aa r X i v : . [ m a t h . D S ] J a n NONSTANDARD EXPANSIVENESS
LUIS FERRARI
Abstract.
Let (
X, d ) be a metric space and f : X → X be a homeomor-phism. We say that a dynamical system ( X, f ) is expansive , with constantof expansivity c ∈ R + , if for all x, y ∈ X , x = y , exists n ∈ Z , such that d ( f n ( x ) , f n ( y )) > c . In this paper we will use the theory of NonstandardAnalysis to study a subfamily of these dynamics, which verify that for all x, y ∈ X , if x = y then the set { n ∈ Z : d ( f n ( x ) , f n ( y ) > c } is infinite. Introduction
Expansive dynamics are an important type of dynamical system. Furthermore,these dynamics have interesting connections with topology and other dynamicalconcepts. An example of this can be seen in the relationship between the topologyof the space and the asymptotic behavior of the dynamics. We know by [9] thatevery expansive homeomorphism on an infinite compact metric space has asymp-totic points; that is, exists x = y such that lim n → + ∞ d ( f n ( x ) , f n ( y )) = 0 (positiveasymptotics) or negative asymptotics ( n → −∞ ). Dynamic systems without doublyasympotic points(positive and negative asymptotic) are a peculiar type of dynamicsystem. All of the examples that we know of expansive homeomorphisms on com-pact metric spaces that are not totally disconnected have asymptotic points. By[6], we know that there are infinite dynamics of this type such that they do notconjugate with each other but in the form of a subshift, and therefore on a totallydisconnected space.Groisman and da Silva [5] introduce the notion of freely expansive dynamic . Let X be a compact metric space and a f : X → X homeomorphism, then the dy-namical system ( X, f ) is freely expansive if there is some c > x, y ∈ X there is some free ultrafilter p over N such that either d ( f p ( x ) , f p ( y )) > c or d ( f − p ( x ) , f − p ( y )) > c , where f p is the p -iterate of f for thefuture, and f − p is the p -iterate of f for the past, the existence of p -iterate is guar-anteed by the compactness of the space. In their notes, the authors demonstratethat these dynamics are equivalent of dynamical systems such that for all different x, y ∈ X the set { n ∈ Z : d ( f n ( x ) , f n ( y ) > c } is infinite when X is compact. An-other important fact is that freely expansive homeomorphism are equivalent to anexpansive homeomorphism without doubly asymptotic points. Why use nonstandard analysis?
Suppose that (
X, f ) is a dynamical system such that for all x, y ∈ X , x = y theset { n ∈ Z : d ( f n ( x ) , f n ( y ) > c } is infinite, and suppose without lost of generalitythat is in a succession of positive integers ( λ n ) n ∈ Z + ; that is, Date : January 5, 2021.2020
Mathematics Subject Classification.
Key words and phrases.
Expansive homeomorphism, doubly asymptotic points, nonstandardanalysis. d ( f λ ( x ) , f λ ( y )) > c, d ( f λ ( x ) , f λ ( y )) > c, . . . , d ( f λ n ( x ) , f λ n ( y )) > c, . . . .Consider the succession of constant succession ( x ) n ∈ Z + , ( y ) n ∈ Z + ( c ) n ∈ Z + , and thesuccession of the moments of separations ( λ n ) n ∈ Z + . In the construction of thenonstandard extension of a set via a free ultrafilter [2], this succession corresponds(respectively) to the points x , y in ∗ X ; nonstandard extension of X , c in ∗ R ; non-standard extension of R , and λ in ∗ Z ; and nonstandard extension of Z , such that ∗ d ( ∗ f λ ( x ) , ∗ f λ ( y )) > c .Where ∗ d : ∗ X × ∗ X → ∗ R is the nonstandard extension of the metric d , and ∗ f : ∗ X → ∗ X the nonstandard extension of the homeomorphism f . Note that wenever use the compactness of X .We will use nonstandard analysis methods to study these dynamics. We will seethat nonstandard analysis is particularly useful to study these dynamics whereasymptotic behavior is crucial. Furthermore, this approach allows us to introducenew concepts and questions.2. Preliminaries of expansive systems
Definition . Let X be a metric space and f : X → X be a homeomorphism. Wesay that f is expansive if exists c ∈ R + such that for all x, y ∈ X , x = y , exists n ∈ Z , such that d ( f n ( x ) , f n ( y )) > c . Definition . Let X and Y be compact metric spaces and f : X → X , g : Y → Y be two homeomorphisms. We say that the dynamics ( X, f ), (
Y, g ) are conjugated if ϕ : X → Y homeomorphisms exist such that ϕ − .g.ϕ = f . Proposition 2.1.
Let X and Y be compact metric spaces, and ( X, f ) , ( Y, g ) beconjugated dynamics. If f : X → X is expansive, then g is also expansive.Proof. Suppose that f is expansive with constant of expansivity α . We will provethat g = ϕ.f.ϕ − is expansive; that is, will find a c ∈ R , c > a, b ∈ Y , different, d ( g n ( a ) , g n ( b )) < c is verified for all n ∈ Z , then a = b . ϕ − is uniform continuity because X is compact, then c ∈ R , c > d ( y, y ′ ) < c , then d ( ϕ − ( y ) , ϕ − ( y ′ )) < α . Suppose that for a, b ∈ Y we have d ( g n ( a ) , g n ( b )) < c for all n ∈ Z . Then, d ( ϕ − ( g n ( a ) , ϕ − ( g n ( b ))) < α for all n ∈ Z ,but a = ϕ ( x ) y b = ϕ ( x ′ ) for some x, x ′ ∈ X . Then, d ( ϕ − ( g n ( ϕ ( x )) , ϕ − ( g n ( ϕ ( x ′ )))) <α , but ϕ − ( g n ( ϕ ( z )) = f n ( z ) for all z ∈ X . Then, d ( f n ( x ) , f n ( x ′ )) < α for all n ∈ Z . If f is α -expansive, then we have x = x ′ and then a = b . (cid:3) Proposition 2.2.
Let X be a compact metric space and f : X → X be expansivehomeomorphisms; then for all k ∈ N , f k is expansive.Proof. Let c be the constant of expansivity of f , for the uniformly continuous weknow that δ > d ( x, y ) < δ , then d ( f i ( x ) , f i ( y )) < c for all i = 1 , . . . k . Suppose that f k is not expansive, then x , y ∈ X m ∈ Z existssuch that d ( f mk ( x ) , f mk ( y )) < δ . Let n ∈ Z , then for the Euclid theorem ex-ists m, i ∈ Z , with 0 ≤ i < k such that n = mk + i , then d ( f n ( x ) , f n ( y )) = d ( f mk + i ( x ) , f mk + i ( y )) = d ( f i ( f mk ( x )) , f i ( f mk ( y ))) < c , then f is not expan-sive, which is absurd. (cid:3) Uniform expansiveness is a useful concept in the theory of expansive dynamics.
Definition . Let X be a compact metric space and f : X → X be homeomor-phisms. We say that the dynamical system ( X, f ) is c -uniformly expansive if forall ǫ > n ǫ ∈ N such that x, y ∈ X , if d ( x, y ) > ǫ then d ( f i ( x ) , f i ( y )) > c ,for some i ∈ Z , | i | < n ǫ . ONSTANDARD EXPANSIVENESS 3
Proposition 2.3.
Let X be a compact metric space and f : X → X be homeomor-phisms. If ( X, f ) is c -expansive, then f is c -uniformly expansive.Proof. Suppose that f is not c -uniformly expansive, then ǫ > n exists x n , y n ∈ X , such that d ( x n , y n ) > ǫ and d ( f i ( x n ) , f i ( y n )) ≤ c for all i ∈ Z , with | i | < n , then how X is compact. Taking subsuccession ifnecessary, we can suppose that lim n → + ∞ x n = x , lim n → + ∞ y n = y , for some x, y ∈ X . Let k ∈ Z , then how lim n → + ∞ x n = x , we have lim n → + ∞ f k ( x n ) = f k ( x ) y lim n → + ∞ f k ( y n ) = f k ( y ), taken δ > n exists such that n > | k | , y d ( f k ( x ) , f k ( x n )) < δ , d ( f k ( y ) , f k ( y n )) < δ for all n ≥ n , then d ( f k ( x ) , f k ( y )) ≤ d ( f k ( x ) , f k ( x n )) + d ( f k ( x n ) , f k ( y n )) + d ( f k ( y n ) , f k ( y )) < δ + c + δ = c + δ , then d ( f k ( x ) , f k ( y )) < c + δ for all δ >
0, then d ( f k ( x ) , f k ( y )) ≤ c , for all k ∈ Z , then f is not c -expansive. (cid:3) Nonstandard expansiveness
Definition . Let f : X → X be homeomorphisms. We say that f is nonstandardexpansive if c ∈ R , c >
0, exists such that for all x, y ∈ X , in contrast n ∈ ∗ Z ∞ exists such that ∗ d (( ∗ f ) n ( x ) , ( ∗ f ) n ( y )) > c , where ∗ Z ∞ is a set of infinite integersof Z . Theorem 1.
Let f : X → X be homeomorphisms, f is nonstandard expansivewith constant of expansivity c if only for all x, y ∈ X , in contrast the set { n ∈ Z : d ( f n ( x ) , f n ( y )) > c } is infinite.Proof. ⇒ : Suppose that { n ∈ Z : d ( f n ( x ) , f n ( y )) > c } is finite, then exists m ∈ N such that( ∀ n ∈ N )( n ≥ m → d ( f n ( x ) , f n ( y )) ≤ c ). Then, for the transfer principle thefollowing formula is true( ∀ n ∈ ∗ N )( n ≥ m → ∗ d ( ∗ f n ( x ) , ∗ f n ( y )) ≤ c ); in particular, for all positive integer n , ∗ d ( ∗ f n ( x ) , ∗ f n ( y )) ≤ c , Analogously { n ∈ Z ≤ : d ( f n ( x ) , f n ( y )) > c } and change f for f − we have that for all infinite negative integer n d ( f n ( x ) , f n ( y )) ≤ c , thenis not nonstandard expansive. ⇐ : Suppose that { n ∈ N : d ( f n ( x ) , f n ( y )) > c } is finite. If it does not work with f − , then for all n ∈ N exists m > n such that d ( f m ( x ) , f m ( y )) > c ) then there ex-ists a function ψ : N → N such that ∀ n ∈ N (( ψ ( n ) > n ) ∧ d ( f ψ ( n ) ( x ) , f ψ ( n ) ( y )) > c ).Then, for transfer principle ∀ n ∈ ∗ N (( ∗ ψ ( n ) > n ) ∧ d ( f ∗ ψ ( n ) ( x ) , f ∗ ψ ( n ) ( y )) > c ).Then, if m ∈ ∗ N ∞ . Then, ∗ ψ ( m ) ∈ ∗ N ∞ . Consequently, f is nonstandard expan-sive. (cid:3) Remark . ( Strongly nonstandard expansive? ) A natural question is what hap-pens if in the definition of nonstandard expansive instead of quantifying on X we quantify on ∗ X ? That is, we could define f : X → X as c -strongly non-standard expansive if only for all x, y ∈ ∗ X , x = y exists n ∈ ∗ Z ∞ such that ∗ d (( ∗ f ) n ( x ) , ( ∗ f ) n ( y )) > c . It is clear that this stronger a priori version implies thenonstandard expansiveness, we will see that the reciprocal is true.For the above theorem, the following formula is true:( ∀ x ∈ X )( ∀ y ∈ X )( x = y → ( ∀ n ∈ N )( ∃ i ∈ Z )( | i | > n ) ∧ ( d ( f i ( x ) , f i ( y )) > c ).Then, for the transfer principle the formula( ∀ x ∈ ∗ X )( ∀ y ∈ ∗ X )( x = y → ( ∀ n ∈ ∗ N )( ∃ i ∈ ∗ Z )( | i | > n ) ∧ ( ∗ d ( f i ( x ) , f i ( y )) > c )is also true. Then, for any x, y ∈ ∗ X and n ∈ ∗ N ∞ exists i ∈ Z , with | i | > n such ∗ N ∞ is the set of infinite numbers of ∗ N L. FERRARI that d ( f i ( x ) , f i ( y )) > c , but n ∈ ∗ N , and then i ∈ ∗ Z ∞ . This proves that they areseparated into infinite numbers.The next two propositions state that the nonstandard expansive dynamics veri-fied the same properties of invariant by potence and conjugation as the expansivedynamics. These facts were proven by Groisman and da Silva [5] in the context ofthe theory of ultrafilters. We give a proof with nonstandard methods. Proposition 3.1.
Let X be a compact metric space. If f : X → X is nonstandardexpansive, then for all k ∈ N , f k is nonstandard expansive.Proof. Let c be the constant of expansivity of f . By uniform continuity, weknow that δ > d ( x, y ) < δ , then d ( f i ( x ) , f i ( y )) < c for all i = 1 , . . . k . Suppose that f k is not nonstandard expansive, then x , y ∈ X ex-ists such that for all m ∈ ∗ Z ∞ , d ( f mk ( x ) , f mk ( y )) < δ . Let n ∈ ∗ Z , thenfor the Euclidean theorem (in a nonstandard version ) m, i ∈ ∗ Z exists, with0 ≤ i < k such that n = mk + i , but if n ∈ ∗ Z ∞ , then m ∈ ∗ Z ∞ , and then d ( f n ( x ) , f n ( y )) = d ( f mk + i ( x ) , f mk + i ( y )) = d ( f i ( f mk ( x )) , f i ( f mk ( y ))) < c ,thus f is not nonstandard expansive, which is absurd. (cid:3) Proposition 3.2.
Let X , Y be a metric compact spaces, and ( X, f ) and ( Y, g ) be conjugated dynamics. If f : X → X and if f is nonstandard expansive, then g : Y → Y is nonstandard expansive.Proof. Let ϕ : X → Y be a homeomorphism and f : X → X be nonstandardexpansive with constant of expansivity α . We will prove that g = ϕ.f.ϕ − is non-standard expansive; that is, we find c ∈ R , c > a, b ∈ Y , if d ( g n ( a ) , g n ( b )) < c is satisfied for all n ∈ ∗ Z ∞ , then a = b .For the transfer principle, if the following diagram is conmute X f / / ϕ (cid:15) (cid:15) X ϕ (cid:15) (cid:15) Y g / / Y thenthe diagram ∗ X ∗ f / / ∗ ϕ (cid:15) (cid:15) ∗ X ∗ ϕ (cid:15) (cid:15) ∗ Y ∗ g / / ∗ Y is conmute. If X is compact and ϕ − : Y → X is uniformly continuous, then c ∈ R , c > d ( y, y ′ ) < c , then d ( ϕ − ( y ) , ϕ − ( y ′ )) < α . However, for the transfer principle ∗ ϕ − , we have ver-ified that for all y, y ′ ∈ ∗ Y if ∗ d ( y, y ′ ) < c , then ∗ d ( ∗ ϕ − ( y ) , ∗ ϕ − ( y ′ )) < α .Suppose that for a, b ∈ Y we have ∗ d ( ∗ g n ( a ) , ∗ g n ( b )) < c for all n ∈ ∗ Z ∞ , then ∗ d ( ∗ ϕ − ( ∗ g n ( a ) , ∗ ϕ − ( ∗ g n ( b ))) < α for all n ∈ ∗ Z ∞ ; but a = ϕ ( x ) and b = ϕ ( x ′ )for some x, x ′ ∈ X , then ∗ d ∗ ( ϕ − ( ∗ g n ( ϕ ( x )) , ∗ ϕ − ( ∗ g n ( ϕ ( x ′ )))) < α ; but ϕ − ( ∗ g n ( ϕ ( z )) = ∗ f n ( z ) for all z ∈ X , and then ∗ d ( ∗ f n ( x ) , ∗ f n ( x ′ )) < α for all n ∈ ∗ Z ∞ . If f is α -expansive, thenwe have that x = x ′ , and then a = b . (cid:3) Proposition 3.3.
Let X be a metric space and f : X → X be homeomorphisms,then x, y ∈ X is positive asymptotic if for all m ∈ ∗ N ∞ , f m ( x ) ≃ f m ( y ) . For all a, b ∈ ∗ Z , b >
0, exist q, r ∈ ∗ Z unique such that a = bq + r , with 0 ≤ r < | b | . Theproof is a simple application of transfer principle Let X be a metric space and x, y ∈ ∗ X , we defined x ≃ y if and only if ∗ d ( x, y ) is aninfinitesimal ONSTANDARD EXPANSIVENESS 5
Proof. ⇒ : If lim n → + ∞ d ( f n ( x ) , f n ( y )) = 0, then for ǫ ∈ R , ǫ > n ǫ ∈ N suchthat( ∀ m ∈ N )( m ≥ n ǫ → d ( f m ( x ) , f m ( y )) < ǫ ). For the transfer principle, the follow-ing formula is true.( ∀ m ∈ ∗ N )( m ≥ n ǫ → d ( f m ( x ) , f m ( y )) < ǫ ), si m ∈ ∗ N ∞ , then m > n ǫ for all ǫ ;and then d ( f m ( x ) , f m ( y )) < ǫ , for all ǫ ; and then f m ( x ) ≃ f m ( y ). ⇐ : Suppose that lim n → + ∞ d ( f n ( x ) , f n ( y )) = 0, then exists ǫ ∈ R , ǫ >
0, such that forall n ∈ N ,( ∃ m ∈ N )(( m ≥ n ) ∧ d ( f m ( x ) , f m ( y )) > ǫ ) is verified. Then by taking a choicefunction ψ : N → N , we have( ∀ n ∈ N )( ψ ( n ) > n ) ∧ ( d ( f ψ ( n ) ( x ) , f ψ ( n ) ( y ) > ǫ ) entonces for the transfer principlethe formula ( ∀ n ∈ ∗ N )( ∗ ψ ( n ) > n ) ∧ ( d ( f ∗ ψ ( n ) ( x ) , f ∗ ψ ( n ) ( y )) > ǫ ) is true. If m ∈ ∗ N ∞ , then ∗ ψ ( m ) ∈ ∗ N ∞ , with d ( f ψ ( n ) ( x ) , f ψ ( n ) ( y ) > ǫ , and then f m ( x ) f m ( y ).Analogously, for x, y negative asymptotic. (cid:3) Lemma Let X be a metric space, a, b ∈ X , a ′ , b ′ ∈ ∗ X , r ∈ R , r > , then thefollowing statements are true. (1) If ∗ d ( a ′ , b ′ ) > r , then d ( a, b ) ≥ r . (2) If d ( a, b ) > r , then st ( ∗ d ( a ′ , b ′ )) ≥ r . Proof. (1) r < ∗ d ( a ′ , b ′ ) ≤ ∗ d ( a ′ .a )+ ∗ d ( a, b )+ ∗ d ( b, b ′ ) = ∗ d ( a ′ , b ′ ) ≤ ∗ d ( a ′ , a )+ d ( a, b ) + ∗ d ( b, b ′ ) = d ( a, b ) + ξ , with ξ infinitesimal, then r < d ( a, b ) + ξ , andthen d ( a, b ) ≥ r , if it is not then d ( a, b ) < r , and then 0 < r − d ( a, b ) < ξ ,which is absurd.(2) r < d ( a, b ) = ∗ d ( a, b ) ≤ ∗ d ( a, a ′ ) + ∗ d ( a ′ , b ′ ) + ∗ d ( b ′ , b ), but ∗ d ( a, a ′ ) ≃ ∗ d ( b, b ′ ) ≃
0, then r < ∗ d ( a ′ , b ′ ) + ξ , with ξ ∈ ∗ R is infinitesimal and notnegative. If ∗ d ( a ′ , b ′ ) is infinite, then it is proven. If this is not the case, then r = st ( r ) ≤ st ( ∗ d ( a ′ , b ′ ) + ξ ) = st ( ∗ d ( a ′ , b ′ )) + st ( ξ ) = st ( ∗ d ( a ′ , b ′ )) + 0 = st ( ∗ d ( a ′ , b ′ )). (cid:3) The following theorem was proven by Groisman and da Silva for the freely ex-pansive dynamics. We will present a proof that gives us a graphic intuition of thetheorem.
Theorem 2.
Let X be a compact metric space and f : X → X be a homeomor-phism, then f is nonstandard expansive if and only if f is expansive and does nothave doubly asymptotic points.Proof. ⇒ : If f is nonstandard expansive, then for all x, y ∈ X exists m ∈ ∗ Z ∞ , suchthat d ( f m ( x ) , f m ( y )) > c . Then for the above proposition, x, y are not asymptotic . ⇐ : The idea of the proof is described in the following: Let r ∈ ∗ R be a nonstandard real number, then st ( r ) is the nonstandard part of r . This isthe unique st ( r ) ∈ R such that r = st ( r ) + ξ , where ξ is an infinitesimal L. FERRARI b b b bb b b bb b b b b b b b b bb b b b b b b bb b b b b b bb b b b b b b b b b b b b b b bb b b b b b b bb x yf ( x ) f ( y ) f ( x ) f ( y ) ∗ f m ( x ) ∗ f m ( y ) x ′ y ′ f n ( x ′ ) f n ( y ′ ) greather than cf n ( ∗ f m ( y )) = ∗ f n + m ( y ) ≃ ≃ ≃ ≃ f n ( ∗ f m ( x )) = ∗ f n + m ( x )If x, y ∈ X , are not doubly asymptotic, then m ∈ ∗ Z ∞ exists such that f m ( x ) f m ( y ). Then α ∈ R exists such that ∗ d ( f m ( x ) , f m ( y )) > α , and X is compact forthe Robinson theorem x ′ , y ′ ∈ X exists such that x ′ ≃ f m ( x ) , y ′ ≃ f m ( y ), andif ∗ d ( f m ( x ) , f m ( y )) > α then d ( x ′ , y ′ ) ≥ α , in particular x ′ = y ′ . However, if f is expansive, then n ∈ Z exists such that d ( f n ( x ′ ) , f n ( y ′ )) > c . However, if f n iscontinuous ,then ∗ ( f n )( f m ( x )) ≃ f n ( x ′ ), ∗ ( f n )( f m ( y )) ≃ f n ( y ′ ). Then, for thelemma 1 part 2, st ( d ( f n ( f m ( x )) , f n ( f m ( y )))) ≥ c , but this implies that d ( f n ( f m ( x )) , f n ( f m ( y ))) > c . If n ∈ Z and m ∈ ∗ Z ∞ , then n + m ∈ ∗ Z ∞ , andthen f is nonstandard expansive. (cid:3) An important example of expansive homeomorphis is the subshift. We know by[1] that every expansive homeomorphism on a totally disconnected compact metricspace is a conjugate of a subshift.
Example Let X ⊂ Σ Z , where Σ = { , , } . On Σ Z , we consider the distance d ( α, β ) = P i ∈ Z | α ( i ) − β ( i ) | −| i | and the shift map σ : Σ Z → Σ Z , σ (( α ( i )) i ∈ Z ) =( α ( i + 1)) i ∈ Z . Let < a < b < be rationally independent real numbers. Considerthe interval exchange map T : I → I , which is defined as follows:Define the three intervals I = [0 , a ) , I = [ a, b ) and I = [ b, . Define the itinerary map I : I → Σ Z as I ( x ) = ( α k ) k ∈ Z if T k ( x ) ∈ I α k for all k ∈ Z . Weobtain the following commutative diagram.For α ∈ Σ Z , we define X α as the closure of the orbit { σ n ( α ) : n ∈ Z } . We saythat x ∈ I is regular if T n ( x ) / ∈ { , a, b } for all n ∈ Z .By - we know that this subshift does not have doubly asymptotic points, and thenby the theorem 2 is nonstandard expansive. The nonstandard analysis allows us to introduce a concept that is analogous touniform expansiveness for expansive nonstandard dynamic. Let X be a metric space, then X is compact if and only if for all y ∈ ∗ X exists x ∈ X , suchthat ∗ d ( x, y ) ≃
0. See [2] Let X be a metric space, f : X → X function and x ∈ X . f is continuous in x if and onlyif for all x ∈ ∗ X , with x ≃ x , ∗ f ( x ) ≃ f ( x ) is verified. See [2] ONSTANDARD EXPANSIVENESS 7
Definition . Let X be a metric space, and f : X → X be a homeomorphism.We say that f is c -uniformly nonstandard expansive, and if for all ǫ > n ǫ ∈ ∗ N ∞ such that for all x, y ∈ X , and d ( x, y ) > ǫ , then d ( f i ( x ) , f i ( y )) > c , forsome i ∈ ∗ Z ∞ , | i | < n ǫ . Theorem 3.
Let X be a metric compact space and f : X → X be homeomorphisms,then the following statements are equivalent. (1) f is c - nonstandard expansive. (2) For all ǫ ∈ R , ǫ > , and for all n ∈ N , exists m ∈ N , m > n such that forall x, y ∈ X , if d ( x, y ) > ǫ then d ( f i ( x ) , f i ( y )) > c for some i ∈ Z with n < | i | < m . (3) For all ǫ ∈ R , ǫ > exists n : N → N succession is strictly monotonicsuch that x, y ∈ X , if d ( x, y ) > ǫ then for all k ∈ N , d ( f i ( x ) , f i ( y )) > c isverified for some i ∈ Z with n ( k ) < | i | < n ( k + 1) . (4) For all ǫ > , n ǫ ∈ ∗ N ∞ exists such that for all x, y ∈ ∗ X , if ∗ d ( x, y ) > ǫ ,then ∗ d ( f i ( x ) , f i ( y )) > c , for some i ∈ ∗ Z ∞ , | i | < n ǫ . (5) f is c -uniformly nonstandard expansive.Proof. ⇒ ǫ > n ∈ N exists suchthat for all m ∈ N , with m > n exists x m , y m ∈ X such that d ( x m , y m ) > ǫ and d ( f i ( x m ) , f i ( y m )) ≤ c for all i ∈ Z with n < | i | < m . If X is compact, then thetaken subsuccession is necessary. We can suppose that ( x m ) m ∈ N converge to x and( y m ) m ∈ N converge to y .Let k ∈ Z such that | k | > n , be δ >
0. If f k is continuous, then f k ( x m )converge to f k ( x ) and f k ( y m ) converge to f k ( y ), then we can find m such that m > n , m > | k | , d ( f k ( x m ) , f k ( x )) < δ y d ( f k ( y m ) , f k ( y )) < δ , d ( f k ( x ) , f k ( y )) ≤ d ( f k ( x ) , f k ( x m )) + d ( f k ( x m ) , f k ( y m )) + d ( f k ( y m ) , f k ( y )) ≤ δ + c + δ = c + δ , then d ( f k ( x ) , f k ( y )) ≤ c for all k ∈ Z with | k | > n , and then x, y they are separated atmost by a finite number of points, and then f is not nonstandard expansive, whichis absurd.2 ⇒ n (1) = 1 and n (2) > n (1) the” m ” of the statement 2. Now suppose that we have defined n (1) , n (2) , . . . , n ( k ), take n ( k + 1) > n ( k ) the ” m ” of the statement 2, then thestatement is proven.3 ⇒ ǫ > n : N → N the succession of statement 3, and a l ∈ ∗ N ∞ .Let x, y ∈ X such that d ( x, y ) > ǫ for the the numerable axiom of choice and thestatement 3, i : N → Z exists such that for all k ∈ N , n ( k ) < | i ( k ) | < n ( k + 1)y d ( f i ( k ) ( x ) , f i ( k ) ( y )) > c . It is possible to extend the function i , such that i : X × X × N → Z . It is verified that if d ( x, y ) > ǫ , i ( x, y, k ) be the oldfunction and for the other case a constant arbitrary. Then, the following for-mula is true ( ∀ x ∈ X )( ∀ y ∈ X )( d ( x, y ) > ǫ → ( ∀ k ∈ N )(( n ( k ) < | i ( x, y, k ) |
Acknowledgements.
This work was carried out in the context of the Mas-ter’s thesis [8] by Facultad de Ciencias (Udelar). I wish to thank ProfessorGroisman for her guidance.
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