Lowering topological entropy over subsets revisited

Let $(X, T)$ be a topological dynamical system. Denote by $h (T, K)$ and $h^B (T, K)$ the covering entropy and dimensional entropy of $K\subseteq X$, respectively. $(X, T)$ is called D-{\it lowerable} (resp. {\it lowerable}) if for each $0\le h\le h (T, X)$ there is a subset (resp. closed subset) $K_h$ with $h^B (T, K_h)= h$ (resp. $h (T, K_h)= h$); is called D-{\it hereditarily lowerable} (resp. {\it hereditarily lowerable}) if each Souslin subset (resp. closed subset) is D-lowerable (resp. lowerable). In this paper it is proved that each topological dynamical system is not only lowerable but also D-lowerable, and each asymptotically $h$-expansive system is D-hereditarily lowerable. A minimal system which is lowerable and not hereditarily lowerable is demonstrated.


Introduction
This paper is a continuation of the research done in [17] by the same authors. Throughout the paper, by a topological dynamical system (t.d.s.) (X, T ) we mean a compact metric space X together with a homeomorphism T : X → X. Let (X, T ) be a t.d.s. and K ⊆ X. Denote by h(T, K) and h B (T, K) the covering entropy and dimensional entropy of K ⊆ X introduced in [2] and [4] respectively. Motivated by [24,25,26,36] in [17] the authors studied the question if for each 0 ≤ h ≤ h(T, X) there is a closed subset of X with entropy h. Inspired by [40,Remark 5.13], in [17] we call (X, T ) (1) lowerable if for each 0 ≤ h ≤ h(T, X) there is a closed K ⊆ X with h(T, K) = h; (2) hereditarily lowerable if each closed subset is lowerable, i.e. for each closed K ⊆ X and any 0 ≤ h ≤ h(T, K) there is a closed K h ⊆ K with h(T, K) = h; (3) hereditarily uniformly lowerable if for each closed subset K ⊆ X and any 0 ≤ h ≤ h(T, K) there is a closed K h ⊆ K such that h(T, K h ) = h and K h has at most one limit point. Then the question is divided further into the following questions in [17]: We remark that the reason we ask Question 3 in such a way is that in [40] the authors showed that if (X, T ) is a t.d.s. and K ⊂ X is a compact infinite subset, then there is a countable subset K ′ ⊂ K (the derived set of which has at most one limit point) with h(T, K ′ ) = h(T, K). In [17] the authors showed that each t.d.s. with finite entropy is lowerable, and that a t.d.s. is hereditarily uniformly lowerable iff it is asymptotically h-expansive. In particular, each hereditarily uniformly lowerable t.d.s. has finite entropy. Moreover, a principal extension preserves the lowerable, hereditarily lowerable and hereditarily uniformly lowerable properties. Though we completely answered Question 3, Question 1 in the case h(T, X) = +∞ and Question 2 still remain open in [17].
Let X be a metric space, the Souslin sets are the sets of the form where E i 1 ,··· ,i k is a closed set for each finite sequence {i 1 , · · · , i k } of positive integers. Observe that each Borel set is Souslin, the pre-image of a Souslin set under a continuous map is Souslin, and if the underlying metric spaces are complete then any continuous image of a Souslin set is Souslin. The well-known result in fractal geometry [10,27] states that (for the definition of Hausdorff dimension see [10,27])  We emphasize that, in fact, [17,Theorem 4.4] tells us that each t.d.s. with finite entropy is D-lowerable.
In this paper, we get complete answers to Questions 1 and 4; and partial answers to Questions 2 and 5 (Question 3 was answered by [17,Theorem 7.7]). Namely, with the help of a relative version of the well-known Sinai Theorem we prove that each t.d.s. is not only lowerable but also D-lowerable. We shall construct a minimal lowerable t.d.s. which is not hereditarily lowerable. Moreover, we also prove that each asymptotically h-expansive t.d.s. is D-hereditarily lowerable. Whereas, there remain some interesting questions unsolved. For example, is there a lowerable t.d.s. with finite entropy which is not hereditarily lowerable?
The paper is organized as follows. In Section 2 the definitions of cover entropy and dimensional entropy of subsets are introduced. In Section 3, a minimal lowerable t.d.s. which is not hereditarily lowerable is presented. Then in section 4 it is proved that each t.d.s. is not only lowerable but also D-lowerable with the help of a relative version of the well-known Sinai Theorem. In the last section, it is shown that each asymptotically h-expansive t.d.s. is D-hereditarily lowerable.
Acknowledgement: We would like to thank Downarowicz, Glasner and Weiss for useful discussions. We also would like to thank the referee for the careful reading and useful comments that resulted in substantial improvements to this paper.

Preliminary
Let (X, T ) be a t.d.s., K ⊆ X and W a collection of subsets of X. We shall write K W if K ⊆ W for some W ∈ W and else K W. If W 1 is another family of subsets of X, W is said to be finer than W 1 (we shall write W W 1 ) when W W 1 for each W ∈ W. We shall say that a numerical function increases (resp. decreases) with respect to (w.r.t.) a set variable K or a family variable W if the value never decreases (resp. increases) when K is replaced by a set K 1 with K 1 ⊆ K or when W is replaced by a family W 1 with W 1 W.
By a cover of X we mean a finite family of Borel subsets with union X, and a partition a cover whose elements are disjoint. Denote by C X (resp. C o X , P X ) the set of covers (resp. open covers, partitions). If α ∈ P X and x ∈ X then let α(x) be the element of α containing x. Given in the sense that each refines the other. For each U ∈ C X and any m, n ∈ Z + with m ≤ n we set U n m = n i=m T −i U. Moreover, if (X, T ) is a t.d.s. then let diam(K) be the diameter of K and put ||W|| = sup{diam(W ) : W ∈ W}, thus if U ∈ C o X then U has a Lebesgue number λ > 0 and so W U when ||W|| < λ.

2.1.
Covering entropy of subsets. Let (X, T ) be a t.d.s., K ⊆ X and U ∈ C X . Set N(U, K) to be the minimal cardinality of sub-families V ⊆ U with ∪V ⊇ K, where ∪V = V ∈V V . We write N(U, ∅) = 1 by convention. Obviously, lowering topological entropy over subsets (II) Clearly h U (T, K) increases w.r.t. U. Define the covering entropy of K by and define the topological entropy of (X, T ) by h top (T, X) = h(T, X). Let (X, T ) and (Y, S) be t.d.s.s. We say that π : (X, T ) → (Y, S) is a factor map if π is a continuous surjection and π • T = S • π. It is easy to check that is a factor map and K ⊆ X; We may also obtain the cover entropy of subsets using Bowen's separated and spanning sets (see [38, P 168−174 ]). Let (X, T ) be a t.d.s. with d a compatible metric on X. For each n ∈ N we define a new metric d n on X by Let r n (d, T, ǫ, K) denote the smallest cardinality of any (n, ǫ)-spanning set for K w.r.t. T and s n (d, T, ǫ, K) denote the largest cardinality of any (n, ǫ)-separated subset of K w.r.t. T . We write r n (d, T, ǫ, ∅) = s n (d, T, ǫ, ∅) = 1 by convention. Put  It is well known that h * (d, T, K) = h * (d, T, K) is independent of the choice of a compatible metric d on the space X. Now, if U ∈ C o X has a Lebesgue number δ > 0 then, for any δ ′ ∈ (0, δ 2 ) and each V ∈ C o X with ||V|| ≤ δ ′ , one has N(U n−1 In this case, it is obvious that h(T, K) = h(T, K).

2.2.
Dimensional entropy of subsets. Now we recall the concept of dimensional entropy introduced and studied in [4].
Let (X, T ) be a t.d.s. and U ∈ C X . For K ⊆ X let For k ∈ N, we define C(T, U, K, k) to be the family of all E, where E is a countable family of subsets of X such that K ⊆ ∪E and E U k−1 0 . Then for each λ ∈ R set here, by convention: 0 · ∞ = 0 and m T, Notice that m T,U (K, λ) ≤ m T,U (K, λ ′ ) for λ ≥ λ ′ and m T,U (K, λ) / ∈ {0, +∞} for at most one λ [4]. We define the dimensional entropy of K relative to U by Note that h B U (T, K) increases w.r.t. U ∈ C X , thus if (X, T ) is a t.d.s. and {U n } n∈N ⊆ C o X satisfies ||U n || → 0 as n → +∞ then lim n→+∞ h B Un (T, K) = h B (T, K). The following result is basic (see [4,Propositions 1 and 2] or [17, Proposition 2.3]). Proposition 2.2. Let (X, T ) be a t.d.s., K 1 , K 2 , · · · , K ⊆ X and U ∈ C X . Then Thus, by Proposition 2.2 (2), h B (T, E) increases w.r.t. E ⊆ X and if E ⊆ X is a non-empty countable set then h B (T, E) = 0. It is worth mentioning that (1) h B U (T, ∅) = h U (T, ∅) = 0 for any U ∈ C X , and so h B (T, ∅) = h(T, ∅) = 0; (2) when ∅ = K ⊆ X, one has h U (T, K) ≥ h B U (T, K) ≥ 0 for any U ∈ C X , and so h(T, K) ≥ h B (T, K) ≥ 0.

2.3.
Hausdorff dimension and dimensional entropy. Let (X, d) be a metric space. We first recall the definition of Hausdorff dimension of a subset A ⊂ X. Fix t ≥ 0. For each δ > 0 and subset A ⊂ X, we define where the infimum is taken over all countable covers {U i : i = 1, 2, · · · } of A of diameter not exceeding δ. Since H t,δ d (A) increases as δ decreases for any A ⊆ X, we can define The Hausdorff dimension is a monotone function of sets, i.e. if A ⊆ B then In the following we investigate the interrelation of Hausdorff dimension and dimensional entropy of a set in some specific t.d.s. Let (X, T ) be a t.d.s. with metric d. We assume that T is Lipschitz continuous with the Lipschitz constant L, i.e. d(T x, T y) ≤ Ld(x, y) for any x, y ∈ X.
The following result is just [ The following result is [11,Lemma 5.4].
for any subset C ⊆ X.
Proposition 2.5. Let T = R/Z be the unit circle of complex plane with the metric for any subset C ⊆ T.  Remark 2.7. Recall that in [40,Remark 5.13] the authors showed that if (X, T ) is a t.d.s. and K ⊂ X is a compact subset, then there is a countable subset K ′ ⊂ K (the derived set of which has at most one limit point) with h(T, K ′ ) = h(T, K). Weiss showed us a proof that when (X, T ) is minimal and X is infinite then there exists a countable subset K with a unique limit point such that h(T, K) = h(T, X). In fact, this can also be obtained by [40, Theorems 4.2 and 5.7].

Negative answers to Question 2
In this section, we shall construct a minimal lowerable t.d.s. which is not hereditarily lowerable. First we give a lowerable t.d.s. which is not hereditarily lowerable and then we make it minimal. We remark that the example we get has infinite entropy, and it is not hard to construct an example which has infinite entropy and at the same time is hereditarily lowerable.
3.1. A general example. First we construct an example (not necessarily minimal) which is lowerable and not hereditarily lowerable. In the next subsection we will modify it such that it is minimal. To do this, we need the following lemma.
(2) Let K ⊆ Y be closed and j ∈ N. By Proposition 2.1 (1) clearly h(T j , π j K) ≤ h(S, K). For the other direction, without loss of generality we assume diam(X i ) ≤ 1 with d i a compatible metric on X i for each i ∈ N. Let ρ be the metric on Y given by . This finishes our proof.
Thus, we have is not lowerable (and so t.d.s. (X ∞ , S) is not hereditarily lowerable) by proving that each closed subset K of E has either infinite topological entropy or zero topological entropy. Let π i : (X ∞ , S) → (X, T i ) be the factor map of the i-th projection map, i ∈ N. We shall prove that if h(T, π 1 K) > 0 then h(S, K) = ∞ and if h(T, π 1 K) = 0 then h(S, K) = 0. In fact, if h(T, π 1 K) = 0 then by Lemma 3.1 one has Now we shall finish our proof by claiming that (X ∞ , S) is lowerable. In fact, for each 0 ≤ h < ∞ we let n ∈ N with h(S n , X n ) > h, where S n = T ×T 2 ×· · ·×T n . Note that [17,Theorem 4.4] states that each t.d.s. with finite entropy must be lowerable, whereas, clearly (X n , S n ) is a t.d.s. with finite entropy, thus there exists closed K h ⊆  [40,Theorem 5.9 and Remark 5.13] state that Let (X, T ) be a t.d.s. Then for any compact K ⊆ X there is a countable compact subset K ′ ⊆ K with h(T, K ′ ) = h(T, K). Thus, the above Proposition 3.2 tells us that these are the best results we may obtain. In view of this, we restate our Question 2 as Question 2': Is there a t.d.s. with finite entropy which is not hereditarily lowerable?
It seems to us that a t.d.s. (X, T ) is not hereditarily lowerable if it has an ergodic measure with infinite entropy.

3.2.
A minimal example. After we finished the construction in the previous subsection, Glasner asked if there is a minimal t.d.s. with the property. We will show that the example in the previous subsection can be made minimal. Recall that a t.d.s. (Y, T ) is called proximal orbit dense, or a POD system if (Y, T ) is totally minimal and whenever x, y ∈ Y with x = y, then for some n = 0, T n y is proximal to x. An interesting property of POD is that (see [21,Corollary 3.5
(2) The only factors of the flow in (1) are the obvious direct factors. If, in addition, (X, T ) is not isomorphic to (X, T −1 ) then (1) and (2) hold for any k i = 0, i = 1, · · · , n with k i = k j for i = j.
A special class of POD is Definition 3.4. A system (X, T ) is said to be doubly minimal if for all x ∈ X and y ∈ {T n x} ∞ −∞ , the orbit of (x, y) is dense in X × X. The first example of non-periodic doubly minimal system was constructed in [18] in the symbolic dynamics. Doubly minimal systems are natural in the sense that: any ergodic system with zero entropy has a uniquely ergodic model which is doubly minimal [39]. The notion of disjointness between two t.d.s. was introduced in [12] and it is easy to see that two minimal t.d.s. are disjoint iff the product system is minimal [1, Proposition 2.5].
Proposition 3.5. There is a minimal t.d.s. which is not hereditarily lowerable.
Proof. Let (Y 1 , S 1 ) be the non-periodic double minimal system constructed in [18], in particular, (Y 1 , S 1 ) is a strictly ergodic t.d.s. with finite entropy. Now let (Y, S) be a minimal t.d.s. with finite positive entropy, which is an extension of (Y 1 , S 1 ) with a factor map π satisfying that {y 1 ∈ Y 1 : π −1 (y 1 ) is a singleton} is a residual subset of Y 1 (see [8,Theorem 3] for the existence of such a t.d.s. (Y, S), as (Y 1 , S 1 ) is a strictly ergodic non-periodic system with finite entropy). Since both (Y, S) and (Y 1 , S 1 ) are minimal, it is well known that the factor map π is almost 1-1 in the sense that the subset {y ∈ Y : π −1 (πy) is a singleton} ⊂ Y is also residual.
Observe that each non-periodic doubly minimal system is not only minimal but also weakly mixing, and so totally minimal. In fact, let (X, T ) be a minimal weakly mixing t.d.s. and m ∈ N, it is well known that the system (X, T m ) is weakly mixing and each point of (X, T m ) is minimal, hence (X, T m ) is a minimal t.d.s.
, then (x, S n 1 y) has a dense orbit and hence is also proximal for any n ∈ Z. Thus (Y 1 , S 1 ), as a non-periodic doubly minimal system, is a POD system. For a given n ∈ N, since ( is minimal, as minimality is preserved by the inverse limit. Then by Proposition 3.2, we get the conclusion.

3.3.
A related result. Finally, we shall present a result related to the property of hereditary lowering. First, let's make some preparations (for details see [7,17,29,31]).
Let π : (X, T ) → (Y, S) be a factor map between t.d.s.s. The relative topological entropy of (X, T ) w.r.t. π is defined as follows: ( Let (X, T ) be a t.d.s. and U 1 , U 2 ∈ C o X . Put N(U 1 |U 2 ) = max{N(U 1 , U 2 ) : U 2 ∈ U 2 }. Then, for each m, n ∈ N, ) : n ∈ N} is sub-additive, and so we may set Define the topologically conditional U 2 -entropy of (X, T ) by and the topologically conditional entropy of (X, T ) by In particular, h * (T, X) ≤ h top (T, X). Observe that, if (X, T ) is zero-dimensional then by a standard construction we can represent (X, T ) as an inverse limit of subshifts over finite alphabets: r with Λ r a finite discrete space, T r is the full shift over Λ Z r and (X r , T r ) is a factor of (X r+1 , T r+1 ) for each r ∈ N. Now let φ r : (X, T ) → (X r , T r ) be the natural homomorphism and U r the clopen generated partition of X r (r ∈ N). Observe that the sequence {h top (T, X|φ r ) : r ∈ N} decreases, there are some easy but useful facts: (1) h top (T r , X r ) = h φ −1 r (Ur) (T, X) and h top (T, X) = lim r→+∞ h top (T r , X r ); (2) h top (T, X|φ r ) ≤ h * (T, X|φ −1 r (U r )) and (3) h * (T, X) = lim r→+∞ h * (T, X|φ −1 r (U r )) ≥ lim r→+∞ h top (T, X|φ r ). Thus we have the following interesting result.
Proof. It's well-known that each t.d.s. with finite entropy has a zero-dimensional principal extension [5, Proposition 7.8], i.e. an extension preserving entropy for each invariant measure; and if π is a principal extension of a system with finite entropy, then π preserves the topologically conditional entropy [22, Theorem 3] and has zero relative topological entropy by conditional variational principles [9, Theorems 3 and 4]. Thus, using Proposition 3.7 (1) we may assume that (X, T ) is zero-dimensional.
We represent (X, T ) by an inverse limit of sub-shifts over finite alphabets (X, T ) = lim ← − (X r , T r ), with φ r : (X, T ) → (X r , T r ) the natural homomorphism for each r ∈ N. Then (3)).
For each r ∈ N we set K r = φ r (K), so h(T, K)−h top (T, X|φ r ) ≤ h(T r , K r ) ≤ h(T, K) (using Proposition 3.7 (1) again). Thus if r ∈ N is large enough then there exists compact K h r ⊆ K r with h(T r , K h r ) = h (using Theorem [17,Theorem 5.4]). Last, put K . We can claim the conclusion by fact (3).

A positive answer to Questions 1 and 4
In this section we shall give a positive answer to Questions 1 and 4. Let (X, T ) be a t.d.s. Denote by M(X) (resp. M(X, T ), M e (X, T )) the set of all Borel probability measures (resp. T -invariant Borel probability measures, ergodic T -invariant Borel probability measures) on X. All of them are equipped with the weak star topology. Denote by B X the set of all Borel subsets of X.
We also need state a relative version of the well-known Sinai Theorem, which is essentially found in [32]. It was made explicit in [33,Theorem 5] and [37] (for another treatment of it see [20]). Before stating it, we need make some preparations.
A relative version of the well-known Sinai Theorem is stated as follows.
the sense of µ) for each k ∈ N, and so C is independent of +∞ i=−∞ T −i α in the sense of µ. Finally, we claim that C is just the sub-σ-algebra we need. Obviously, T −1 C = C. Now we are going to show h µ (T, X|C) = h. On one hand, where the last identity follows from the fact that C is independent of +∞ i=−∞ T −i α. On the other hand, for each n ∈ N by the relative Pinsker formula

By (4.3)
Proof. When h = h(T, X), we may take K h = X. When h < h(T, X), by the classical variational principle (see for example [ Remark 4.5. We should remark that in [36] Shub and Weiss presented a t.d.s. with infinite entropy such that its each non-trivial factor has infinite entropy.

A partial answer to Question 5
In this section, we shall give a partial answer to Question 5 by proving that each asymptotically h-expansive (equivalently, hereditarily uniformly lowerable) t.d.s. is D-hereditarily lowerable.
Lemma 5.1. Let (X, T ) be a symbolic t.d.s. Then (X, T ) is not only hereditarily lowerable but also D-hereditarily lowerable.
In fact, we can construct a D-hereditarily lowerable t.d.s. which is not asymptotically h-expansive but quasi-asymptotically h-expansive.
Example 5.5. There is a D-hereditarily lowerable t.d.s. which is not asymptotically h-expansive but quasi-asymptotically h-expansive.
n } × C n ⊆ R 2 for each n ∈ N. Now for each n ∈ N we let T n : X n → X n be a minimal subshift such that h(T n , Φ ǫn (x n )) ≥ log 2 for some x n ∈ X n (we may assume that lim n→+∞ x n = x 0 ) and let T 0 : X 0 → X 0 be the identity map. Last, (X, T ) is defined naturally. We may add the assumptions on the defined (X n , T n ), n ∈ N such that (X, T ) forms a t.d.s.
Remark 5.6. In the above example we can choose (C 0 n , S n ), n ∈ N such that h top (T, X) = +∞, and so one has that there is a D-hereditarily lowerable t.d.s. (with entropy infinite) which is not anti-asymptotically h-expansive but quasi-asymptotically hexpansive. We do not know if there is such an example which is minimal.

5.2.
Each anti-asymptotically h-expansive t.d.s. is asymptotically h-expansive. In this subsection we will show that asymptotical h-expansiveness and anti-asymptotical h-expansiveness are equivalent properties. For that, we need some notions and results in [5].
Given a t.d.s. (X, T ), we will say a sequence of partitions {α k } of X is refining if the maximum diameter of elements of α k goes to zero with k; and for each k the partition α k+1 refines α k . The partitions have small boundaries if their boundaries have measure zero for all µ in M(X, T ). For a finite entropy t.d.s. (X, T ) admitting a nonperiodic minimal factor, by [25,Theorem 6.2] and [26,Theorem 4.2] we know that (X, T ) has the so called small boundary property, which is equivalent to the existence of a basis of the topology consisting of sets whose boundaries have measure zero for every invariant measure. Moreover, it is easy to construct the refining sequence of partitions with small boundaries for (X, T ) (see [5, Theorem 7.6 (3)]).
Definition 5.7. Let (X, T ) be a finite entropy t.d.s. admitting a nonperiodic minimal factor. An entropy structure for (X, T ) is a sequence H of functions {h k } defined on M(X, T ) in the following way: suppose {α k } is a refining sequence of finite Borel partitions with small boundaries, then h k : M(X, T ) → R is obtained by setting µ → h µ (T, α k ) for each k ∈ N.
Proposition 5.8. Let (X, T ) be a finite entropy t.d.s. admitting a nonperiodic minimal factor. The following statements are equivalent for (X, T ) with entropy structure H: (1) (X, T ) is asymptotically h-expansive.
Definition 5.9. A function f : K → R defined on a compact metric space K is upper semicontinuous (u.s.c.) if one of the following equivalent conditions holds: (1) f = inf i∈I f i for some family {f i } i∈I of continuous functions.
(2) f = lim i→+∞ g i , where {g i } is a nonincreasing sequence of continuous functions.
For u.s.c functions, the following properties hold: a) The infimum of any family of u.s.c. functions is u.s.c. (by Definition 5.9 (1)). b) Both the sum and the supremum of finitely many u.s.c. functions are u.s.c.
(by Definition 5.9 (2)). c) Every u.s.c. function from a compact metric space to R is bounded above and attains its maximum.
The following result is [5, Proposition 2.4]. Proof. Remark that each asymptotically h-expansive t.d.s. admits a principal extension to a symbolic t.d.s. which has zero relative topological entropy (cf the proof of Theorem 5.2). Thus an asymptotically h-expansive t.d.s. is clearly antiasymptotically h-expansive by definitions. Conversely, let (X, T ) be an anti-asymptotically h-expansive t.d.s. Suppose (Z, R) is an aperiodic minimal zero entropy system. Let Y = X × Z and S = T × R. Then (Y, S) is an anti-asymptotically h-expansive t.d.s., since (Z, R) is asymptotically h-expansive. Now (Y, S) is a finite entropy t.d.s. admitting a nonperiodic minimal factor (Z, R). Suppose {α k } is a refining sequence of finite Borel partitions of (Y, S) with small boundaries. Define H by setting h k : µ → h µ (S, α k ) for each k ∈ N. Then H is an entropy structure of (Y, S).
For m ∈ N, let π m : (X m , T m ) → (Y, S) be a factor map such that (X m , T m ) is a symbolic t.d.s. and h top (T m , X m |π m ) < 1 m . Then π m induces a continuous map π * m : M(X m , T m ) → M(Y, S) satisfying π * m ν(A) = ν(π −1 m A) for any ν ∈ M(X m , T m ) and any Borel subset A of Y .
Let β m be a generating clopen partition of (X m , T m ). Now we consider the function g m k : M(X m , T m ) → R with g m k (ν) = h ν (T m , X m ) − h π * m ν (S, α k ) for each ν ∈ M(X m , T m ). Then for k ∈ N, The last equality follows from the fact that the sequence a n (ν) := H ν ( is subadditive, i.e. a n 1 +n 2 (ν) ≤ a n 1 (ν) + a n 2 (ν). Since β m and π −1 m α k have small boundaries, is a continuous function on M(X m , T m ) for each N ∈ N. Thus the function g m k is u.s.c. by Definition 5.9 (1).