entropy and denote it by h pre,µ (f) where µ is a Borel invariant probability measure for f. If one maximizes this quantity over all invariant measures, it is clear that one gets a topological invariant, but it was suprising to us that this can be defined in terms of our quantity h pre (f). In addition to this variational principle, we will show that the quantities h pre (f) and h pre,µ (f) also satisfy power and product rules analogous to the standard topological and metric entropy quantities, that the map µ h pre,µ (f) is affine, and that there are analogs of the Shannon-Breiman-McMillan and ergodic decomposition theorems for the metric pre-image entropy. 2 Statement of Results Given f : X X as above, ɛ > 0 and n N, let us define h pre (f) = lim lim sup ɛ 0 n n sup r(n, ɛ, f k x). (4) x X,k n It is clear the h m (f) h pre (f) h top (f), and that h pre (f) = 0 if f is a homeomorphism. Our first results are the following. Theorem 2. Let f : X X and g : Y Y be continuous self-maps of the compact metric spaces X, Y, respectively. Then,. The pre-image entropy h pre (f) is independent of the choice of metric on X. 2. (Power Rule) For any positive integer τ, we have h pre (f τ ) = τ h pre (f). 3. (Product Rule) h pre (f g) = h pre (f) + h pre (g). 4. (Topological Invariance) If f is topologically conjugate to g, then h pre (f) = h pre (g).

5 February, Remark. We have already mentioned that, for forward expansive maps, Fiebig, Fiebig, and Nitecki showed that h p (f) = h m (f) = h top (f). Hence, also for these maps we have h pre (f) = h top (f) and our pre-image entropy is nothing new. On the other hand, analogous to the case of h m above, our product rule implies that if T = f g where f is any endomorphism and g is an automorphism with h top (g) > 0, then h pre (T ) = h pre (f) + h pre (g) = h pre (f) h top (f) since h pre (g) = 0. But, h top (T ) = h top (f) + h top (g) > h top (f). Thus there are many natural examples of endomorphisms T with h pre (T ) < h top (T ). Next, we consider the analogous notion of metric pre-image entropy. Let (f, X, B, µ) denote a measure preserving transformation of the probability space (X, B, µ). That is, (X, B, µ) is a measure space with µ(x) =, and, X = X X2 is a disjoint union of measurable sets in which X 2 is a possibly empty at most countable set of atoms and (X, B X, µ X ) is isomorphic (mod 0) to a subinterval of the real closed unit interval with Lebesgue measure and the σ algebra of Lebesgue measurable sets. The map f : X X is such that f (E) B for each E B, and µ(f (E)) = µ(e) for each E B. Following standard terminology, we call (f, X, B, µ) a measurepreserving transformation. Set B = n 0 f n B. We call B the infinite past σ algebra related to B. As usual (e.g. see Petersen [9]), given a subset A B, we set µ(a B ) = E(χ A B ) where χ A denotes the characteristic function of A, and E(ψ A) denotes the conditional expectation of the function ψ given the sub-σ-algebra A. For finite partitions α, β, we set α β = {A B : A α, B β}. If 0 j n, are positive integers, we let α n j = i=n i=j f i α, and α n = α n 0. Define the information function Iα B of α given the infinite past σ algebra B to be Iα B = log µ(a B )χ A. A α Set H µ (α B ) = Iα B dµ = log µ(a B )µ(a). A α It is standard (see e.g. [9]) that the quantity H µ (α B ) is increasing in the first variable and decreasing in the second variable. That is, if β is a partition which refines α and A is a sub σ algebra of B, then,

6 February, H µ (α B ) H µ (β A). Also, H µ (f α f B ) = H µ (α B ). It follows that the numbers a n = H µ (α n B ) form a subadditive sequence. Indeed, for positive integers n, m, using f n B = B, we have a n+m = H µ (α n+m B ) = H µ (α n f n α m ) B ) H µ (α n B ) + H µ (f n α m B ) = H µ (α n B ) + H µ (f n α m f n B ) = H µ (α n B ) + H µ (α m B ) = a n + a m. Consequently, there is a well-defined number h µ (α B ) given by h µ (α B ) = lim n n H µ(α n B ) = inf n n H µ(α n B ). We define the metric (or measure-theoretic) pre-image entropy of f with respect to µ and B to be h pre,µ (f) = sup α h µ (α B ). It is easy to see that h pre,µ (f) is an invariant of measure-theoretic conjugacy. That is, if (f, X, B, µ) and (g, Y, B, ν) are measure preserving transformations, and π : X Y is a bimeasurable bijection (mod 0) such that gπ = πf, then h pre,µ (f) = h pre,ν (g). This is the measure-theoretic analog of statement 4 in Theorem 2.. Next, we have the measure-theoretic analog of statements 2 and 3. Theorem 2.2 Let (f, X, B, µ) and (g, Y, B, ν) be measure preserving transformations. Then,. (Power Rule) For any positive integer τ, we have h pre,µ (f τ ) = τ h pre,µ (f).

7 February, (Product Rule) h pre,µ ν (f g) = h pre,µ (f) + h pre,ν (g). Our next three results are analogs of well-known theorems concerning metric entropy adapted to the setting of metric pre-image entropy. Theorem 2.3 ( Affinity of metric pre-image entropy)let (X, B) be a measurable space, f : X X be a measurable transformation, and let µ and ν be two f-invariant probability measures so that both (X, B, µ) and (X, B, ν) are Lebesgue spaces together with the possible exception of countable sets of atoms. Let q [0, ]. Then, and h pre,qµ+( q)ν (α, f) = qh pre,µ (α, f) + ( q)h pre,ν (α, f), (5) h pre,qµ+( q)ν (f) = qh pre,µ (f) + ( q)h pre,ν (f). (6) Theorem 2.4 (Shannon-Breiman-McMillan Theorem for metric pre-image entropy) Let (f, X, B, µ) be an ergodic measure preserving transformation of the probability space (X, B, µ), and let α be a finite measurable partition. Then, lim n n I α n B = h µ(α B ), (7) where the convergence is µ almost everywhere and in L (µ). Theorem 2.5 (Ergodic Decomposition for metric pre-image entropy) Let (f, X, B, µ) be a measure preserving endomorphism of the probability space (X, B, µ), and let {ν x, x X} denote the ergodic components of µ. Let α be a finite measurable partition of X. Then, the mappings x h νx (α B ) and x h pre,νx (f) are in L (µ) and we have h µ (α B ) = h νx (α B )dµ(x), (8) and h pre,µ (f) = h pre,νx (f)dµ(x). (9)

8 February, Remark. The previous two theorems actually work if B is replaced by any invariant sub-σ algebra A; i.e. f A = A. Next, we present the main result of this paper: that the topological and metric pre-image entropies are related by a variational principle. We consider measure preserving transformations (f, X, B, µ) in which X is a compact metric space, f : X X is continuous, and B is the σ algebra of Borel subsets of X. As usual in this case we call µ an f invariant Borel probability measure. Theorem 2.6 (Variational Principle for pre-image entropy) Let f : X X be a continuous self-map of the compact metric space X, and let M(f) denote the set of f invariant Borel probability measures. Then, Remarks. h pre (f) = sup h pre,µ (f). (0) µ M(f). Another type of pre-image topological entropy could be defined as h pre(f) = lim lim sup ɛ 0 n n sup r(n, ɛ, f k x). x X,k It is clear that h pre (f) h pre(f), but it is not obvious whether one might have strict inequality. It turns out that one can follow through the proofs of Theorem 2.6 with this new quantity and show that it also equals sup µ h pre,µ (f). Hence, it makes no difference whether one uses k n or k in the definition of h pre (f). 2. It is obvious from the Variational Princlple that one can compute h pre (f) by restricting to any set which contains the supports of the f invariant probability measures. We do not know if this is true for the quantities h p (f), h m (f) of Hurley. 3. It will be clear to experts that many of our methods of proof here are obtained by making appropriate modifications of many known techniques in ergodic theory and topological dynamics. However, the proof of the part of the variational principle which asserts that the pre-image entropy is no larger than the supremum of the metric pre-image entropies of invariant meaures is considerably more delicate than the standard result for topological entropy.

9 February, Proof of Theorem 2. Before proceding to the proof, we recall some concepts from entropy theory. Let f, (X, d), d f,n be as above. Let ɛ > 0 and n > 0. A subset F X is an (n, ɛ, f) separated if whenever x, y F and x y, we have d f,n (x, y) > ɛ. Given a subset K X, we let r(n, ɛ, K, f) denote the maximal cardinality of an (n, ɛ, f) separated subset of K. A subset E K is an (n, ɛ, K, f)-spanning set if, for every x K, there is a y E such that d f,n (x, y) ɛ. Let s(n, ɛ, K, f) be the minimal cardinality of any (n, ɛ, K, f)-spanning set. It is standard that for any subset K X, s(n, ɛ, K, f) r(n, ɛ, K, f) s(n, ɛ, K, f). () 2 Next, using techniques as in Bowen [], we have the following. If n, n 2, l are positive integers such that l n, then r(n + n 2, ɛ, f l K, f) s(n, ɛ 2, f l K, f) s(n 2, ɛ 2, f l+n K, f) r(n, ɛ 2, f l K, f) r(n 2, ɛ 2, f l+n K, f). (2) Also, observe that if m > 0, n > 0, E is an (n, ɛ, f) separated subset of f k+m x, and F f m n x is such that f m maps F bijectively onto E, then, F is an (m + n, ɛ, f) separated subset of f k x. Hence, r(n, ɛ, f k+m x, f) r(m + n, ɛ, f k x, f). (3) It will be convenient to use the notation so that h pre (f, ɛ) = lim sup n n log sup k n,x X h pre (f) = lim ɛ 0 h pre (f, ɛ) = sup ɛ>0 r(n, ɛ, f k x, f) h pre (f, ɛ). When the metric d needs to be explicitly mentioned, we write h pre (f, d), h pre (f, ɛ, d), r(n, ɛ, f k x, f, d),

10 February, etc. Let us prove that h pre (f) is independent of the metric on X. Let d, d 2 be two metrics on X. Then, by compactness of X, for every ɛ > 0 there is a δ > 0 such that, for all x, y X, if d (x, y) < δ, then d 2 (x, y) < ɛ. It follows that r(n, ɛ, f k x, f, d 2 ) r(n, δ, f k x, f, d ) for all n, ɛ, k, x. This gives that h pre (f, ɛ, d 2 ) h pre (f, δ, d ). Letting δ 0 gives h pre (f, ɛ, d 2 ) h pre (f, d ). Now, letting ɛ 0 gives h pre (f, d 2 ) h pre (f, d ). Interchanging d and d 2 then gives the opposite inequality, proving that h pre (f, d ) = h pre (f, d 2 ). Now, let us proceed to prove the Power Rule. Write g = f τ. Assertion : For any ɛ > 0 we have h pre (g, ɛ) τ h pre (f, ɛ). Let k n and x X. It is clear that Hence, we have r(n, ɛ, g k x, g) r(τn, ɛ, f τ k x, f). h pre (g, ɛ) = lim sup n n log sup r(n, ɛ, g k x, g) k n,x X lim sup n n log sup r(τn, ɛ, f τ k x, f) k n,x X τ = lim sup n nτ log τ k τ sup r(τn, ɛ, f τ k x, f) n,x X τ lim sup n nτ log k τ sup r(τn, ɛ, f k x, f) n,x X = τ lim sup n nτ log k τ sup r(τn, ɛ, f k x, f) n,x X τ lim sup n n log sup r(n, ɛ, f k x, f) k n,x X = τ h pre (f, ɛ). proving Assertion. Assertion 2: Given ɛ > 0, let δ > 0 be such that if d(x, y) < δ, then d(f j x, f j y) < ɛ for j [0, τ). Then, 4 h pre (g, δ) τ h pre (f, ɛ).

11 February, 2004 Proof. Let n > 0, k n. From the definition of δ, we have that if l, s are positive integers such that l τs, then r(τs, ɛ 4, f l x, f) r(s, δ, f l x, g). (4) Write k = τn 2 +l 2 with 0 l 2 < τ and n l 2 = τn +l with 0 l < τ. Let C(j) denote a constant depending on the positive integer j. From (), (2), (3), and (4), we have r(n, ɛ, f k x, f) s(n l 2, ɛ 2, f k x, f)s(l 2, ɛ 2, f k+n l 2 x, f) C(l 2 )s(n l 2, ɛ 2, f k x, f) = C(l 2 )s(τn + l, ɛ 2, f k x, f) C(l 2 )s(τn, ɛ 4, f k x, f)s(l, ɛ 4, f k+τ n x, f) C(l 2 )C(l )r(τn, ɛ 4, f k x, f) = C(l 2 )C(l )r(τn, ɛ 4, f τ n 2 l 2 x, f) = C(l 2 )C(l )r(τn, ɛ 4, f τ (n 2+)+τ l 2 x, f) C(l 2 )C(l )r(n, δ, f τ (n 2+) (f τ l 2 x), g) = C(l 2 )C(l ) r(n, δ, g (n 2+) (f τ l 2 x), g). Note that τ(n 2 + ) > k n = τn + l + l 2 τn, so n 2 + > n. As n, so does n. Hence, using that log(c(l n 2)C(l )) 0 as n, we get h pre (f, ɛ) = lim sup n lim sup n = lim sup n = τ lim sup n n log sup r(n, ɛ, f k x, f) k n,x X n log sup C(l 2 )C(l )r(n, δ, g k x, g) k n,x X log sup r(n, δ, g k x, g) τn + l + l 2 k n,x X log sup r(n, δ, g k x, g) n k n,x X

12 February, = τ h pre(g, δ). Letting ɛ 0 in Assertion, we get h pre (g) τ h pre (f). Also, letting δ 0 and then taking ɛ 0 in Assertion 2, we get h pre (g) τ h pre (f), completing the proof of the Power Rule in Theorem 2.. Next, we go to the Product Rule. Let (X, d ), (Y, d 2 ) be compact metric spaces with continuous maps f : X X, g : Y Y. Give X Y the metric d((x, y ), (x 2, y 2 )) = max(d (x, x 2 ), d 2 (y, y 2 )). We have two things to prove: and h pre (f g) h pre (f) + h pre (g), (5) h pre (f g) h pre (f) + h pre (g). (6) Given (x, y) X Y, n > 0, k n, let E (f g) k (x, y) be a maximal (n, ɛ, f g) separated set. Let E be a miminal (n, ɛ, f) spanning 2 set in f k x, and let E 2 be a minimal (n, ɛ, g) spanning set in 2 g k y. For each (u, v) E, there is a pair (x (u), y (v)) E E 2 such that d (f j u, f j x (u)) ɛ 2 and d 2(f j v, f j y (v)) ɛ 2 for 0 j < n. If (u, v ), (u 2, v 2 ) are in E and are such that x (u ) = x (u 2 ) and y (v ) = y (v 2 ), then d (f j u, f j u 2 ) ɛ and d 2 (g j v, g j v 2 ) ɛ for all 0 j < n. It follows that (u, v ) and (u 2, v 2 ) remain ɛ close for all interates (f g) j with 0 j < n. Since E was (n, ɛ, f g) separated, we have (u, v ) = (u 2, v 2 ). Thus, the map (u, v) (x (u), y (v)) is injective. This implies that r(n, ɛ, (f g) k (x, y), f g) r(n, ɛ 2, f k (x), f) r(n, ɛ 2, g k (y), g). Let a n (ɛ) = sup{r(n, ɛ, (f g) k (x, y), f g) : (x, y) X Y, k n},

13 February, b n (ɛ) = sup{r(n, ɛ, f k (x), f) : x X, k n}, and c n (ɛ) = sup{r(n, ɛ, g k (y), g) : y Y, k n}. Then, we have so, a n (ɛ) b n ( ɛ 2 ) c n( ɛ 2 ) (7) h pre (ɛ, f g) = lim sup n lim sup n n log a n(ɛ) n log b n( ɛ ) + lim sup 2 n = h pre ( ɛ 2, f) + h pre( ɛ 2, g). n log c n( ɛ 2 ) Letting ɛ 0 implies (5). For the converse statement, it is tempting to try the following. Let n > 0, k n, ɛ > 0. If E is a maximal (n, ɛ, f)-separated set in f k x, and E 2 is a maximal (n, ɛ, g)-separated set in g k y, then E E 2 is clearly an (n, ɛ, f g) separated set in (f g) k (x, y). This gives r(n, ɛ, f k (x), f) r(n, ɛ, g k (y), g) r(n, ɛ, (f g) k (x, y), f g), and If b n (ɛ) c n (ɛ) a n (ɛ). lim n log a n(ɛ), lim n log b n(ɛ), and lim n log c n(ɛ) all existed, then this would give that h pre (ɛ, f)+h pre (ɛ, g) h pre (ɛ, f g), and (6) would follow. In general, these limits do not exist, and this argument breaks down.

14 February, Instead, we bring the Variational Principle and the product rule for metric pre-image entropy to the rescue. Let ɛ > 0, and let µ, ν be invariant measures so that Then, we have h pre,µ (f) > h pre (f) ɛ, and h pre,ν (g) > h pre (g) ɛ. h pre (f g) h pre,µ ν (f g) = h pre,µ (f) + h pre,ν (g) h pre (f) ɛ + h pre (g) ɛ. Since ɛ is arbitrary, (6) follows. Finally, we prove the topological invariance statement. Let φ : X Y be a topological conjugacy from f to g. That is, φ is a homeomorphism and gφ = φf. Let d be the metric on X and d 2 be the metric on Y. It is obvious that the induced metric φ d on Y defined by φ d (y, y 2 ) = d (φ y, φ y 2 ) gives the same pre-image entropy to g as h pre (f). Since, the pre-image entropy is independent of the choice of metric on Y, we have h pre (f) = h pre (g) as required. This completes the proof of Theorem 2. 4 Proofs of Theorems 2.2 and 2.3 Here and in the sequel we will assume without further mention that all sets considered are appropriately measurable. The Power Rule: Let g = f τ with τ > 0, and consider the two σ algebras B = n f n (B), B = n g n B = n f nτ (B). It is obvious that B B since the latter is an intersection of fewer sets. On the other hand, if l k, then f l (B) f k (B). So, for each n > 0 we have B f nτ (B) f n (B). Hence, B f n (B) for all n > 0. This gives

15 February, B = B. (8) Let α be a finite partition, and let β = τ i=0 f i α. Then, So, n i=0 g i (β) = nτ i=0 f i (α). h µ (α B nτ, f) = lim n nτ H µ( = lim n nτ H µ( = τ lim n n H µ( i=0 n i=0 n i=0 = τ h µ(β B, g) f i α B ) g i β B ) g i β B ) τ h pre,µ(g). Taking the supremum over all α then gives h pre,µ (f) τ h pre,µ(g). On the other hand, returning to h µ (α B, f) = τ h µ(β B, g) and using h µ (β B, g) h µ (α B, g), we get γ. h µ (α B, f) τ h µ(α B, g). This time taking the supremum over all α gives h pre,µ (f) τ h pre,µ(g). Product Rule: Given a collection γ of sets, let B(γ) denote the σ algebra generated by

16 February, Observe that if γ, γ 2,... is an increasing sequence of finite partitions of X such that B( γ i ) = B (i.e, the sequence γ i generates B), then for any finite partition α, we have H µ (α B ) = lim j lim i H µ (α f j γ i ). Let B X denote the σ algebra on X and B Y denote that on Y. For U = X, Y, let A U denote the collection of finite partitions of B U. Let γ γ 2... and δ δ 2... be increasing sequences of finite partitions such that γ i A X, i γ i generates B X, δ i A Y, and i δ i generates B Y. Then, for any partitions α A X, β A Y we have H µ (α B X) = lim j lim i H µ (α f j (γ i )), and H ν (β B Y ) = lim j lim i H ν (α g j (δ i )), H µ ν (α β B X Y ) = lim j lim i H µ ν (α β f j (γ i ) g j (δ i )). Here, of course, we define α β = {A B : A α, B β}. Recall the standard formula H µ (α β) = H µ (α β) H µ (β) (9) for a measure µ and finite partitions α, β. Let us use this formula with the notation that A, B, C, D run through α, β, γ, δ, respectively, to get H µ ν (α β γ δ) = A,B,C,D µ(a C)ν(B D) log(µ(a C)ν(B D)) + C,D µ(c)ν(d) log(µ(c)ν(d)) = µ(a C) log(µ(a C)) ν(b D) log(ν(b D)) A,C B,D + C µ(c) log µ(c) + D ν(d) log ν(d) = H µ (α γ) + H ν (β δ).

17 February, Putting the above formulas together gives that for any finite partitions α of X, β of Y, we have H µ ν (α β B X Y ) = H µ (α B X) + H ν (β B Y ). This implies that, for any positive integer n, we have n H µ ν ( (f g) k (α β) BX Y ) = H µ ν ( k=0 = H µ ( n k=0 n k=0 f k α n k=0 g k β B X Y ) n f k α BX) + H ν ( g k β BY ). k=0 Multiplying both sides by n and taking the limit as n gives h pre,µ ν (α β, f g) = h pre,µ (α, f) + h pre,ν (β, g) for any α A X, β B Y. Since h pre,µ ν (f g) can be computed as the supremum over product partitions α β, this proves the Product Rule in Theorem 2.2. Proof of Theorem 2.3: It clearly suffices to prove formula (5). Let m = qµ + ( q)ν. Recall from page 6 of [2] that, for any finite partition α, we have 0 H m (α) qh µ (α) ( q)h ν (α) log 2. (20) Let γ γ 2... be an increasing sequence of finite partitions such that i γ i generates the σ algebra B. Then, for any positive integer n and ρ = m, ν, µ, we have By (20), we get H ρ (α n B ) = lim j lim i H ρ (α n f j γ i ). 0 H m (α n f j γ i ) qh µ (α n f j γ i ) ( q)h ν (α n f j γ i ) log 2, (2) and

18 February, [ H m (f j γ i ) qh µ (f j γ i ) ( q)h ν (f j γ i ) ] log 2. (22) The second term of (22) is non-positive, so adding it to the second term of (2) does not increase the latter s value, so H m (α n f j γ i ) qh µ (α n f j γ i ) ( q)h ν (α n f j γ i ) log 2. Similarly, adding the second term of (2) to that of (22) does not decrease the latter s value, so log 2 H m (α n f j γ i ) qh µ (α n f j γ i ) ( q)h ν (α n f j γ i ). Putting these two inequalities together gives log 2 H m (α n f j γ i ) qh µ (α n f j γ i ) ( q)h ν (α n f j γ i ) log 2. Letting i and then j gives log 2 H m (α n B ) qh µ (α n B ) ( q)h ν (α n B ) log 2. Now, dividing by n and letting n gives that as required. h m (α B ) = qh µ (α B ) + ( q)h ν (α B ) 5 Proofs of Theorems 2.4 and 2.5 The proof of Theorem 2.4 is a more or less straightforward adaptation of the proof of the standard Shannon-Breiman-McMillan Theorem in [9]. We sketch the ideas. For a finite partition α, let B(α) denote the σ algebra generated by α. For notational convenience, denote Iα A also by I(α A) for a partition α and sub σ algebra A. Let f n = I(α B(α n ) B ) for n > 0, and f 0 = I(α B ). First observe that

19 February, I(α n 0 B ) = I(α n B ) + I(α B(α n ) B ) = I(α n 0 B ) f + I(α B(α n ) B ) = n f n k f k (23) k=0 Integrating both sides, and using f k invariance of µ gives H µ (α n 0 B ) = This gives the formula n f n k f k = k=0 n f n k = k=0 n f k. n H µ (α n 0 B ) = H µ (α B ) + H µ (α B(α k ) B ). (24) k= Since the sequence H µ (α B(α k ) B ) converges non-increasingly to some number h 0, the associated Cesaro sequence n+ n n H µ (α B(α k ) B ) k= also converges to h 0 as n. But, multiplying both sides of (24) by and taking the limit, we then get k=0 h 0 = n lim n + H µ(α n 0 B ) = h µ (α B ). Now, the proof of the Shannon-Breiman-McMillan theorem in [9] gives that the sequence {I(α α k ), k } is an L bounded martingale. Since I(α B(α k ) B )dµ = H µ (α B(α k ) B ) H µ (α B(α k )) = I(α B(α k ))dµ, we have that {I(α B(α k ) B ), k } is also an L bounded martingale.

20 February, Hence the last sequence converges almost everywhere and in L to some integrable function h. Its Cesaro sums must also converge in both senses to h as well. Note, also, that hdµ = h 0. Again, following the method in [9], we have I(α n 0 B ) lim n n + = n lim n + = n + n f n k f k k=0 n (f n k h) f k + n h f k. k=0 n + k=0 Since f n h, the first term in the last equality approaches 0, and, ergodicity of the µ gives that the second term approaches hdµ = h 0. Hence, I(α n B ) lim n n completing the proof of Theorem 2.4. = lim n I(α n 0 B ) n + = h 0, Proof of Theorem 2.5: The ergodic components {ν x } satisfy the following property. For each φ L (µ), there is an f invariant set E X of full µ measure such that. ν x (E) = for all x E, 2. φ L (ν x ) for x E, 3. the function x φ(y)dν x (y) is in L (µ), and 4. φ(x)dµ(x) = ( φ(y)dν x (y))dµ(x). Let α be a finite measurable partition, and let n > 0. Applying the above to the function Iα n B, we get n H(αn B ) = n = n = n I(α n B )(x)dµ(x) ( I(α n B )(y)dν x (y))dµ(x) H νx (α n B )dµ(x).

21 February, Note that n H ν x (α n B ) = n H ν x ( n = n n k=0 n k=0 n k=0 f k α B ) H νx (f k α B ) H νx (α B ) = H νx (α B ) and the latter function of x is bounded and measurable (it is bounded above by log card α). Letting n and using the bounded convergence theorem gives (8). Now, taking the supremum over all α gives (9) and completes the proof of Theorem Proof of Theorem 2.6 We adapt the well-known arguments of Misiurewicz (as in pages of [9]) to pre-image entropy. Step. h pre,µ (f) h pre (f) for all µ M(f): Let µ M(f). If E, E 2,..., E s is a finite disjoint collection of compact subsets of X, we call the partition s α = {E, E 2,..., E s, X \ E i } i= a compact partition. It is easy to verify that h pre,µ (f) is the supremum of h pre,µ (α, f) where α varies over the compact partitions. Thus, using a standard technique, it suffices to prove that, for any compact partition α, h pre,µ (α, f) h pre (f) + log 2. (25) Indeed, once this is done, it follows that h pre,µ (f) h pre (f) + log 2

22 February, for every f. Applying this to f N for large N, gives or, Nh pre,µ (f) = h pre,µ (f N ) h pre (f N ) + log 2 h pre,µ (f) h pre (f) + log 2 N. Letting N then, gives h pre,µ (f) h pre (f). For (25), it suffices to show that there is an ɛ > 0 such that for any n > 0 and any k > 0, we have H µ (α n f k B) n log 2 + log sup r(n, ɛ, f k x). (26) x X Let ɛ be such that any 4ɛ ball meets at most two elements of α. Let β β 2... be a non-decreasing sequence of finite partitions with diameters tending to zero. Thus, and, B = β j, j= H µ (α n f k B) = lim j H µ (α n f k β j ). So, it suffices to show that, for sufficiently large j, we have H µ (α n f k β j ) n log 2 + log sup r(n, ɛ, f k x). (27) x X Let ɛ = ɛ (n, ɛ) > 0 be small enough so that if d(x, y) < ɛ, then d(f i x, f i y) < ɛ for 0 i < n. The collection {f k x : x f k X} is an uppersemicontinuous decomposition of X. Hence, for each x f k X there is an ɛ 2 (x, k, ɛ ) > 0 such that if d(y, x) < ɛ 2 (x, k, ɛ ), y f k X, and y f k y, then there is an x f k x such that d(y, x ) < ɛ. Let U be the collection of open ɛ 2 (x, k, ɛ ) balls in f k X as x varies in f k X, and let ɛ 3 be a Lebesgue number for U. Since diam(β j ) 0 as j, we may choose j 0 large enough so that if j j 0 and B is an element of β j, then diam(b) < ɛ 3. Assume j j 0.

23 February, For a set C f k β j, let µ C denote the conditional measure of µ restricted to C. Let a n C denote the set {A C : A α n, A C }. We have H µ (α n f k β j ) = C f k β j H µc (α n C)µ(C) max H µc (α n C) C f k β j max C f k β j log card α n C Fix a B β j so that C = f k B is non-empty. For each A α n C, let x A be an element of A. Since f k x A B and diam(b) < ɛ 3, there is a u B f k X such that if y B f k X, then d(u B, y) < ɛ 2 (u B, k, ɛ ). In particular, d(f k x A, u B ) < ɛ 2 (u B, k, ɛ ). Hence, there is a point φ (A) f k u B such that d(x A, φ (A)) < ɛ, so d(f i x A, f i φ (A)) < ɛ for 0 i < n. Let E C be a maximal (n, ɛ)-separated set in f k u B. Since E C spans f k u B, there is a point φ 2 (A) E C such that, for 0 i < n, and, hence, d(f i φ (A), f i φ 2 (A)) ɛ, d(f i x A, f i φ 2 (A)) 2ɛ, We have defined a map φ 2 from α n C into E C. Hence, To prove (27), we will show card(α n C) card E C (max y EC φ 2 (y)).

24 February, for each y E C. Let A, Ã be such that φ 2(x A ) = φ 2 (Ã). Then, for 0 i < n, we have card(φ 2 (y)) 2 n (28) d(f i x A, f i xã) 4ɛ. It follows that f i x A is in the 4ɛ ball about f i xã. Since each 4ɛ ball meets at most two elements of α, it follows that there are at most 2 n choices for A, Ã. This proves (27) and finishes the proof of Step. Step 2. h pre (f) sup h pre,µ (f). (29) µ M(f) Given ɛ > 0, we wish to produce an f invariant probability measure µ such that h pre,µ h pre (f, ɛ). (30) Again we want to follow the idea of the Misiurewicz proof of the standard variational principle (as in [9], pages ), but we must make important modifications. Although it is not strictly necessary, the reader will find it helpful in what follows to be familiar with the proof in the aforementioned reference. We will motivate our construction by making frequent reference to that proof. When it is convenient to refer to the arguments in [9], we simply call them the SVP arguments. Choose sequences n i, k i > n i, x i X, such that h pre (f, ɛ) = lim log r(n i, ɛ, f k i x i, f). i n i Let E i denote a maximal (n i, ɛ) separated set in f k i x i such that card E i = r(n i, ɛ, f k i x i, f). Thus, we have h pre (f, ɛ) = lim log card E i. (3) i n i Letting δ x denote the point mass at point x X, let

25 February, let µ i = δ x, card E i x E i ν i = n i n i j=0 f j µ i, and, let ν = lim i ν i where we have passed to subsequences assuring that the last limit of measures exists. Let α be a finite ν continuity partition such that diam(α) < ɛ. That is, for each A α, µ( A) = 0, and diam(a) < ɛ. Motivated by the SVP arguments in [9], we wish to show that, for every positive integer m, Indeed, from this, we have H ν (α m B ) mh pre (f, ɛ). (32) h pre,ν (f) h pre,ν (α, f) = inf m H ν (α m B ) m h pre (f, ɛ), as required. If we were in the SVP case and not conditioning on B, the result that H ν (α m ) mh(f, ɛ) follows from the following two facts: and H ν (α m ) = lim i H νi (α m ), (33) H νi (α m ) n i n i l=0 H f l µ i (α m ) = n i n i l=0 H µi (f l α m ). (34) Statement (33) follows from the fact that the map µ µ(a) is continuous at ν, and statement (34) follows from the concavity of the function t t log t arising in the entropy formula

26 February, H νi (α m ) = ν i (A) log ν i (A). (35) A α m We need to establish the analogs of (33) and (34) after conditioning on B. To do this we will make use of the Rokhlin theory of measurable partitions (see [5] for definitions and basic facts about this theory). This theory concerns itself with complete measure spaces, so we will actually deal with a slightly larger σ algebra BC in which ν is complete. Let C denote the subcollection of B consisting of ν null sets. That is, E C if and only if E B and ν(e) = 0. Since the countable union of elements in C is again in C, it follows that for any σ algebra A of subsets of X, there is an enlarged σ algebra A C defined by A A C if and only if there are sets B, M, N such that A = B M, B A, N C, and M N. The σ algebra BC is simply the standard ν completion of B. We will also consider the σ algebras B k = (f k B) C for k. Since f (C) C, we have that B B Letting B = k B k, it is easy to check that and B B C B, f l B k B l+k l. Note that the σ algebras B k, B are not necessarily ν complete. Lemma 6. Under the above notation, we have and H ν (α m B ) lim sup H νi (α m B ), (36) i H νi (α m B k i ) n i H n f l µ i (α m B k i ). (37) i l=0

27 February, Remark. In the course of the proof of the lemma, we will see that while the proof of estimate (36) is rather straightforward, that of (37) is considerably more complicated. It is not true that H νi (α m 0 A) n i H n f l µ i (α m 0 A). (38) i for all average measures ν i and all invariant sub σ algebras A. Indeed, statement (37) holds because the measure ν i is closely related to the measurable partition {f k i x, x X}. Proof of (32) assuming Lemma (6.): Since B B, we have that l=0 H ν (α m B ) H ν (α m B ). (39) Further, since B is the decreasing limit of the subalgebras B k, we clearly have H ν (α m B ) H ν (α m B k i ), for all i. Also, since µ i is supported on f k i x i, the canonical system of conditional measures induced by µ i on the measurable partition {f k i x, x X} reduces to a single measure on the set f k i x i which we may identify with µ i. Now, each element A B k i can be expressed as the disjoint union A = B C with B f k i B = {f k i x, x X}, and C C. Since µ i is supported on elements of f k i B, we have µ i (C) = 0. Hence, for any finite partition γ, we have H µi (γ B k i ) = H µi (γ f k i x i ). Since each element of α m f k i x i has at most one point, we have H µi (α m f k i x i ) = log card E i. In the sequel, for non-negative integers a b, let us denote by [a, b) the set of integers j such that a j < b.

28 February, For 0 j < m, let and, let C(i, j) = {n [0, n i ) : n j mod(m)}, We use the standard notations α t s = ρ(i) = max C(i, j). j=t j=s f j α, α s = α s 0. Since we can express [0, n i ) as the disjoint union [0, n i ) = 0 j<m and, for each fixed 0 j < m, we can write C(i, j), (40) we get α ni = α j t C(i,j) f t α m α n i n i ρ(i), (4) Thus, log card E i = H µi (α n i f k i x i ) H µi ( t C(i,j) Using (37) we have H µi (α j f k i x i ) + H µi ( t C(i,j) + H µi (α n i n i ρ(i) f k i x i ). f t α m f k i x i ) f t α m f k i x i ) log card E i 2m log card α. (42)

29 February, H νi (α m B ) H νi (α m B k i ) H n i ni l=0 f l µ i (α m B k i ) n i n i l=0 = n i n i l=0 n i n i l=0 n i H f l µ i (α m B k i ) H µi (f l α m f l B k i ) H µi (f l α m B k i ) 0 j<m l C(i,j) m n i inf 0 j<m H µ i ( = m n i inf 0 j<m H µ i ( From these inequalities, we get that H µi (f l α m B k i ) l C(i,j) l C(i,j) f l α m B k i ) f l α m f k i x i ) m n i [log card E i 2m log card α]. lim sup H νi (α m B m ) lim sup [log card E i 2m log card α] i i n i m = lim [log card E i 2m log card α] i n i = mh pre (f, ɛ). This and (36) proves (32). Proof of Lemma 6.: The triple (X, BC, ν) is a Lebesgue space together with a (possibly empty) at most countable set of atoms. Hence, we can find an increasing sequence β β 2... of finite partitions consisting of elements of BC such that BC = β j j=

30 February, and each β j is a ν continuity partition. Then, H ν (α m B C ) = inf j H ν (α m β j ). (43) Also, since α m and all of the β js are ν continuity partitions, we have and This and the standard formulas and give H ν (α m β j ) = lim i H νi (α m β j ) H ν (β j ) = lim i H νi (β j ). H ν (α m β j ) = H ν (α m β j ) H ν (β j ) H νi (α m β j ) = H νi (α m β j ) H νi (β j ) H ν (α m β j ) = lim i H νi (α m β j ). (44) Now, letting ɛ > 0 and using (43), we can pick j such that Then, using (44), we have H ν (α m B C ) > H ν (α m β j ) ɛ. H ν (α m BC ) H ν (α m β j ) ɛ = lim H νi (α m β j ) ɛ i lim sup H νi (α m BC ) ɛ. i lim sup H νi (α m B ) ɛ. i Since H ν (α m B ) H ν (α m B C ) and ɛ was arbitrary, we get (36).

31 February, Next, we proceed to the proof of (37). The measure ν i is supported on the set X = n i l=0 f k i (f l x i ), Case : For every pair of integers 0 r < s k i, f r x i f s x i. In this case, let ξ = {f k i (f l x i ), 0 l < n i }. This is a finite partition of X, and each of the measures f l µ i is supported on the element f k i (f l x i ) of the partition ξ. Moreover, f µ l i coincides with the conditional measure ν i,c of ν i on the element C = f k i (f l x i ) ξ. Note that ν i (C) = n i for each C ξ. We have H νi (α m B k i ) = H νi (α m ξ) = C ξ H νi,c (α m C)ν i (C) = n i n i l=0 = n i n i l=0 H f l µ i (α m f k i (f l x i )) H f l µ i (α m B k i ) So, in case, we actually have equality in (37). Case 2. There are integers 0 s < t < n i such that f s x i = f t x i. In this case, let τ be the least positive integer such that there is a t > τ such that f t x i = f τ x i, and let τ be the least integer such that τ > τ and f τ x i = f τ +τ xi. Then, {f k i (f l x i ) : 0 l < n i } is the disjoint union 0 l<τ +τ This time, let ξ be the partition f k i (f l x i ). ξ = {C = f k i (f l x i ) : 0 l < τ + τ}, and, again, let ν i,c denote the conditional measure of ν i on C.

32 February, For each τ l < τ + τ, let D(l) be the set of integers in [τ, n i ) which are congruent to l mod τ, and let d(l) = card D(l).. Then, for 0 l < τ, f µ l i is supported on the single element f k i (f l x i ) of B k i which gives In addition, we have H f l µ i (α m f k i (f l x i )) = H f l µ i (α m B k i ). (45) and ν i,f k i(f l x i ) = f l µ i (46) ν i (f k i (f l x i )) = n i. (47) On the other hand, for τ l < τ + τ, and s D(l), f µ s i is supported on the single element f k i (f l x i ) of B k i which gives In addition, we have H f s µ i (α m f k i (f l x i )) = H f s µ i (α m B k i ). (48) and ν i,f k i(f l x i ) = d(l) s D(l) f s µ i (49) ν i (f k i (f l x i )) = d(l) n i. (50) From concavity of s s log s, we get, for τ l < τ + τ, H νi,f k i (f l xi ) (αm f k i (f l x i )) d(l) Applying the above statements gives s D(l) H f s µ i (α m f k i (f l x i )). (5) H νi (α m B k i ) = H νi (α m ξ) = C ξ H νi,c (α m C)ν i (C)

33 February, = H νi,f k (αm i (f l xi f k i (f l x ) i ))ν i (f k i (f l x i )) 0 l<τ + H νi,f k (αm i (f l xi f k i (f l x ) i ))ν i (f k i (f l x i )) τ l<τ +τ H n f l µ i (α m f k i (f l x i )) i 0 l<τ + H f s n µ i (α m f k i (f l x i )) i τ l<τ +τ s D(l) = H n f l µ i (α m B k i ) i 0 l<τ = n i n i + n i τ l<τ +τ l=0 s D(l) H f l µ i (α m B k i ) H f s µ i (α m B k i ) completing the proofs of (37), Lemma 6., and Theorem 2.6. References [] R. Bowen. Entropy-expansive maps. Trans. A.M.S, 64: , 972. [2] M. Denker, C. Grillenberger, and K. Sigmund. Ergodic Theory on Compact Spaces, volume 527 of Lecture Notes in Mathematics. Springer Verlag, 976. [3] D. Fiebig, F. Fiebig, and Z. Nitecki. Entropy and preimage sets. preprint, [4] M. Hurley. On topological entropy of maps. Jour. Ergodic Theory and Dyn. Sys., 5: , 995. [5] Ya.G. Sinai I.P. Cornfeld, S.V. Fomin. Ergodic Theory. Springer-Verlag, 982. [6] R. Langevin and F.. Przytycki. Entropie de l image inverse d une application. Bull. Soc. Math. France, 20: , 99.

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