Sensitive dependence on initial conditions and chaotic group actions
aa r X i v : . [ m a t h . D S ] J u l Sensitive dependence on initial conditions andchaotic group actions
Fabrizio PoloJune 27, 2018
Abstract
A continuous action of a group G on a compact metric space has sensitive dependence on initial conditions if there is a number ε > U we can find g ∈ G such that g.U hasdiameter greater than ε. We prove that if a G action preserves a prob-ability measure of full support, then the system is either minimal andequicontinuous, or has sensitive dependence on initial conditions. Thisgeneralizes the invertible case of a theorem of Glasner and Weiss. Weprove that when a finitely generated, solvable group, acts and certaincyclic subactions have dense sets of minimal points, then the systemhas sensitive dependence on initial conditions. Additionally, we showhow to construct examples of non-compact monothetic groups, andtransitive, non-minimal, almost-equicontinuous, recurrent, G -actions. By a topological dynamical system we shall mean a continuous action of atopological group G on a compact metric space ( X, d ) , i.e., a continuousmap π : G × X → X : ( g, x ) g.x with the property that g. ( h.x ) = gh.x. Our theorems make no reference to the topology on G. So, one may assume G is discrete. We denote such a system by the pair ( X, G ) . If T : X → X is continuous, we may also call ( X, T ) a topological dynamical system. Inthis case we are referring to the obvious action Z . Occasionally we refer totheorems in which T is not assumed to be invertible. It will be made clearwhen this is the case.To understand the following property of dynamical systems is the pri-mary goal of this paper: Definition 1.1.
A topological dynamical system (
X, G ) is said to have sensitive dependence on initial conditions (or is sensitive ) if there is some1 > x ∈ X and for every open neighborhood U of x thereexists y ∈ U and g ∈ G such that d ( g.x, g.y ) > ǫ. It is worth noting that one need not (overtly) mention points in the defini-tion of sensitive dependence on initial conditions; see the following equivalentdefinition: there exists ε > U, sup g ∈ G diam( g.U ) >ε. Definition 1.2.
A point x ∈ X is called an equicontinuous point if forall ε > δ > y ∈ X, d ( x, y ) < δ implies d ( g.x, g.y ) < ε for all g ∈ G. A point which is not an equicontinuous pointwill be called a sensitive point.
A system (
X, G ) is said to be equicontinuous if the set of all maps { x g.x : g ∈ G } is an equicontinuous family. If ( X, G ) is equicontinuous then itis easily seen that every point is equicontinuous. Conversely, if every pointis equicontinuous then for each ε > x there exists δ x > g.B δ x ( x ) has diameter less than ε for all g ∈ G. From the balls B δ x ( x ) we can choose a finite subcover U . Let δ be a Lebesgue coveringnumber for U . Then, if d ( x, y ) < δ, there is some U ∈ U containing x and y. So, d ( g.x, g.y ) ≤ diam( g.U ) < ε for all g. This proves that every point isequicontinuous if and only if the system is equicontinuous.Now we define a new metric d ∞ on X by d ∞ ( x, y ) = sup g ∈ g d ( g.x, g.y ) . Consider the identity map Id : (
X, d ) → ( X, d ∞ ) . When G is a monoid, Id − is a contraction and hence continuous. If ( X, G ) is equicontinuous, it is easyto see that the Id is a homeomorphism. Also, if (
X, d ∞ ) is compact thenId has compact domain and Hausdorff range and so is a homeomorphism.As we will see in the proof of Theorem 4, a sensitive point is precisely adiscontinuity point for Id . Logically speaking, the assertion that a system have sensitive depen-dence on initial conditions is stronger than the assertion that every point issensitive (they differ in order of quantifiers.) But, by a category argument(see the excellent article [1]), one can prove
Proposition 1.3.
A transitive system is sensitive at each point if and onlyif it has sensitive dependence on initial conditions.
Definition 1.4.
A system is called almost equicontinuous if it has a denseset of equicontinuous points.It follows from Corollary 2 in [3] that for a single transitive map, eitherthe system is sensitive, or the set of equicontinuous points is exactly the same2s the set of transitive points. This is commonly known as the Auslander-Yorke dichotomy theorem. It holds in the non-invertible case as well.The following standard definitions are discussed in much greater detailelsewhere. For instance, [11] and [6] are good references. From now on wetake G to be a group. We say ( X, G ) is topologically transitive (or transitive for short) if for any open
U, V there exists g ∈ G such that U ∩ g.V = ∅ . In our setting, this is equivalent to the existence of a dense set of points x ∈ X such that G.x is dense in X (see Proposition 3.2 from [9].)A system is called minimal if every point has dense orbit. A point willbe called minimal if its orbit closure is a minimal system. When the actinggroup is discrete, a point p ∈ X is called a periodic point if G.p is finite.Periodic points are easily seen to be minimal. Saying that x is a minimalpoint is equivalent to requiring that for every open neighborhood U of x,R := { g ∈ G : g.x ∈ U } is left syndetic . That is, there is a finite subset F of G such that F R = G. In a group, any set which contains an infinite set of the form SS − = { st − : s, t ∈ S } is called a ∆ set. A set is called ∆ ⋆ if it intersects every ∆set. A ∆ ⋆ set which is symmetric (i.e. equal to it’s inverse) is always synde-tic. To see this, suppose the set L ⊂ G is symmetric but not syndetic. Let g = 1 G and assume we have already chosen distinct elements g , . . . , g n − with the property that g i g − j / ∈ L when i = j. Since L is not syndetic we canchoose g n such that { g n g − , . . . , g n g − n − } misses L. Since L is symmetric wealso know { g g − n , . . . , g n − g − n } misses L. Inductively, we see L ∩ SS − = ∅ where S = { g n } n . That is, L is not ∆ ⋆ . A probability measure µ on the Borel σ -algebra B = B ( X ) is invariant if µ ( A ) = µ ( g − .A ) for all A ∈ B , g ∈ G. All measures should be assumed tobe Borel measures. The measure µ is said to be ergodic if any invariant sethas measure one or zero.In the literature on sensitivity, many results have a common theme: un-der some additional hypothesis, an almost equicontinuous system must beequicontinuous. This gives a dichotomy: any system satisfying the addi-tional hypothesis is either equicontinuous or sensitive. Topological hypothe-ses are popular (see [4], [1], and [7].) Measure theoretic hypotheses alsoappear (see [7]) The next two theorems are particularly nice examples. Theorem 1.5 (Glasner, Weiss [7]) . If ( X, T ) is a transitive topological sys-tem equipped with an invariant probability measure µ of full support theneither ( X, T ) has sensitive dependence on initial conditions or it is minimaland equicontinuous. heorem 1.6 (Akin, Auslander, Berg [1] ) . Let ( X, T ) be a (possibly non-inervtible) transitive system. If the set of all minimal points is dense theneither the system has sensitive dependence on initial conditions or X is aminimal equicontinuous system. Both of these theorems are generalizations of their predecessor (see [4]):
Theorem 1.7 (Banks, Brooks, Cairns, Davis, Stacey [4]) . Let ( X, T ) be a(possibly non-inervtible) transitive system. If the set of periodic points isdense then either the system has sensitive dependence on initial conditionsor X is a finite set. A periodic point is a minimal, so 1.6 implies 1.7. Theorem 1.5 implies 1.7because the existence of a dense set of periodic points allows one to easilyconstruct an invariant measure of full support by adding weighted countingmeasures on periodic orbits.In this paper we derive similar results for the actions of more generalgroups. Our main results are Theorems 1.8 and 1.11. Theorem 1.8 general-izes the invertible case of Theorem 1.5.
Theorem 1.8.
Let ( X, G ) be a transitive system. If X admits an invariantmeasure of full support then the system is either minimal and equicontinuousor sensitive. A close examination of the proof in [4] of Theorem 1.7 reveals that (withobvious modifications) it already works for any group action. However,Theorem 2.5 from [1] is more general so we include it here for completeness.
Theorem 1.9.
Let G be a group and ( X, G ) be transitive system having adense set of minimal points. Then, either ( X, G ) has sensitive dependenceon initial conditions or X is minimal, equicontinuous. Heuristically speaking when G is a large group, the requirement that theaction have dense periodic or minimal points is quite strong. Does a resultlike Theorem 1.9 hold if we require only dense minimal points for certainsubactions? Indeed, when G is solvable there is such a theorem. In order tostate it we need another definition. Definition 1.10.
If a finite set S generates a solvable group G we will say S is nice if G ( n ) is generated by S ∩ G ( n ) (where G = G (0) and G ( n +1) =[ G ( n ) , G ( n ) ] . ) 4 heorem 1.11. Let S be a nice generating set for a finitely generated solv-able group G which acts transitively on a compact metric space X. Given s ∈ S, write h s i for the subgroup generated by s. If the system ( X, h s i ) hasa dense set of minimal points for each s, then ( X, G ) is either minimal,equicontinuous or it has sensitive dependence on initial conditions. In this section we demonstrate the existence of many transitive, almostequicontinuous systems which are not minimal, equicontinuous. We proveanalogues of Theorems 4.2 and 4.6 from [2]. The dichotomy type theoremsdiscussed in the introduction use additional hypotheses to eliminate the pos-sibility of almost equicontinuous but not equicontinuous systems. Withoutthe knowledge that such examples exist, the strength of this kind of theoremis questionable.
Lemma 2.1.
Let ( X, G ) be a topological dynamical system with an equicon-tinuous point x. If y ∈ X, g n ∈ G and g n .y → x then y is also equicontinuousand has the same orbit closure as x. Proof.
Given ε > U of x such that diam( g.U ) < ε for all g ∈ G. Fix h such that h.y ∈ U. Then V := h − .U is a neighborhoodof y with diam( g.V ) < ε for all g ∈ G. Thus y is an equicontinuity point.Since d ( g.x, gh − .y ) < ε for all g ∈ G, we see that the orbit of x is ε -densein the orbit of y and the orbit of y is ε -dense in the orbit of x. Thus, takingclosures of each orbit yields equal sets.
Proposition 2.2.
Let ( X, G ) be a system with a transitive point x. Thefollowing are equivalent:1. ( X, G ) is almost equicontinuous.2. x is an equicontinuous point.3. For all ε > there exists δ > such that for all g, h ∈ G, d ( h.x, x ) < δ implies d ( gh.x, g.x ) < ε. d and d ∞ induce the same topology on the set of transitive points.Proof. Assume 2. Then any translate of x is an equicontinuous point. 1follows. Now we prove 3 implies 2. Fix ε > δ > d ( h.x, x ) < δ implies d ( gh.x, g.x ) < ε/ g ∈ G. Now fix g ∈ G d ( x, y ) < δ/ . Choose h to make h.x close enough to y that d ( gh.x, g.y ) < ε/ d ( x, h.x ) < δ. Then d ( g.x, g.y ) ≤ d ( g.x, gh.x ) + d ( gh.x, g.y ) < ε/ ε/ . We will now prove 1 implies 4 and 4 implies 3. For 1 implies 4, It sufficesto show that a sequence x n of transitive points converges under d to anothertransitive point x , if and only if the same is true under d ∞ . One direction isobvious. For the other direction, suppose x n → x under d. By assumption x is transitive and hence equicontinuous by Lemma 2.1. For any ε > , when n is sufficiently large, we have d ( g.x n , g.x ) < ε for all g ∈ G. That is, d ∞ ( x n , x ) ≤ ε. Therefore d ∞ ( x n , x ) → , as desired.To see that 4 implies 3, suppose h n ∈ G are such that d ( h n .x, x ) → d ∞ ( h n .x, x ) → Proposition 2.3.
Suppose G acts transitively, by isometries on a possiblynon-compact metric space X and ι : X → X is a uniformly continuousmetric G -compactification. I.e., ( X, G ) is a compact metric system and ι is auniformly continuous, G -equivariant homeomorphic embedding of X , ontoa dense subset of X. Then ( X, G ) is an almost equicontinuous, transitivesystem with ι ( X ) contained in the transitive points of X. Conversely, everyalmost equicontinuous, transitive system ( X, G ) arises in this way: X maybe taken to be the set of transitive points equipped with the d ∞ metric.Proof. Since G acts transitively, by isometries on X , it acts minimally.Thus every point y ∈ X has orbit dense in X and so ι ( y ) =: x has orbitdense in X. We now must show that x is an equicontinuous point. Fix ε > , and let x ′ := ι ( y ′ ) . Using continuity of ι − and uniform continuity of ι wecan choose δ > d ( x, x ′ ) < δ then for all g ∈ G, g.y and g.y ′ aresufficiently close that d ( ι ( g.y ) , ι ( g.y ′ )) = d ( g.x, g, x ′ ) < ε. In other words, x is an equicontinuous point.For the converse, let ( X, G ) be an almost equicontinuous, transitive sys-tem and let X be the set of all transitive points equipped with the d ∞ metric. Then the inclusion ι : X → X is a contraction and hence uniformlycontinuous. By Proposition 2.2 part 4, ι is a homeomorphic embedding. So,in fact, ι is a uniformly continuous G -compactification, as desired.According to Proposition 2.3, constructing almost equicontinuous sys-tems, is equivalent to constructing equicontinuous compactifications of tran-sitive isometric G -actions. One simple way to construct an almost equicon-tinuous G action is to let X = G with the metric d ( g, h ) = 1 if and only if6 = h. This is an invariant metric giving the discrete topology. One couldthen take the one-point compactification of G and extend the left multi-plication action of G by fixing the point at infinity. The infinite point issensitive and all other points are equicontinuous and transitive.From a topological perspective, this example is not very interesting. No-tice that, except for the point at infinity, none of the transitive points arerecurrent. I.e. it is not true that for every transitive point x, every neigh-borhood U ∋ x, and every compact K ⊂ G we can find g / ∈ K with g.x ∈ U. If G = Z we can construct more examples by taking X = X to be somecompact monothetic group (a group with a dense cyclic subgroup.) Theseare well known and abundant. They occur precisely as Pontryagin duals ofsubgroups of the circle equipped with the discrete topology. Again, theseexamples are not very dynamically interesting because they are minimal andequicontinuous.If we take X to be a non-compact monothetic group then we can easilypick a metric on it with respect to which Z acts by isometries. Then any uni-form compactification ι : X → X gives a transitive almost-equicontinuoussystem which is not minimal, equicontinuous. In fact, it is not hard tosee that any transitive isometric Z -action on a complete metric space X isactually a monothetic group. This is explored in detail in [2].Now we will show how to construct an example of a G -system which istransitive, almost equicontinuous, and recurrent, but not minimal, equicon-tinuous. If one analyzes the procedure, we exploit the existence of non-compact monothetic groups. Some examples of such groups are known (see[10]) and any of them may be used in our construction. We will show adifferent method for constructing such groups. First we prove some obvi-ous propositions which reduce the problem of defining isometric transitive G -actions to defining norms on G. Definition 2.4. A symmetric norm on a group G is a function g
7→ k g k ∈ [0 , ∞ ) such that k g k = 0 if and only if g = 1 , k g k = k g − k , and k gh k ≤k g k + k h k . Two norms k · k i , i = 1 , uniformly equivalent if for all ε > δ > i, j k g k i < δ implies k g k j < ε. Proposition 2.5.
Transitive, isometric, free G -actions on complete, pointedmetric spaces are in one-to-one correspondence with symmetric norms on G. Proof.
First, suppose G acts transitively, isometrically, and freely on a com-plete pointed metric space ( X, x ) . Define k g k = d ( x, g.x ) . Since the ac-tion is free x = g.x unless g = 1 . Since the action is isometric, k g − k =7 ( x, g − .x ) = d ( g.x, x ) = k g k . Finally k gh k = d ( x, gh.x ) ≤ d ( x, g.x )+ d ( g.x, gh.x ) = d ( x, g.x )+ d ( x, h.x ) = k g k + k h k . Now suppose we have a symmetric norm k · k on G. Then we can definea left invariant metric on G by d ( g, h ) = k g − h k . Let X be the completionof G with respect to this metric and choose x = 1 for the base point. Then G obviously acts on X transitively, isometrically, and freely.A point x is said to be recurrent if for every neighborhood U ∋ x, outsideevery compact subset of G we can find g such that g.x ∈ U. A G -action issaid to be recurrent if for every point is recurrent. Proposition 2.6.
Fix a symmetric norm k·k on G. The associated isometricaction on the completion X of G is recurrent if and only if there exist g n ∈ G, g n → ∞ such that k g n k → . Furthermore X is compact if and only iffor all ε > the set { g ∈ G : k g k < ε } is left syndetic.Proof. Assume k g n k → . Fix any point x ∈ X and write x = 1 for thebase point of X. Choose g ∈ G with g.x ∈ B ε ( x ) . Then d ( x, gg n g − .x ) ≤ d ( x, g.x ) + d ( g.x , gg n .x ) + d ( gg n .x , gg n g − .x ) < ε + d ( x , g n .x ) + d (( gg n g − ) g.x , ( gg n g − ) .x ) < ε + d ( x , g n .x ) + ε. But the middle term is equal to k g n k which tends 0 . So x is recurrent.For the converse, we use the same argument but reverse the roles of x and x . We assume g n .x → x and h n .x → x . We see that we can make h n g n h − n leave every compact set, while, at the same time k h n g n h − n k = d ( x , h n g n h − n .x ) → . If X is compact, then ( X, G ) is a minimal equicontinuous system. Sothe set of return times R of x to B ε ( x ) is left syndetic. But R is precisely { g ∈ G : k g k < ε } . Conversely, if this set is left syndetic, then we can choosea finite set F ⊂ G such that F R = G. It follows that for every ε > x ∈ X we can find f ∈ F and g ∈ R such that d ( x, f g.x ) < ε. But d ( x, f.x ) < d ( x, f g.x ) + d ( f g.x , f.x ) < ε + d ( g.x , x ) = ε + k g k < ε. This proves that the finite set
F.x is 2 ε -dense. It follows that X is compact.8uppose ϕ is any symmetric nonnegative function on G which takes thevalue 0 at 1 . Define k h k = inf { ϕ ( h ) + · · · + ϕ ( h n ) : h · · · h n = h } . Certainly k k = 0 and k h − k = k h k . Fix h, h ′ ∈ G. Given ε > h , . . . , h n + k such that h · · · h n = h and h n +1 · · · h n + k = h ′ and ϕ ( h ) + · · · + ϕ ( h n ) < k h k + ε and ϕ ( h n +1 ) + · · · + ϕ ( h n + k ) < k h ′ k + ε. Then hh ′ = h · · · h n + k so k hh ′ k ≤ ϕ ( h ) + · · · ϕ ( h n + k ) < k h k + k h ′ k + 2 ε. It follows that k hh ′ k ≤ k h k + k h ′ k . Except for the possibility that k g k = 0for some g = 1 , the function k · k as defined above is a symmetric norm.Write G ε for the subgroup generated by { g ∈ G : ϕ ( g ) < ε } . It is easyto see that if, g / ∈ G ε then k g k ≥ ε. So, if for each g ∈ G, g = 1 there exists ε > g / ∈ G ε , then k · k is a symmetric norm. This condition is farfrom necessary. For instance, suppose G = Z = h t i acts on the circle X = T by an irrational rotation. Define k g k to be the distance from 1 to g. . Noticethat for any n, the subaction generated by t n is also a minimal. It followsthat for any ε > n, k such that t n and t kn +1 both have normless than ε. So t ∈ G ε and G ε = G. Nonetheless, k · k is a symmetric norm.Assume G has an element t of infinite order and define a function ϕ on G as follows. Let ϕ (1) = 0 . For n ≥ , let ϕ ( t n ! ) = ϕ ( t − n ! ) = ( n + 1) − . Elsewhere, let ϕ ≡ . Use ϕ as above to construct k · k . Suppose k g k < n − and g = 1 . Then we can write g as a product of elements of the form t k ! where k ≥ n. So g = t m and m = a k ! + a k ! + · · · + a l k l ! where the k i ≥ n are distinct, and each a i is a nonzero integer. The assumption on k g k tellsus that we can do this efficiently so that P i | a i | ( k i + 1) − < n − . Put the k i in decreasing order and observe that P li =2 | a i | < k n − . If a > k + 1 thenwe could1. replace a by a − ( k + 1)2. introduce k = k + 13. and introduce a = 1 . This would give us another way of representing m which reduces the valueof P i | a i | ( k i + 1) − . Loosely speaking, instead of taking very many large’steps’, we could have taken one even larger step.9 symmetric statement can be made if a < − k − . So, let us assume | a | ≤ k + 1 . Then | m − a k ! | ≤ | a | k ! + · · · + | a l | k l ! ≤ ( k − l X i =2 | a i |≤ ( k − k n = k ! n . In particular m lies in the interval [ k !( a − n − ) , k !( a + n − )] . Assemblingthese results over all possible values of k and | a | ≤ k + 1 tells us that m must lie in I := { i : k t i k < n , i > } ⊆ [ k ≥ n k +1 [ a =1 [ k !( a − n − ) , k !( a + n − )] . When n is sufficiently large, I misses any given interval around 0 . Also I isnot syndetic. Therefore, transferring the statement about exponents to thegroup itself, we see { g ∈ G : k g k < n , g = 1 } = { t i : i ∈ I or − i ∈ I } is not syndetic. Furthermore, given any element g ∈ G, g = 1 , we can choose n large enough that this set does not contain g. Let X be the completion of G with respect the metric induced by k · k . It is not necessary, but illuminating to observe that this is a disjoint union ofnon-compact monothetic groups, one for each coset of h t i . By Propositions2.5 and 2.6, G acts transitively, isometrically, freely, and recurrently on thenon-compact metric space X . Now we may choose any uniformly continuouscompactification ι : X → X and apply Proposition 2.3 to get a transitive,recurrent, non-minimal, almost equicontinuous system ( X, G ) . In this section we prove Theorem 1.8. First, we need a Lemma (which isinteresting in its own right.)
Lemma 3.1.
Let ( X, G ) be a system, and let µ be an ergodic measure of fullsupport. Given A ⊆ X of positive measure and x ∈ X, write R = R ( x, A ) = { g ∈ G : g.x ∈ A } . Then for µ -almost every x ∈ X, RR − is ∆ ⋆ . The author would like to thank Vitaly Bergelson for help with this proof.10 roof.
Let S be an infinite subset of G and choose g = g S , h = h S ∈ S such that µ ( g − .A ∩ h − .A ) > . Let T ( g, h ) be the full measure set S k ∈ G ( kg − .A ∩ kh − .A ) . If we make such choices for every infinite subset S, countability of G × G tells us, we have an at most countable collection T ( g S , h S ) of full measure sets. Let Y = T S T ( g S , h S ) . Then µ ( Y ) = 1 . Take x ∈ Y. Then for any infinite S ⊆ G there exist g = g S , h = h S ∈ S and k ∈ G such that k.x ∈ g − .A ∩ h − .A . Therefore gk.x ∈ A and hk.x ∈ A. Therefore RR − ∋ gk ( hk ) − = gh − ∈ SS − . The proof of Theorem 1.8 was motivated by the techniques in [7].
Proof. (of Theorem 1.8) Suppose (
X, G ) is not sensitive. That is, it has anequicontinuous point x. Fix ε > δ = ε. Now choose δ i , i = 1 , , d ( x, y ) < δ i then d ( g.x, g.y ) < δ i +1 for all g ∈ G.
2. 3 δ i < δ i +1 . Let A be the δ ball around x. Since µ has full support we can choosesome ergodic component ν with ν ( A ) > . Now we can apply Lemma 3 to ν and A to deduce the existence of a point y ∈ A with the property that R := R ( y, A ) satisfies RR − is ∆ ⋆ . For g ∈ R we have d ( g.x, y ) ≤ d ( g.x, g.y ) + d ( g.y, y ) < δ + δ < δ . So, d ( x, g.x ) ≤ d ( x, y ) + d ( y, g.x ) < δ + 2 δ < δ . Taking h ∈ R we get d ( x, hg − .x ) ≤ d ( x, h.x ) + d ( hg − . ( g.x ) , hg − .x ) < δ + δ < ε. We have proven that RR − ⊆ R ( x, B ε ( x )) . Since ε was arbitrary, itfollows that the set of return times of x to any neighborhood of itself is∆ ⋆ . In particular, R ( x, B δ ( x )) is ∆ ⋆ . Given g ∈ R ( x, B δ ( x )) , we know d ( x, g − .x ) = d ( g − . ( g.x ) , g − .x ) < ε. It follows that R ( x, B δ ( x )) ∪ R ( x, B δ ( x )) − ⊆ R ( x, B ε ( x )) . In other words, R ( x, B ε ( x )) contains a symmetric ∆ ⋆ and somust be (left) syndetic (as explained in the introduction.) But, this is equiv-alent to the assertion that x have minimal orbit closure (see for instance [6].).Since x is transitive, we conclude that ( X, G ) is minimal. By Lemma 2.1),(
X, G ) is equicontinuous. 11n the case that µ is an ergodic measure, there is an alternate proofof Theorem 1.8 that relies on Proposition 2.3. The method is completelydifferent and requires a simple lemma. First define the function s ( x ) = inf ε> sup { diam( g.B ε ( x )) : g ∈ G } = inf { d ∞ -diam( U ) : x ∈ U open } We call s ( x ) the sensitivity constant at x. Notice that x is a sensitivepoint if and only if s ( x ) > . Lemma 3.2.
The function s defined above is upper semi-continuous andhence measurable. If ( X, G ) admits an ergodic measure µ of full support forwhich the set of all sensitive points has positive measure then the system hassensitive dependence on initial conditions.Proof. Suppose U is a neighborhood of x such that d ∞ -diam( U ) < s ( x ) + ε. Then for any y ∈ U, s ( y ) ≤ d ∞ -diam( U ) < s ( x ) + ε. Therefore s is uppersemi-continuous. Measurability follows.Since s is an invariant function it is constant µ -almost everywhere. Sinceit was assumed to take positive values on a set of positive measure, it mustbe equal to some c > , µ -almost everywhere. Thus s − ( c ) is a dense set.Let U be any open subset of X. Then U contains an element of s − ( c ) . Bydefinition of s we can find g ∈ G such that diam( g.U ) > c/ . So, (
X, G ) issensitive.Let (
X, G ) be an almost-equicontinuous system and let µ be an ergodicprobability measure of full support. Then the set X of equicontinuity pointsis an invariant set and so must have measure 1 or 0 . If it has measure 0 thenalmost every point is sensitive. By Lemma 3.2, (
X, G ) is sensitive. If X has measure 1 then by Proposition 2.3 we can think of µ as a Borel measureon ( X , d ∞ ) . If ( X , d ∞ ) is not compact then for some ε > x , x , · · · ∈ X with d ∞ ( x i , x j ) > ε when i = j. Cover X by countablymany balls of d ∞ -radius ε/ . One of them must have positive measure. Callit B and choose g n ∈ G such that x n ∈ g n .B. Since G acts on ( X , d ∞ ) byisometries, the balls g n .B are disjoint. This is ludicrous, since they musteach have positive measure.So, ( X , d ∞ ) must be compact. Continuity of ( X , d ∞ ) → ( X, d ) tells us X is compact as a subspace of X. Density tells us X = X. The identityis then a homeomorphism and (
X, G ) is isomorphic to an isometric system.I.e. (
X, G ) is minimal and equicontinuous.12
Periodic points and minimal subsystems
In [5], Devaney suggests three properties which define the essence of chaos.According to him, a dynamical system (i.e. a continuous map T : X → X )should be called chaotic if it is1. topologically transitive,2. has a dense set of periodic points,3. has sensitive dependence on initial conditions.It was first observed in [4] that these requirements are not independent.In fact, they prove that the first two conditions imply the third (this is thecontent of Theorem 1.7.) In [7], Glasner and Weiss derive this as a corollaryof Theorem 1.8. They also produce a remarkably simple direct proof (theirCorollary 1.4.) Unfortunately, it is unclear to this author how to adapt thesecond argument to the case of a non-abelian acting group.Now we set out to prove Theorem 1.11. Lemma 4.1.
Assume S is a nice generating set for the solvable group G. Then there is some bound N so that any g ∈ G can be written in the form g = s e i s e i · · · s e N i N where each s i ∈ S, e i ∈ Z . Proof. If G is abelian this is obvious, so assume G has higher solvability de-gree. Enumerate S = { s , s , . . . s n } . We know we can write g = s e i s e i · · · s e k i k for some k. Rearrange this word by collecting like terms and introducingcommutators where necessary. That is, rearrange the word into the form: g = s f c s f c s f c · · · c n − s f n n c n where c j ∈ [ G, G ] . By induction on degree of solvability, there is some con-stant M such that each c i contributes at most M terms of the form s ba . So, g can be written with no more than nM + n = | S | ( M + 1) terms. Lemma 4.2.
Suppose x is an equicontinuous point in a topological dynam-ical system ( X, G ) . If x is a limit of minimal points x n then x is minimal. This lemma appears in [1].
Proof.
Fix ε > δ > ε/ g.B δ ( x ) ⊆ B ε/ ( g.x ) . Choose y := y n ∈ B δ ( x ) and let R = { g ∈ G : g.y ∈ B ε/ ( x ) . Since y n is minimal, R is syndetic. If g ∈ R then d ( g.x, x ) ≤ d ( g.x, g.y ) + d ( g.y, x ) ≤ ε/ ε/ . So R ⊆ { g ∈ G : g.x ∈ B ε ( x ) } , which proves the latterset is left syndetic. Thus, x is minimal.13 roof. (of Theorem 1.11) Suppose the system is not sensitive. Then it hasan equicontinuous point. By Theorem 1.6, any transitive point must beequicontinuous. Let x be such a point. Fix y ∈ X = G.x.
Choose a sequence g n ∈ G such that g n .x → y. Lemma 4.1 tells us that each g n can be writtenwith at most N terms of the form s e . By passing to a subsequence wecan find a sequence s , s , . . . , s k ∈ S such that each g n is of the form g n = s e n,k k s e n,k − k − · · · s e n, (we allow the possibility that s i = s j for unequal i, j. )Let X be the orbit closure of x under h s i . Pass to a subsequencealong which s e n, .x converges to some point x ∈ X . The points which areminimal under the action of h s i are dense by assumption. So, by Lemma4.2, x is also minimal under this action. Thus we can find powers of s which move x arbitrarily close to x. Applying Lemma 2.1 we see that x isanother transitive equicontinuous point. Since x is an equicontinuous point,as d ( s e n, .x, x ) → d ( s e n,k k · · · s e n, . ( s e n, .x ) , s e n,k k · · · s e n, .x ) → . This proves that s e n,k k · · · s e n, .x → y. Repeat this process: let X be the orbit closure of x under h s i . Passto a subsequence along which s e n, .x converges to some point x ∈ X . Argue as above to conclude that X is minimal under the action of h s i , x is transitive equicontinuous, and satisfies s e n,k k · · · s e n, .x → y. Continuing in this way we get a transitive equicontinuous point x k − , whose orbit closure X k under the action of s k is minimal and contains y. Then we can find some powers of s k which move y as close as we like to x k − . By Lemma 2.1, y is a transitive, equicontinuous point.We have shown that every point is transitive and equicontinuous. Equiv-alently, ( X, G ) is a minimal equicontinuous system.
Corollary 4.3.
Let
G, S be as in Theorem 1.11 and assume the set ofperiodic points for ( X, h s i ) is dense for each s ∈ S (where h s i is the groupgenerated by s. ) If ( X, G ) is not sensitive then X must be a finite set onwhich G acts transitively. It would be advantageous to drop the condition that S be nice in The-orem 1.11. Unfortunately, the following example demonstrates that thiscondition (or something like it) is unavoidable. Let G be the solvable group Z ⋊ Z / Z and let a and b be the generators of the factors Z and Z / Z re-spectively. Then S := { a, ab } is another generating set for G. Let X be theone point compactification of G and extend the left multiplication action of14 on itself to an action on X by fixing the point at infinity. The system( X, G ) is transitive but not minimal.Each s ∈ S has order two. So, every point is minimal for the sys-tem ( X, h s i ) . However, (
X, G ) does not have sensitive dependence on ini-tial conditions. In fact, the only sensitive point is the point at infinity.All other points are isolated and therefore equicontinuous. This is nota counterexample to Theorem 1.11 because S is not nice. Suppose weadd more generators to S to make it nice. For instance we could take S = { a, ab, [ a, ab ] } . Then the hypotheses of the theorem are not met: forany x ∈ X, lim | n |→∞ [ a, ab ] n .x = ∞ , which proves the only minimal point of( X, h [ a, ab ] i ) is ∞ (clearly not dense.)Acknowledgements: The author would like to thank Vitaly Bergelson forproposing this avenue of research, Cory Christopherson for proof-reading anearly version, and Michael Hochman, who made me aware of [1]. The authorwould especially like to thank Rafal Pikula for thoroughly checking the finaldraft. References [1] E. Akin, J. Auslander, and K. Berg. When is a transitive map chaotic? In
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