AN URYSOHN-TYPE THEOREM UNDER A DYNAMICAL CONSTRAINT

We address the following question raised by M. Entov and L. Polterovich: given a continuous map $f:X\to X$ of a metric space $X$, closed subsets $A,B\subset X$, and an integer $n\geq 1$, when is it possible to find a continuous function $\theta:X\to\mathbb{R}$ such that 
 \[ \theta f-\theta\leq 1, \quad \theta|A\leq 0, \quad\text{and}\quad \theta|B> n\,? \] 
 To keep things as simple as possible, we solve the problem when $A$ is compact. The non-compact case will be treated in a later work.


INTRODUCTION
We will consider a continuous map f : X → X of a metric space X . Suppose that A, B ⊂ X are closed subsets, with A compact, and n ≥ 1 is an integer. In December 2014, M. Entov and L. Polterovich asked the author when is it possible to find a continuous function θ : X → R such that θ f − θ ≤ 1, θ|A ≤ 0, and θ|B > n.
They observed that the condition B ∩ n i =0 f i (A) = is obviously necessary. They needed the converse for their work [2]. This is the content of the following theorem. THEOREM 1.1 (Discrete case). Suppose f : X → X is a continuous map defined on a metric space X . Suppose that A, B ⊂ X are closed subsets, with A compact, and also that B ∩ n i =0 f i (A) = , where n ≥ 0. Then we can find a Lipschitz function θ : X → [0, n + 1] such that θ f − θ ≤ 1 everywhere, the function θ is identically 0 on a neighborhood of A, and θ is identically n + 1 on a neighborhood of B .
If X is locally compact metric and σ-compact, the map f : X → X is proper, and B is also compact, then we can further assume that θ has compact support. COROLLARY 1.2. Under the hypothesis of Theorem 1.1, if we further assume that X is a smooth manifold, then for every > 0, we can find a C ∞ function θ : X → [0, n + 1] which is identically 0 in a neighborhood of A, identically n + 1 in a neighborhood of B , and satisfies θ f − θ ≤ 1 + . Moreover, if B is also compact, we can assume that the smooth function θ also has a compact support.
In fact, in the paper [2], M. Entov and L. Polterovich were interested in the flow version of the theorem above, which we will obtain from the discrete case. We will only consider the case of autonomous flows, i.e., families of maps φ t : We can find a continuous function θ : X → [0, ∞), with θ identically 0 on a neighborhood of A, and θ identically constant and > T on a neighborhood of B , such that the map t → θφ t (x) has a continuous derivative in t for every x ∈ X with If X is locally compact metric and σ-compact, and B is also compact, then we can further assume that θ has compact support. 3, if we further assume that X is a smooth manifold and the flow φ t is C 1 , generated by the vector field V , then we can find such a function θ which is moreover C ∞ and satisfies V θ ≤ 1, where V θ is the derivative of θ in the direction of V .
The proof of these last two corollaries is deduced from the discrete case with the same idea that L. Buhovsky, M. Entov and L. Polterovich used to prove the corollary under the stronger hypothesis B ∩ T t =−T φ t (A) = , see [1].

REMARK 1.5. When
A is not compact, a slightly modified version of the results above remain true. To keep things simple, we postpone the statements and proofs for this non-compact case to another paper.

THE DISCRETE CASE
We will use the method from our previous work with Pierre Pageault, see [3]. Let us fix a metric defining the topology on X . For every k > 0, we define the cost c k : X × X → R by Recall that a chain in X is a sequence of points (x 0 , . . . , x n ), where n ≥ 1. We define the cost C k (x 0 , . . . , x n ) of the chain (x 0 , . . . , x n ) by As in [3], it is not difficult to obtain the following properties of Γ k . PROPOSITION 2.1. The function Γ k satisfies the following properties: In particular, the function Γ k is continuous, and uniformly Lipschitz in the second variable with Lipschitz constant k.
Proof of Theorem 1.1. For > 0, we recall that for a subset S ⊂ X , its closedneighborhood isV Note that ϕ k |V 1/k (A) ≡ 0. The function ϕ k is k-Lipschitz. We now estimate from below the values of ϕ k onV 1/k (B ). It is not difficult to show that This implies the inequality Choosing y = x, we obtain Therefore Since ϕ k |V 1/k (A) ≡ 0, we do have θ k |V 1/k (A) ≡ 0. We now show that for k large enough, we have θ k |V 1/k (B ) ≥ n + 1.
We argue by contradiction. If we assume that θ k |V 1/k (B ) ≥ n + 1 is not true for k large enough, we can find sequences k → ∞, and z ∈V 1/k (B ), such that θ k (z ) < n + 1.

(2.2)
Therefore, extracting if necessary, we can assume that n = m ≤ n, with m independent of . Since k → ∞, the inequalities (2.2) imply d ( f (y i ), y i +1 ) → 0, as → ∞, for i = 0, . . . , m − 1, and also d (y 0 , A) → 0 as → ∞. In particular, we can find a sequence x ∈ A, such that d (y 0 , x ) → 0. By compactness of A, extracting further if necessary, we can assume x → x ∈ A. Hence y 0 → x ∈ A, and from d ( f (y i ), y i +1 ) → 0, by induction, we obtain y i +1 → f i +1 (x), for i = 0, . . . , m − 1. Since B is closed, and y m = y n = z ∈V 1/k (B ), we get f m (x) = lim →∞ y m ∈ B . But x ∈ A and m ≤ n. This contradicts the hypothesis B ∩ n i =0 f i (A) = . Obviously, for k large, the function θ = min{θ k , n + 1} satisfies all the conditions in the first part of the theorem.
To prove the second part, we consider the Alexandrov (one point) compact-ificationX = X ∪ {∞}. Since X is locally compact, metric, and σ-compact, the compactificationX is compact and metric. 1 Since the map f : X → X is proper, it can be extended continuously toX , with f (∞) = ∞. If we setÃ = A ∪ {∞}, then bothÃ and B are compact subsets ofX , and they satisfy By the first part of the theorem, we can find a continuous function θ :X → [0, n + 1] which is equal to 0 in a neighborhood ofÃ, equal to n +1 in a neighborhood of B , and satisfies θ f − θ ≤ 1. Since ∞ ∈Ã, this implies that the restriction θ|X has compact support.
Proof of Corollary 1.2. This is a simple approximation argument. We pick up the continuous function θ : X → [0, n + 1] given by Theorem 1.1. For any given > 0, we can find a smooth function θ : X → [0, n + 1] such that θ − θ ∞ ≤ /2. Using θ f − θ ≤ 1, we obtain θ f − θ ≤ 1 + . Since θ ≡ 0 (resp. θ ≡ n + 1) in a neighborhood of the closed set A (resp. B ), choosing θ carefully, we can assume that θ ≡ 0 in a neighborhood of A, and θ ≡ n + 1 in a neighborhood of B . By a similar careful choice of θ , we can assume that θ has compact support if θ has itself compact support.

THE FLOW CASE
In this section, we prove Corollary 1.3. The proof of this corollary is deduced from the discrete case using the well-known way to smooth a continuous function of one variable by averaging on an interval. In [1], L. Buhovsky, M. Entov and L. Polterovich used the same method to prove Corollary 1.2 under the stronger hypothesis B ∩ T t =−T φ t (A) = . The easiest way to smooth a function θ along the orbits of a flow φ t is to consider the function where > 0 is fixed. The following computation gives the derivative at s = t of s → θ (φ s (x)): Therefore if we let h → 0, we obtain that the derivative in t of t → θ (φ t (x)) exists and is given by is open, the continuity of the flow φ t and the compactness of A allow us to find α > 0 such that We can find an integer N ≥ 1 and ∈ [0, α] such that where the last equality follows from (3.4). Therefore, in view of (3.3), we obtaiñ (3.5) By Theorem 1.1, we can find a continuous function θ : X → [0, N + 1] and neighborhoods VÃ and VB ofÃ andB , respectively, such that We now introduce the function θ defined as above by From (3.1), we know that θ(φ t (x)) is differentiable in t , and the derivative is given by Since θφ − θ ≤ 1 by (3.6), we conclude that for all x ∈ X and all t ∈ R. We now show that θ is identically 0 (resp. T +α) in a neighborhood of A (resp. B ). Since t =0 φ t (A) =Ã (resp. t =0 φ t (B ) ⊂B ) and VÃ (resp. VB ) is a neighborhood ofÃ (resp.B ), using compactness of [0, ], we can find a neighborhood V A (resp. V B ) of A (resp. B ) such that By the choice of θ and the definition of θ , we obtain that θ ≡ 0 on V A and θ ≡ (N + 1) = T + α on V B . This finishes the proof that θ is the function θ we are looking for in the first part of the corollary.
It remains to prove the last statement about compact support. Using the assumptions of the last part of the corollary, we note that, by the last part of Theorem 1.1, we can further choose θ whose support Supp(θ) is compact. Obviously, from its definition, the function θ is identically 0 outside of the compact Proof of Corollary 1. 4. We indicate here the changes that have to be done in the proof of Corollary 1.3. The choices of α, , N , f = φ ,Ã, andB are the same. Given δ > 0, instead of applying Theorem 1.1, we apply Corollary 1.2, to obtain a C ∞ function θ δ : X → [0, N + 1], and neighborhoods VÃ and VB ofÃ andB , respectively, such that θ δ |VÃ ≡ 0, θ δ |VB ≡ N + 1, and θ δ f − θ δ = θ δ φ − θ δ ≤ 1 + δ. Since θ δ is C ∞ , and φ t is C 1 , the function θ δ is also C 1 . This implies Moreover, like in the proof above of Corollary 1.3, we can show that the function θ δ takes values in [0, T + α] and is identically 0 (resp. T + α) in a neighborhood of A (resp. B ). Since θ δ is C 1 , we can approximate it, in the fine Whitney C 1 topology, by a C ∞ functionθ δ : X → [0, T + α], which is still identically 0 (resp. T + α) in a neighborhood of A (resp. B ) and satisfies |Vθ δ −V θ δ | ≤ δ everywhere. Therefore Vθ δ ≤ 1 + 2δ. For δ > 0 small enough, we still have (1 + 2δ) −1 (T + α) > T , and therefore the function θ = (1 + 2δ) −1θδ indeed satisfies the conclusion of the corollary.