Equicontinuous mappings on finite trees
EEQUICONTINUOUS MAPPINGS ON FINITE TREES
GERARDO ACOSTA AND DAVID FERN ´ANDEZ-BRET ´ON
Abstract. If X is a finite tree and f : X −→ X is a continuous function, as the Main Theorem of thispaper (Theorem 8), we find eight conditions, each of which is equivalent to the fact that f is equicontinuous.Our results either generalize the ones shown by Vidal-Escobar and Garc´ıa-Ferreira in [18], or complementthose of Bruckner and Ceder ([3]), Mai ([11]) and Camargo, Rinc´on and Uzc´ategui ([5]). Some of themethods, however, have not been used previously in this context (for example, in one of our proofs we applythe Ramsey-theoretic result known as Hindman’s theorem). To name just a few of the results obtained:the equicontinuity of f is equivalent to the fact that there is no arc A ⊆ X satisfying A (cid:40) f n [ A ] for some n ∈ N . It is also equivalent to the fact that for some nonprincial ultrafilter u , the function f u : X −→ X iscontinuous (in other words, failure of equicontinuity of f is equivalent to the failure of continuity of every element of the Ellis remainder E ∗ ( X, f )). Introduction
For a metric space X, this paper deals with continuous functions f : X −→ X, whose family of iterates isequicontinuous. Such functions represent well-behaved, non-chaotic, dynamical systems (equicontinuityis diametrically opposite to what is known as sensitivity to initial conditions, see [1, Theorem 2.4]). Wedefine the notions of dynamical systems that will be used. Definition 1. (1) A discrete dynamical system is an ordered pair (
X, f ) such that X is a metric space and f : X −→ X is a continuous function.(2) If ( X, f ) is a discrete dynamical system, we define f as the identity function on X , and, for each n ∈ N , f n = f n − ◦ f .(3) If X, Y are metric spaces and F is a family of functions from X to Y , we say that F is equicontin-uous at x ∈ X if for every ε > δ > d ( x, y ) < δ implies d ( f ( x ) , f ( y )) ≤ ε for all y ∈ X and every f ∈ F ; and if F is equicontinuous at every x ∈ X , we say that F is equicontinuous .(4) If X is a metric space, the function f : X −→ X is equicontinuous at x ∈ X if its family ofiterates, { f n (cid:12)(cid:12) n ∈ N } , is equicontinuous at x ; and if f is equicontinuous at every x ∈ X we say thatit is equicontinuous .The definition of equicontinuity makes sense for every uniform space, but in this paper we will only considermetric spaces. Note that, upon fixing x, ε, δ , equicontinuity of a family of functions F is a pointwise closedcondition; consequently if F is equicontinuous at x then so is F , where F is the closure of F in Y X withthe product topology. Note also that, if X is compact, then by the usual argument, equicontinuity impliesuniform equicontinuity (i.e., given ε >
0, a δ > x ∈ X ).The following is also an important notion in dynamical systems. Mathematics Subject Classification.
Primary 54A20, 54D80, 54H15, 54H20. Secondary 54C05, 54D05, 54D30, 54E45,54F15.
Key words and phrases.
Dendrites, Discrete Dynamical Systems, Ellis semigroup, Equicontinuous functions, Finite Graphs,Finite Trees, Ramsey Theory.This research was carried out as part of the second author’s postdoctoral stay (supported by a Fellowship from DGAPA-UNAM) under the mentoring of the first author. a r X i v : . [ m a t h . GN ] A ug G. ACOSTA AND D. FERN ´ANDEZ
Definition 2.
Let (
X, f ) be a discrete dynamical system, where X is compact.(1) The Ellis semigroup (also called the enveloping semigroup ) of (
X, f ) is defined as E ( X, f ) = { f n (cid:12)(cid:12) n ∈ N } , the closure in X X (with the product topology) of the family { f n (cid:12)(cid:12) n ∈ N } . Note that,as X X is compact (by Tychonoff’s theorem), so is E ( X, f ).(2) The
Ellis remainder of the dynamical system (
X, f ) is E ( X, f ) ∗ = ∞ (cid:92) n =1 { f k (cid:12)(cid:12) k ≥ n } . Note that E ( X, f ) = E ( X, f ) ∗ ∪ { f n (cid:12)(cid:12) n ∈ N } .Composition of functions, as a binary operation, is what makes E ( X, f ) a semigroup . In fact, E ( X, f ) is acompact right-topological semigroup (a semigroup equipped with a topology making all right translationscontinuous). Since X is a metric space, by the observation immediately after Definition 1, equicontinuityof f is equivalent to equicontinuity of the family E ( X, f ), and either of these is equivalent to the samestatement with uniform equicontinuity instead of equicontinuity (cf. [8, Theorem 3.3]).The seemingly abstract object E ( X, f ) can be made more concrete by means of ultrafilters: for everyultrafilter u on N , define the ultrafilter-limit function f u by letting f u ( x ) = u - lim n ∈ N f n ( x ). Then by [8,Theorem 2.2] we have E ( X, f ) = { f u (cid:12)(cid:12) u is an ultrafilter on N } , and consequently E ( X, f ) ∗ = { f u (cid:12)(cid:12) u is a nonprincipal ultrafilter on N } . Full definitions of ultrafilters, both principal and nonprincipal, as well as of u -limits will be provided inSection 3. We now introduce some more concepts. Definition 3.
Let (
X, f ) be a discrete dynamical system.(1) A point x ∈ X is a fixed point if f ( x ) = x ; the set of fixed points of f is denoted by Fix( f ) . (2) A point x ∈ X is a periodic point if f n ( x ) = x , for some n ∈ N , in which case the least such n iscalled the period of x . The set of all periodic points of f is denoted by Per( f ).(3) We say that f is periodic if there exists n ∈ N such that f n is the identity function on X , and wesay that f is pointwise-periodic if Per( f ) = X. It is immediate from Definition 3 that Per( f ) = (cid:83) ∞ n =1 Fix( f n ).We primarily deal with continua (compact, connected and metric spaces). A simple closed curve is acontinuum homeomorphic to the unit circle S , and an arc is a continuum homeomorphic to the unitinterval [0 , . Other examples of continua are graphs (compact, connected one-dimensional polyhedra), dendrites , finite trees and k -ods , for each k ∈ N with k ≥ . We give the proper definitions of the last threein Section 2. For the moment it is convenient to note that a 2-od is an arc, k -ods are finite trees and finitetrees are dendrites.In the early nineties, Bruckner and Hu ([4]) and Bruckner and Ceder ([3]) carried out a very deep andcomplete study of equicontinuity of functions defined on arcs, obtaining the following result. Theorem 4 (Subset of [3], Theorem 1.2) . If X is an arc and f : X −→ X is a continuous function, thenthe following are equivalent: (1) f is equicontinuous; (2) the restriction f (cid:22) (cid:84) ∞ m =1 f m [ X ] is the identity function; (3) Fix( f ) = (cid:84) ∞ m =1 f m [ X ]; QUICONTINUOUS MAPPINGS ON TREES 3 (4) Fix( f ) is connected. Attempting to generalize this result from arcs to finite trees is futile, if taken too literally. Allowing,however, exponents other than 2 in the theorem above yields valid characterizations: we prove that, for anarbitrary finite tree X and a continuous function f : X −→ X , equicontinuity of f is equivalent to each ofthe following conditions: that the restriction f (cid:22) (cid:84) ∞ m =1 f m [ X ] is periodic, that Fix( f n ) = (cid:84) ∞ m =1 f m [ X ] forsome n , and that Fix( f n ) is connected for all n ; furthermore, any of these is also equivalent to the setPer( f ) being connected.Further interesting results regarding equicontinuity of a continuous function f : X −→ X have beenobtained by Mai ([11]) in the case where X is a graph, and by Camargo, Rinc´on and Uzc´ategui ([5]) inthe case where X is a dendrite. The former shows in [11, Theorem 5.2] that, if X is a graph, then f isequicontinuous if and only if (cid:84) ∞ m =1 f m [ X ] = Rec( f ) (here Rec( f ) denotes the set of recurrent points of f ,to be defined later in Definition 12; for the moment just note that Per( f ) ⊆ Rec( f )); the latter proves in [5,Theorem 4.12] that, if X is a dendrite, then f is equicontinuous if and only if cl X (Per( f )) = (cid:84) ∞ m =1 f m [ X ]plus an extra condition having to do with the ω -limit sets of f . Obtaining a simultaneous strengthening ofthese two results at the expense of considering a less general class of spaces, we prove that, in the casewhere X is a finite tree, f is equicontinuous if and only if Per( f ) = (cid:84) ∞ m =1 f m [ X ].Another concept that will play a central role in this paper is that of an expanding arc. To motivate thisconcept consider a nonnegative α ∈ R and the function f α : R −→ R defined by(1) f α ( x ) = αx, for each x ∈ X. It is readily checked that f α is equicontinuous if and only if 0 ≤ α ≤
1, whereas if α > f α failsto be equicontinuous at every x ∈ R . Intuitively speaking, functions that expand the real line fail tobe equicontinuous. Note that for the function f α defined in (1) we have I (cid:40) f nα [ I ] for all n ∈ N , where I = [0 ,
1] is the unit interval. This leads to the following definition.
Definition 5.
Let (
X, f ) be a discrete dynamical system, and let A ⊆ X be a subspace homeomorphic toan arc. We will say that A is an f -expanding arc if there exists an n ∈ N such that A (cid:40) f n [ A ].Hence the function f α defined in (1) is equicontinuous if and only if [0 ,
1] is f α -expanding. Surprisingly,something like this very simple characterization still holds in more general situations. Given a continuousfunction f : X −→ X , where X is an arc, from the proof of [3, Theorem 1.2] it follows that if f is notequicontinuous, then X contains an f -expanding arc. Vidal-Escobar and Garc´ıa-Ferreira proved that if X is a k -od with k ≥ X is an arc and f is surjective ([18, Theorem 3.1]), then f isequicontinuous if and only if X contains no f -expanding arcs. Hence the next result follows from the citedresults in [3] and [18]. Proposition 6.
Let f : X −→ X be a continuous function that is not equicontinuous. If X is a k -od forsome k ≥ , then X contains an f -expanding arc. In this paper, we generalize Proposition 6 from k -ods to arbitrary finite graphs. Our proof of thisgeneralization uses at a crucial point a highly nontrivial Ramsey-theoretic result (Hindman’s theorem).Hence, our result is not a direct use of the proofs presented in [3, Theorem 1.2] and [18, Theorem 3.7].Another result of Vidal-Escobar and Garc´ıa-Ferreira ([18, Theorem 3.7]) is that for each continuousfunction f : X −→ X , where X is a k -od with k ≥
3, the following statement holds: if f u is continuous for every nonprincipal ultrafilter u then f is equicontinous (note that the converse implication is trivially trueas a consequence of the observation right after Definition 2). Hence if f is not equicontinuous then forsome nonprincipal ultrafilter u, f u is not continuous. It [18] the authors consider the possibility that, forsome nonprincipal ultrafilter v, distinct from u, f v might be continuous.In this paper we show that the possibility mentioned in the previous line cannot occur by proving that thestatement from the preceding paragraph is true with every replaced by some , even if X is a finite treerather than just a k -od. As a consequence of this, if f fails to be equicontinuous with X a finite tree, then G. ACOSTA AND D. FERN ´ANDEZ every element g ∈ E ∗ ( X, f ) fails to be continuous. Thus, for continuous functions f : X −→ X on a finitetree X , we have a strong dichotomy by means of which either every element of E ∗ ( X, f ) is continuous, orevery element of E ∗ ( X, f ) is discontinuous, according to whether or not f is equicontinuous. This is adirect generalization of a result of Szuca ([17, Theorem 2]), who obtains the same dichotomy for functionsin an arc. This result is therefore worth stating explicitly. Theorem 7.
Let ( X, f ) be a discrete dynamical system, where X is a finite tree. Then, either everyelement of E ( X, f ) ∗ is continuous, or every element of E ( X, f ) ∗ is discontinuous. We now state the Main Theorem of this paper.
Theorem 8.
Let X be a finite tree and f : X −→ X be a continuous function. Then, the following areequivalent: (a) f is equicontinuous; (b) there is an n ∈ N such that the restriction of f n to (cid:84) ∞ m =1 f m [ X ] is the identity function; (c) there exists an n ∈ N such that Fix( f n ) = (cid:84) ∞ m =1 f m [ X ];(d) Per( f ) = (cid:84) ∞ m =1 f m [ X ];(e) there is no f -expanding arc in X ;(f) for every n ∈ N , the set Fix( f n ) is connected; (g) the set Per( f ) is connected; (h) for every nonprincipal ultrafilter u , the function f u is continuous ( i.e., every element of E ( X, f ) ∗ is continuous ) ; (i) for some nonprincipal ultrafilter u , the function f u is continuous ( i.e., some element of E ( X, f ) ∗ is continuous ) .Remark . Some remarks about the equivalences from Theorem 8:(1) The equivalence between (e), (f) and (g) will be established not only for finite trees, but forarbitrary dendrites.(2) The equivalence between (a) and (d) still holds if X is merely a dendrite with finitely manybranching points: the implication from (a) to (d) follows from [5, Theorem 4.12] together with [16,Lemma 2.6]; whereas the reverse implication is [5, Theorem 4.14] together with [11, Theorem 5.2].(3) The equivalence between (a) and (c) was established by T. Sun in [14], who also showed that in (c)one can take n = k !, where k is the number of endpoints of the finite tree X .(4) Further conditions equivalent to equicontinuity of a function f on a space X have been establishedin [15, Theorem 2, p. 62] for X a finite tree, in [11, Theorem 5.2] when X is a finite graph, andin [5, Theorem 4.12] in the case of X an arbitrary dendrite.We also show that (with the exception of (a) ⇐⇒ (d)) neither of these characterizations of equicontinuityholds for dendrites that are not finite trees. More specifically, we provide examples of dendrites andfunctions in each of these dendrites that together show that none of the conditions (b)-(i) in Theorem 8above is equivalent to condition (a) on arbitrary dendrites.The paper is structured around the equivalence that constitutes its main result (Theorem 8). In Section 2we begin by proving the equivalence of items (a), (b), (c) and (d), which is a fairly elementary result, andthe rest of the section is devoted to the study of expanding arcs, starting with the equivalence of (e) and(f), and concluding with the implication from (e) to (a). Then, in Section 3, we establish the equivalencebetween (e) and (g), in order to later on focus on ultrafilter-limit functions to establish that (i) implies(e) (this finishes the main theorem, since the implication from (h) to (i) is obvious and that from (a) to QUICONTINUOUS MAPPINGS ON TREES 5 (h) is well-known). Finally, in Section 4 we describe the examples that exhibit the failure of all thesecharacterizations in the context of arbitrary dendrites, and state some questions that remain open.2.
Equicontinuity and expanding arcs
Given a subset A of a space X, we denote by either A or cl X ( A ) , the closure of A in X. The interior of A in X is denoted by int X ( A ) . We begin by stating some standard results that will be used.
Proposition 10.
Let ( A n ) n ∈ N be a decreasing sequence of closed subsets of a compact space X , and let A = (cid:84) ∞ n =1 A n . Then, (1) if U is an open set containing A , then A n ⊆ U for all sufficiently large n ; (2) if each A n is nonempty, then so is A ; (3) if every A n is connected, then so is A ; (4) if f : X −→ Y is a continuous function, then (cid:84) ∞ n =1 f [ A n ] = f [ A ] .Proof. Parts (1) and (3) follow from [7, Corollary 3.1.5 and Corollary 6.1.19]. Part (2) follows from (1)with U = ∅ . To show part (4), it is enough to verify that (cid:84) ∞ n =1 f [ A n ] ⊆ f [ A ]. Let y ∈ (cid:84) ∞ n =1 f [ A n ] . Since( f − [ y ] ∩ A n ) n ∈ N is a decreasing sequence of closed, nonempty subsets of X ; by (2), f − [ y ] ∩ A (cid:54) = ∅ andthen y ∈ f [ A ] . (cid:3) Note that, if (
X, f ) is a dynamical system with X compact, then by Proposition 10, (cid:84) ∞ m =1 f m [ X ] is anonempty compact subspace of X satisfying f (cid:34) ∞ (cid:92) m =1 f m [ X ] (cid:35) = ∞ (cid:92) m =1 f m [ X ] . This means that the restricted function f (cid:22) (cid:84) ∞ m =1 f m [ X ] is onto (cid:84) ∞ m =1 f m [ X ]. In the case where X is aconnected space, so is (cid:84) ∞ m =1 f m [ X ], again by Proposition 10.2.1. Basic lemmas, definitions, and the first equivalences.
Before delving deep into the study ofdendrites and finite trees, we state two general lemmas (on arbitrary metric spaces) containing someuseful consequences of the failure of equicontinuity of a function. First note that, after some elementarymanipulation of the definition of equicontinuity, it is not hard to see that a continuous function f : X −→ X fails to be equicontinuous at the point x ∈ X if and only if there exists an ε >
0, a sequence of points( x k ) k ∈ N converging to x , and an increasing sequence of indices ( n k ) k ∈ N such that d ( f n k ( x k ) , f n k ( x )) > ε for all k ∈ N . In this case we will say that ε , ( x k ) k ∈ N , and ( n k ) k ∈ N witness the failure of equicontinuity of f at x . Lemma 11.
Let X be a metric space, let f : X −→ X be a continuous function, and suppose that f failsto be equicontinuous at x ∈ X . Then, for every n ∈ N , (1) f fails to be equicontinuous at f n ( x ) , and (2) there exists an ≤ i < n such that f n fails to be equicontinuous at f i ( x ) .Proof. Suppose that ε >
0, the sequence of points ( x k ) k ∈ N , and the sequence of indices ( n k ) k ∈ N witnessthe failure of equicontinuity of f at x , and let n ∈ N . To prove (1), assume without loss of generality that n > n ; then, the sequence ( f n ( x k )) k ∈ N (which converges to f n ( x ) by continuity of the function f n ), andthe increasing sequence ( n k − n ) k ∈ N of natural numbers, along with ε , witness the failure of equicontinuityof f at f n ( x ). This shows (1). For (2), apply the pigeonhole principle to assume, without loss of generality,that there is a fixed 0 ≤ i < n such that n k ≡ i mod n for all k ∈ N . Let m k be such that n k = nm k + i ;then, the sequence ( f i ( x k )) k ∈ N , which converges to f i ( x ), along with the increasing sequence ( m k ) k ∈ N ofnatural numbers and ε , witness the failure of equicontinuity of f n at f i ( x ). (cid:3) G. ACOSTA AND D. FERN ´ANDEZ
Before considering the specific case of dendrites, we introduce some more definitions and a general resultthat shall be used later.
Definition 12.
Let X be a metric space, and let f : X −→ X be a continuous function.(1) The ω -limit set of f at x ∈ X, is the set of all points y ∈ X for which there is an increasingsequence ( n i ) i ∈ N with lim i →∞ f n i ( x ) = y ; this set is denoted by ω ( x, f ).(2) A point x ∈ X is a recurrent point if x ∈ ω ( x, f ); the set of recurrent points of f is denoted byRec( f ).It is immediate that every periodic point is recurrent; the converse is not necessarily true. The nextproposition follows from known results and will be used for our Main Theorem. Proposition 13. If X is a finite tree and f : X −→ X is an equicontinuous surjective function, then f isa homeomorphism which furthermore is periodic.Proof. By [11, Proposition 2.4] and [4, Corollary 8], cf. [11, Corollary 3.2], f is a homeomorphism thatis pointwise-recurrent , i.e., such that x ∈ ω ( x, f ) for each x ∈ X . Hence, by [12, Theorem 4.4], f isperiodic. (cid:3) We will mention some standard facts about dendrites that will be used throughout the paper.
Definition 14. A dendrite is a locally connected continuum without simple closed curves.We proceed to mention several important well-known facts about dendrites. First of all, recall thatdendrites have the fixed point property, that is, whenever X is a dendrite and f : X −→ X is a continuousfunction, then Fix( f ) (cid:54) = ∅ ([13, Theorem 10.31]). Another important fact is that every subcontinuum of adendrite is again a dendrite ([13, Corollary 10.6]). We also use that every connected subset of a dendriteis arcwise connected ([13, Proposition 10.9]).If X is a dendrite and x, y ∈ X , then there is a unique (closed) arc in X joining x and y ; such an arcwill always be denoted by xy . Since continuous images of connected sets must be connected, for anycontinuous function f : X −→ X and every x, y ∈ X we have that f ( x ) f ( y ) ⊆ f [ xy ], by uniqueness of thearc f ( x ) f ( y ).Whenever X is a dendrite and Y is a subcontinuum of X , then there exists a retraction r Y : X −→ Y, called the first point function , such that for x ∈ X and y ∈ Y , r Y ( x ) is the first point in the arc xy (equipping such an arc with a linear order where x ≤ y ) that belongs to Y. The mapping r Y does notdepend on the specific y ∈ Y (see [13, Lemmas 10.24, 10.25 and Terminology 10.26]).Finally, the last well-known fact that we will use is that every dendrite has the property of being uniformlylocally arcwise connected , that is, for every ε > δ > d ( x, y ) < δ and x (cid:54) = y , the arc xy must have diameter < ε (as a matter of fact, every compact, connected and locallyconnected metric space has this property, which in this more general context must be phrased as: for every ε > δ > d ( x, y ) < δ and x (cid:54) = y, then there is an arc joining x and y with diameter < ε ; see [19, Theorem 31.4]). Proposition 15.
Let X be a dendrite, let f : X −→ X be a continuous function, and let x ∈ X . If Y ⊆ X \ { x } is a connected component of X \ { x } such that f ( x ) ∈ Y , then Y ∩ Fix( f ) (cid:54) = ∅ .Proof. Notice that cl X ( Y ) = Y ∪ { x } is a subcontinuum of X –hence cl X ( Y ) is itself a dendrite. Weconsider the first point function r cl X ( Y ) : X −→ cl X ( Y ) and note that, for y / ∈ cl X ( Y ), it must be the casethat r cl X ( Y ) ( y ) = x . Since cl X ( Y ) is a dendrite, it has the fixed point property; therefore the continuousfunction r cl X ( Y ) ◦ ( f (cid:22) cl X ( Y )) : cl X ( Y ) −→ cl X ( Y ) has a fixed point y . It is now easy to check that wemust have f ( y ) = y . (cid:3) QUICONTINUOUS MAPPINGS ON TREES 7
The following is another definition that will be crucial throughout the paper.
Definition 16.
Let X be a dendrite and k ∈ N . (1) The order of a point x ∈ X is the number of connected components of X \ { x } ;(2) a point x ∈ X is(a) an endpoint if its order is 1,(b) an ordinary point if its order is 2,(c) a branching point if its order is ≥ k ≥ X is a k -od if it contains exactly one branching point (called the vertex of X ), whichhas order k ; a 2 -od is simply defined to be an arc (we do not specify a vertex in this case);(4) X is a finite tree if it has only finitely many branching points and each of these branching pointshas a finite order.In a general topological space X , the order of a point x ∈ X is defined as the least cardinal number κ such that, for every open neighbourhood U of x, there exists another open neighbourhood V with x ∈ V ⊆ U and | ∂ ( V ) | ≤ κ (where ∂ ( V ) denotes the boundary of V in X ), cf. [13, Definition 9.3]; this willbe important towards the end of Section 4. If, however, the topological space X under consideration isa dendrite, then Definition 16 agrees with the general definition just mentioned (see [13, Lemma 10.12,Theorem 10.13 and Corollary 10.20.1]).We now show the equivalence of the first four conditions in Theorem 8. Proposition 17.
Let X be a finite tree and let f : X −→ X be a continuous function. Then the followingconditions are equivalent: (a) f is equicontinuous; (b) for some n ∈ N , the restriction f n (cid:22) (cid:84) ∞ m =1 f m [ X ] is the identity function; (c) for some n ∈ N , Fix( f n ) = (cid:84) ∞ m =1 f m [ X ] ; (d) Per( f ) = (cid:84) ∞ m =1 f m [ X ] .Proof. We consider first the case where f is surjective. Note that in such situation, X = (cid:84) ∞ m =1 f m [ X ] and f n (cid:22) (cid:84) ∞ m =1 f m [ X ] = f n for each n ∈ N . Moreover (b) asserts that f is periodic, (c) that Fix( f n ) = X forsome n ∈ N , and (d) that f is pointwise-periodic. Now we start the proof.( a ) ⇒ ( b ) : This implication follows from part (2) of Proposition 13.( b ) ⇒ ( c ) ⇒ ( d ) : Obvious.( d ) ⇒ ( a ) : By [5, Theorem 4.14], this implication holds not only on finite trees, but on every dendriteand with f being any continuous (surjective) function.We now consider the case of an arbitrary (not necessarily surjective) continuous function f : X −→ X. Since every finite tree is, in particular, a finite graph, we may use [11, Theorem 5.2] to see that f isequicontinuous if and only if so is f (cid:22) (cid:84) ∞ m =1 f m [ X ], and since the latter function is onto (cid:84) ∞ m =1 f m [ X ] (andsince Fix( f n ) = Fix( f n (cid:22) (cid:84) ∞ m =1 f m [ X ]) and Per( f ) = Per( f (cid:22) (cid:84) ∞ m =1 f m [ X ])), then the theorem followsfrom the surjective case. (cid:3) Expanding arcs.
We now establish a part of the equivalence of the Main Theorem.
Lemma 18.
Let X be a dendrite, and let f : X −→ X be a continuous function. Then the following areequivalent: G. ACOSTA AND D. FERN ´ANDEZ ( e (cid:48) ) X contains an f -expanding arc; ( f (cid:48) ) for some n ∈ N , the set Fix( f n ) is disconnected; ( j (cid:48) ) there exist points x, y ∈ X and n ∈ N such that x = f n ( x ) , y (cid:54) = f n ( y ) , and y ∈ xf n ( y ) .Proof. ( e (cid:48) ) ⇒ ( f (cid:48) ) Let ab be an f -expanding arc and fix an n ∈ N such that ab (cid:40) f n [ ab ]. We consider three casesaccording to whether both, exactly one, or none of a, b are fixed points for f n . Case 1: If f n ( a ) = a and f n ( b ) = b , use the fact that ab (cid:40) f n [ ab ] to get a c ∈ ab \ { a, b } such that f n ( c ) / ∈ ab . In particular, c / ∈ Fix( f n ) with c ∈ ab and a, b ∈ Fix( f n ), showing that Fix( f n ) is notarcwise connected, so Fix( f n ) is disconnected. Case 2: If f n ( a ) = a but f n ( b ) (cid:54) = b , use the fact that ab ⊆ f n [ ab ] to get a c ∈ ab \ { a, b } such that f n ( c ) = b . Let Z be the connected component of X \ { c } containing a. If b ∈ Z then, since Z isarcwise connected, we have c ∈ ab ⊂ Z ⊂ X \ { c } , a contradiction. Hence, b / ∈ Z . Letting Y (cid:54) = Z be the connected component of X \ { c } containing b , Proposition 15 guarantees the existence of a d ∈ Fix( f n ) ∩ Y . Then we have c / ∈ Fix( f n ), a, d ∈ Fix( f n ), and c ∈ ad , showing that Fix( f n ) isdisconnected. Case 3: : If f n ( a ) (cid:54) = a and f n ( b ) (cid:54) = b . Then (since ab ⊆ f n [ ab ]) we can find x, y ∈ ab \ { a, b } with f n ( x ) = a and f n ( y ) = b . Note that x (cid:54) = y . Equip ab with a linear order via a homeomorphism: [0 , −→ ab mapping 0 to a and 1 to b . If x < y , then a couple of applications of Proposition 15yield fixed points c, d ∈ Fix( f n ) such that x, y ∈ cd ; since x, y / ∈ Fix( f n ), this shows that Fix( f n ) isdisconnected. If, on the other hand, we have y < x , then notice that y ∈ ab = f n ( x ) f n ( y ) ⊆ f n [ xy ],so there is a y (cid:48) ∈ xy with f n ( y (cid:48) ) = y ; also, x ∈ yb = f n ( y (cid:48) ) f n ( y ) ⊆ f n [ yy (cid:48) ] and so there isan x (cid:48) ∈ yy (cid:48) with f n ( x (cid:48) ) = x . This way we have obtained x (cid:48) < y (cid:48) with f n ( x (cid:48) ) = f n ( x ) = a and f n ( y (cid:48) ) = f n ( y ) = b , thus, a couple of applications of Proposition 15 yield two fixed points c, d ∈ Fix( f n ) with x (cid:48) , y (cid:48) ∈ cd ; the fact that x (cid:48) , y (cid:48) / ∈ Fix( f n ) implies then that Fix( f n ) isdisconnected, and we are done.( f (cid:48) ) ⇒ ( j (cid:48) ) Let n ∈ N and a, b ∈ Fix( f n ) be such that ab (cid:42) Fix( f n ). Then there is a y ∈ ab with f n ( y ) (cid:54) = y .Considering the first point function r ab : X −→ ab and the point z = r ab ( f n ( y )), we must have that either y ∈ az or y ∈ bz . Since the situation is entirely symmetric, assume without loss of generality that y ∈ az and let x = a . Then we have f n ( x ) = x and y ∈ xz ⊆ xz ∪ zf n ( y ) = xf n ( y ).( j (cid:48) ) ⇒ ( e (cid:48) ) Under the assumptions we have xy (cid:40) xf n ( y ) = f n ( x ) f n ( y ) ⊆ f n [ xy ] and so xy is an f -expanding arc. (cid:3) We now proceed to prove that the failure of equicontinuity of a function on a finite tree implies theexistence of an f -expanding arc. The next result allows us to restrict any function without expanding arcsfrom a finite tree to a simpler subcontinuum. Lemma 19.
Let X be a finite tree, let f : X −→ X be a continuous function and let x ∈ Fix( f ) . If X hasno f -expanding arcs, then there is an f -invariant subspace Y ⊆ X ( that is, f [ Y ] ⊆ Y ) , where Y is a k -odfor some k ≥ , such that x ∈ int X ( Y ) .Proof. Let n be the order of x ∈ X. Since X (being a finite tree) only has finitely many branching points,there is an ε > x contains no branching points other than x . Thismeans that this closed ball can be written as (cid:83) ni =1 I i , where the I , . . . , I n are (free) arcs having only thepoint x in common. By continuity of f, f , . . . , f n , we can pick a δ with 0 < δ ≤ ε such that, if d ( y, z ) ≤ δ ,then d ( f i ( y ) , f i ( z )) < ε for every 0 ≤ i ≤ n . Let Z be the closed ball of radius δ centred at x , and define Y = Z ∪ f [ Z ] ∪ · · · ∪ f n [ Z ] . QUICONTINUOUS MAPPINGS ON TREES 9
We have that x belongs to the interior of Y ; moreover, by the choice of δ and Z , we have that Y ⊆ (cid:83) ni =1 I i .Since x is a fixed point for f , whenever y ∈ Y we must have the whole arc xy contained in Y . Hence Y is a k -od for some k ≥ k = n and x is the vertex of Y if n ≥ k = 2 if Y is an arc, i.e., n ∈ { , } ; moreover x is an interior point of the arc Y if n = 2, and it is an endpoint of the arc Y if n = 1). It remains to showthat Y is an f -invariant subspace, so let y ∈ Y and let us argue that f ( y ) ∈ Y . We can write y = f m ( z )for some z ∈ Z (this includes the case m = 0, interpreted as y = z ∈ Z ); there is nothing to do if m < n , soassume that y = f n ( z ) for z ∈ Z . Looking at the finite sequence of points z, f ( z ) , . . . , f n ( z ), the pigeonholeprinciple guarantees the existence of 0 ≤ i < j ≤ n and l ∈ { , . . . , n } such that f i ( z ) , f j ( z ) ∈ I l . Letus linearly order the arc I l by copying the order of [0 ,
1] along a homeomorphism mapping x to 0. If f i ( z ) < f j ( z ), this would mean that xf i ( z ) (cid:40) xf j ( z ) = f j − i ( x ) f j − i ( f i ( z )) ⊆ f j − i [ xf i ( z )]is an f -expanding arc, a contradiction. Therefore we must have f j ( z ) ≤ f i ( z ), implying that f j ( z ) ∈ xf i ( z ) = f i ( x ) f i ( z ) ⊆ f i [ xz ]and so there exists a point z (cid:48) ∈ xz ⊆ Z with f j ( z ) = f i ( z (cid:48) ). Hence y = f n ( z ) = f n − j + i ( z (cid:48) ) and so f ( y ) = f n − ( j − i )+1 ( z (cid:48) ) ∈ Y , and we are done. (cid:3) Remark . Suppose that X is a metric space, f : X −→ X is a continuous function, and Y ⊆ X is an f -invariant subspace with x ∈ int X ( Y ). If f is not equicontinuous at x , then the restriction f (cid:22) Y : Y −→ Y also fails to be equicontinuous at x : given ε >
0, if δ > f (cid:22) Y at x , thenmin { δ, d ( x, X \ int X ( Y )) } witnesses the equicontinuity of f : X −→ X at x . In particular, if we applyLemma 19 to a fixed point x such that either f fails to be equicontinuous at x , or x is an accumulationpoint of the set of points where f is not equicontinuous, then the subcontinuum Y that we obtain byLemma 19 will satisfy that f (cid:22) Y is not equicontinuous.For our next proof we will use a Ramsey-theoretic result known as Hindman’s theorem, so we proceed toexplain the relevant concepts and notations. Given a sequence ( n k ) k ∈ N of elements of N , its set of finitesums is defined as FS( n k ) k ∈ N = (cid:40)(cid:88) k ∈ F n k (cid:12)(cid:12)(cid:12)(cid:12) F ⊆ N is finite and nonempty (cid:41) = (cid:8) n k + · · · + x k m (cid:12)(cid:12) m ∈ N and k < · · · < k m (cid:9) , the set of all numbers that can be obtained by adding a finite amount of terms of the sequence ( n k ) k ∈ N without repetitions. The result known as Hindman’s theorem ([9, Theorem 3.1]) states that for any finitepartition of N , there exists an infinite (strictly increasing) sequence ( n k ) k ∈ N such that the set FS( n k ) k ∈ N is completely contained in a single cell of the partition.We will use a slightly stronger form of the aforementioned theorem. Given two (strictly increasing)sequences of natural numbers ( n k ) k ∈ N and ( m k ) k ∈ N , we say that the sequence ( m k ) k ∈ N is a sum subsystem of the sequence ( n k ) k ∈ N if FS( m k ) k ∈ N ⊆ FS( n k ) k ∈ N . With this terminology, we record the result that willbe used later. Its proof can be found in [10, Corollary 5.15]. Theorem 21 (Hindman) . For every infinite ( strictly increasing ) sequence ( n k ) k ∈ N and for every finitepartition { A , . . . , A m } of the set FS( n k ) k ∈ N , there exists an i ∈ { , . . . , m } and a sum subsystem ( m k ) k ∈ N of the sequence ( n k ) k ∈ N such that FS( m k ) k ∈ N ⊆ A i . Since N = FS(2 k − ) k ∈ N , the original version of Hindman’s theorem follows immediately from Theorem 21above. Theorem 22.
Let X be a finite tree and let f : X −→ X be a continuous function. If f is not equicontin-uous, then X has an f -expanding arc. Proof.
Take an x ∈ X such that f is not equicontinuous at x . We have two cases. Case 1:
The point x is eventually periodic (i.e., there is k ∈ N such that f k ( x ) is a periodic point;equivalently, the set { f n ( x ) (cid:12)(cid:12) n ∈ N } is finite). This means that, replacing x by some f k ( x ) if necessary(and using clause (1) of Lemma 11), we may assume that x is a periodic point, say with period n . Now useclause (2) of Lemma 11 to find an i < n such that f n fails to be equicontinuous at y = f i ( x ), and noticethat y is a fixed point for f n . If X has an f n -expanding arc, then this is also an f -expanding arc andwe are done. If, on the contrary, there are no f n -expanding arcs, then we can use Lemma 19 to obtainan f n -invariant subspace Y ⊆ X such that Y is a k -od for some k ≥
2, and y ∈ int X ( Y ). By Remark 20,the restricted function f n (cid:22) Y : Y −→ Y fails to be equicontinuous at y , and so by Proposition 6, Y mustcontain an f n -expanding arc I . Then I ⊆ X is also an f -expanding arc, and we are done. Case 2:
The point x is not eventually periodic. Then the set { f n ( x ) (cid:12)(cid:12) n ∈ N } is infinite. The space X is afinite tree and so it can be written as a union of finitely many maximal free arcs I , . . . , I t such that anytwo distinct I i , I j have at most one (branching) point in common. Now we define subsets A , A , . . . , A t of N as follows: n ∈ A iff f n ( x ) is a branching point of X and, for each i ∈ { , . . . , t } , n ∈ A i iff f n ( x ) ∈ I i and f n ( x ) is not a branching point of X. Clearly A i ∩ A j = ∅ for every i, j ∈ { , , . . . , t } with i (cid:54) = j. Given i ∈ { , , . . . , t } the set A i can be empty. Since the set { f n ( x ) (cid:12)(cid:12) n ∈ N } is infinite, there exist m , m , . . . , m r ∈ { , , . . . t } such that A m j (cid:54) = ∅ for every j ∈ { , , . . . , r } and A i = ∅ for each i ∈ { , , . . . , t } \ { m , m , . . . , m r } . Hence { A m , A m , . . . , A m r } is a finite partition of N . Theorem 21provides us with a j ∈ { , , . . . , r } and an infinite strictly increasing sequence n < · · · < n k < n k +1 < · · · of natural numbers, such that the set FS( n k ) k ∈ N ⊆ A m j . Since the f n ( x ) as n ∈ N varies are pairwisedistinct and X only has finitely many branching points, A is finite. Therefore m j (cid:54) = 0 and so { f n ( x ) (cid:12)(cid:12) n ∈ FS( n k ) k ∈ N } ⊆ I m j . Use a homeomorphism of [0 ,
1] onto I m j to equip I m j with a linear order ≤ . We now partition theset FS( n k ) k ≥ according to whether f n ( x ) < f n + n ( x ) or f n + n ( x ) < f n ( x ); a further application ofTheorem 21 allows us to obtain a sum subsystem ( n k ) k ≥ of ( n k ) k ≥ such that FS( n k ) k ≥ is contained inone piece of this partition. This means that there is an R ∈ { >, < } such that, f n + n ( x ) R f n ( x ) , for every n ∈ FS( n k ) k ≥ . Continuing this process by induction, we obtain, for each K ∈ N , a sum subsystem ( n K +1 k ) k ≥ K +1 of( n Kk ) k ≥ K +1 and an R K ∈ { >, < } such that f n KK + n ( x ) R K f n ( x ) , for each n ∈ FS( n K +1 k ) k ≥ K +1 . Now, an application of the pigeonhole principle allows us to obtain an infinite increasing sequence ( K k ) k ∈ N such that all the R K k are equal, say, without loss of generality, to > . What this means is that, if wedefine the sequence ( n k ) k ∈ N by n k = n K k K k , then for every K ∈ N and each n ∈ FS( n k ) k ≥ K +1 we have f n ( x ) < f n K + n ( x ).Now, for each K ∈ N we define a point y K ∈ I m j by y K = sup { f n ( x ) (cid:12)(cid:12) n ∈ FS( n k ) k ≥ K } . Since f n ( x ) < f n K + n ( x ) for every n ∈ FS( n k ) k ≥ K +1 , we must have y K +1 ≤ y K for every K ∈ N . We maynow break the proof into two further subcases (recall that d is the metric on X ). Subcase 2.A:
There is a K ∈ N such that y K +1 = y K . Let y = y K = y K +1 and note that, for every N ∈ N , there is an l N ∈ FS( n k ) k ≥ K +1 with f l N ( x ) < y and d ( f l N ( x ) , y ) < N .
QUICONTINUOUS MAPPINGS ON TREES 11
We have f l N ( x ) < f n K + l N ( x ) < y ; in particular, we also have d ( f n K + l N ( x ) , y ) < N . It follows thatlim N →∞ f l N ( x ) = y and lim N →∞ f n K ( f l N ( x )) = y. By continuity of the function f n K , we may conclude that f n K ( y ) = y . Thus, all the points y, f ( y ) , . . . , f n K − ( y )are fixed points of the function f n K . If X contains an f n K -expanding arc, then this arc is also f -expanding and we are done, so assume otherwise. Then we may apply Lemma 19 to each of thepoints y, f ( y ) , . . . , f n K − ( y ) to get f n K -invariant subcontinua Y , Y , . . . , Y n K − ⊆ X such that, for every i ∈ { , , . . . , n K − } , we have f i ( y ) ∈ int X ( Y i ) and each Y i is a k i -od forsome k i ≥
2. Let ε > i < n K , the ball centered at f i ( y ) with radius ε iscontained in Y i . By the continuity of the functions f, f , . . . , f n K − we get a δ > d ( z, y ) < δ , then d ( f i ( z ) , f i ( y )) < ε. Hence, for each i < n K , if d ( z, y ) < δ then f i ( z ) ∈ int X ( Y i ).Note that y contains points of the form f n ( x ) arbitrarily close, and all the points of the form f n ( x )are points where the function f is not equicontinuous (by clause (1) of Lemma 11). Hence we canfind a z with d ( z, y ) < δ such that f is not equicontinuous at z ; now use clause (2) of Lemma 11 toget i < n K such that f n K is not equicontinuous at f i ( z ) ∈ int X ( Y i ). Since Y i is f n K -invariant, wemay conclude that f n K (cid:22) Y i is not an equicontinuous function (see Remark 20). Since Y i is a k i -od,by Proposition 6, the subcontinuum Y i of X must have an f n K -expanding arc, and we are done. Subcase 2.B: y K +1 < y K for every K ∈ N . Then let y = inf { y K (cid:12)(cid:12) K ∈ N } . For each K ∈ N fix an m K ∈ FS( n k ) k ≥ K such that y K +1 < f m K ( x ) < y K . If for some K ∈ N , it is not the case that y < f n K ( y ), then we must have yf m K +1 ( y ) (cid:40) f n K ( y ) f n K + m K +1 ( x ) ⊆ f n K [ yf m K +1 ( x )]and therefore there is an f -expanding arc and we are done, so assume that for all K ∈ N wehave y < f n K ( y ). The points f m k ( x ) for k > K are arbitrarily close to y and they all satisfy f n K ( f m k ( x )) = f n K + m k ( x ) ≤ y K , so by continuity of f n K we have f n K ( y ) ≤ y K .We define connected subspaces Y , Y ⊆ X as follows. Y is the connected component of X \{ f m ( x ) } that does not contain y , and Y is the connected component of X \ { f n ( y ) } containing y . Since y < f n ( y ) ≤ y < f m ( x ) < f n + m ( x ) , and all such points belong to the maximal free arc I m j of X , we can write X = Y ∪ f n ( y ) f m ( x ) ∪ Y , and the union is disjoint. Since f n + m ( x ) ∈ Y , by Proposition 15 there is a z ∈ Y ∩ Fix( f n );now if we let K be sufficiently large that y K < f n ( y ) then we will have f n K ( y ) ∈ Y and so againby Proposition 15 there exists a z ∈ Y ∩ Fix( f n K − n ). Letting N = ( n K − n ) n , we get that z , z ∈ Fix( f N ), and f m ( x ) ∈ z z \ Fix( f N ). Hence Fix( f N ) is a disconnected set, and so byLemma 18, X must have an f -expanding arc. (cid:3) The Ellis remainder and ultrafilter-limits
In this section we introduce the notion of ultrafilter-limits and point out the relation of this concept withthat of the Ellis remainder, with the objective of establishing the equivalence of items (a), (e), (h) and (i)from Theorem 8. We begin by recalling the relevant definitions regarding ultrafilters.
Definition 23. (1) An ultrafilter on N is a family u of subsets of N such that (a) u is nonempty and ∅ / ∈ u ;(b) if A, B ∈ u, then A ∩ B ∈ u ;(c) if A ∈ u and A ⊆ B ⊆ N , then B ∈ u ;(d) whenever N = A ∪ B , then either A ∈ u or B ∈ u ; moreover, if A and B are disjoint thenexactly one of the two options holds.(2) An ultrafilter u on N is principal if there exists an n ∈ N such that u = { A ⊆ N (cid:12)(cid:12) n ∈ A } ; otherwisewe say that u is nonprincipal .(3) We use the symbol β N to denote the set of all ultrafilters on N , and we denote with N ∗ the set ofall nonprincipal ultrafilters on N .(4) Given a metric space ( X, d ), a sequence ( x n ) n ∈ N of points on X , and an ultrafilter u ∈ β N , we saythat x is the u ultrafilter-limit of ( x n ) n ∈ N , in symbols x = u -lim n →∞ x n , if for every ε > { n ∈ N (cid:12)(cid:12) d ( x, x n ) < ε } ∈ u .(5) Given a metric space X , a function f : X −→ X , and an ultrafilter u ∈ β N , we define the u ultrafilter-limit function f u : X −→ X (also called the u -th iterate of f ) by f u ( x ) = u -lim n →∞ f n ( x ).Given a compact metric space X, a continuous function f : X −→ X and x ∈ X , it can be shown that ω ( x, f ) = { f u ( x ) (cid:12)(cid:12) u ∈ N ∗ } . A few comments about the above definitions are in order. For each n ∈ N , it is common to identify thenatural number n with the principal ultrafilter u n = { A ⊆ N (cid:12)(cid:12) n ∈ A } ; this way we can think of N asa subset of β N , and we have N ∗ = β N \ N . Furthermore, one can topologize β N by declaring the sets¯ A = { u ∈ β N (cid:12)(cid:12) A ∈ u } to be open, for each A ⊆ N ; this endows β N with a compact Hausdorff topologycontaining N as a discrete dense subspace ([10, Lemma 3.17 and Theorems 3.18 and 3.28]). Regarding theconcept of a u -limit, it is worth pointing out that, in a compact metric space X , every sequence ( x n ) n ∈ N ofpoints will have a unique u -limit (for every u ∈ β N )([10, Theorem 3.48]). Moreover, if u n is the principalultrafilter { A ⊆ N (cid:12)(cid:12) n ∈ A } , then u n -lim m →∞ x m = x n ; similarly (and as a consequence of the above), for afunction f : X −→ X we will have that f u n = f n . Thus, no confusion should arise if we sometimes abusenotation and write n instead of u n .Furthermore, it is possible to equip β N with a right-topological semigroup operation, denoted by +. Thatis, + is an associative binary operation on β N such that, for each fixed u ∈ β N , the function v (cid:55)−→ u + v iscontinuous. The operation is given by the formula u + v = { A ⊆ N (cid:12)(cid:12) { n ∈ N (cid:12)(cid:12) { m ∈ N (cid:12)(cid:12) n + m ∈ A } ∈ v } ∈ u } . This operation extends the usual sum on N , in the sense that, if n, m ∈ N and u n , u m are the correspondingprincipal ultrafilters, then u n + u m = u n + m , although + is not commutative on all of β N . It is possible toverify that, for any u, v ∈ β N , we have f u ◦ f v = f u + v (see [2, p. 38]).As we mentioned in the Introduction, the equation E ( X, f ) = { f u (cid:12)(cid:12) u ∈ β N } , shown in [8, Theorem 2.2] and which holds for every continuous function f : X −→ X on a compact metricspace X , is the main reason why obtaining information about the ultrafilter-limit functions f u has a greatdeal of importance within the study of the dynamical system ( X, f ). At this moment, we aim to provethat the existence of expanding arcs implies the discontinuity of ultrafilter-limit functions. We begin byintroducing a definition that will help to expedite the statement of the subsequent lemmas.
Definition 24.
Let X be a metric space. QUICONTINUOUS MAPPINGS ON TREES 13 (1) Let I ⊆ X be an arc, and let ( x n ) n ∈ N be a sequence of elements of I . We say that the sequence is I -monotone if it is monotone (i.e., either increasing or decreasing) when viewed as a sequence onthe unit interval [0 ,
1] via a homeomorphism : I −→ [0 , x n ) n ∈ N ismonotone if x n +1 ∈ x n x n +2 for each n ∈ N (noting that x n x n +2 ⊆ I ).(2) If g : X −→ X a continuous function, a sequence ( x n ) n ∈ N of elements of some arc I ⊆ X is said tobe g -backwards if it is I -monotone and for each n ∈ N we have g ( x n +1 ) = x n . Remark . Note that, by compactness of an arc and monotonicity of backward sequences, any g -backwardsequence on a dendrite is always convergent. Furthermore, the limit of the sequence is a fixed point of g ,since g ( lim n →∞ x n ) = lim n →∞ g ( x n ) = lim n →∞ x n − = lim n →∞ x n . Lemma 26.
Let X be a dendrite, f : X −→ X be a continuous function, and suppose that there is an f -expanding arc I ⊆ X . Then the following two conditions hold: (1) for some m ∈ N there exists an f m -backward sequence ( y n ) n ∈ N in I ; (2) the set Per( f ) is disconnected.Proof. If there is an f -expanding arc I in X then, by Lemma 18, there exist points x, y ∈ X and n ∈ N such that f n ( x ) = x , f n ( y ) (cid:54) = y , and y ∈ xf n ( y ) ⊆ f n [ xy ], so we can find a y ∈ xy \ { y } such that f n ( y ) = y . Now y ∈ xy = f n ( x ) f n ( y ) ⊆ f n [ xy ], so we can find a y ∈ xy \ { y } such that f n ( y ) = y .Continuing by induction, if we already know y , . . . , y k with f n ( y i ) = y i − and y i ∈ xy i − \ { y i − } , then y k ∈ xy k − = f n ( x ) f n ( y k ) ⊆ f n [ xy k ], and so there exists a y k +1 ∈ xy k \ { y k } such that f n ( y k +1 ) = y k .This way we obtain a sequence ( y n ) n ∈ N which is f n -backward. So (1) holds.To show (2), we use the points x, y and the sequence ( y n ) n ∈ N obtained in (1). Since y ∈ xf n ( y ), byProposition 15 there is a point z ∈ Fix( f n ) such that y ∈ xz ; then we have x, z ∈ Per( f ), so it sufficesto show that xz \ Per( f ) (cid:54) = ∅ . If y / ∈ Per( f ) we are done, so assume that y = Per( f ), say with period k .Then { f n ( y ) (cid:12)(cid:12) n ∈ N } = { y, f ( y ) , . . . , f k − ( y ) } ; since the y n are pairwise distinct we can choose an n ∈ N such that y n / ∈ { y, f ( y ) , . . . , f k − ( y ) } . Then f m ( y n ) ∈ { y n − , . . . , y , y, f ( y ) , . . . , f k − ( y ) } for all m ∈ N , thus for each m ∈ N we have f m ( y n ) (cid:54) = y n and so y n / ∈ Per( f ). Since y n ∈ xz , the proof is finished. (cid:3) Corollary 27.
For a dendrite X and a continuous function f : X −→ X , the following are equivalent: (e) there is no f -expanding arc in X ; (g) the set Per( f ) is connected.Proof. We prove both implications of this biconditional by contrapositive. Suppose that Per( f ) isdisconnected, and find x, y ∈ Per( f ) such that there exists a z ∈ xy \ Per( f ). If x has period n and y hasperiod m , then we have x, y ∈ Fix( f nm ); as z is not periodic, we have z ∈ xy \ Fix( f nm ). Hence the setFix( f nm ) is disconnected and so, by Lemma 18, X contains an f -expanding arc. Conversely, if X containsan f -expanding arc, use Lemma 26. (cid:3) Now, in order to use g -backward sequences to deduce discontinuity of elements in the Ellis remainder, wewill introduce a fairly stronger definition that allows us to work in a slightly more general context. Inwhat follows, it will be convenient that the indexing of our sequences starts at 0 rather than at 1. Definition 28.
Let X be a compact metric space, and let g : X −→ X be a continuous function. Asequence ( x n ) n ∈ N ∪{ } of elements of X will be said to be g -divergent if the following three conditionshold:(1) x = lim n →∞ x n exists in X ; (2) for each n ∈ N , g ( x n +1 ) = x n (this implies that g ( x ) = x );(3) there exists an open neighbourhood U ⊆ X of x such that U ∩ { g n ( x ) (cid:12)(cid:12) n ∈ N } = ∅ .It is not hard to see that g -divergent sequences can only exist if g fails to be equicontinuous. As a matterof fact, much more is true, as seen in the following theorem. Theorem 29.
Let X be an arbitrary compact metric space and let g : X −→ X be a continuous function.If there is an m ∈ N such that X contains a g m -divergent sequence, then for every nonprincipal ultrafilter u ∈ N ∗ , the function g u is discontinuous.Proof. Let ( x n ) n ∈ N ∪{ } be the hypothesized g m -divergent sequence, let x = lim k →∞ x k , and let U be anopen set containing x such that U ∩ { g mn ( x ) (cid:12)(cid:12) n ∈ N } = ∅ .Now let u ∈ N ∗ be an arbitrary nonprincipal ultrafilter. There exists a unique 0 ≤ i < m such that m N + i ∈ u , so that m N ∈ u − i . This means that it makes sense to consider the Rudin–Keisler image v of the ultrafilter u − i under the mapping : m N −→ N given by mk (cid:55)−→ k . So we have that mv + i = u (where mv denotes the Rudin–Keisler image of the ultrafilter v under the mapping k (cid:55)−→ mk ).Define a new sequence ( y n ) n ∈ N ∪{ } by letting y n = g m − i ( x n ), and let y = g m − i ( x ). Since the sequenceof x n converges to x and g m − i is a continuous function, the sequence of y n will converge to y . We nowproceed to observe that g u ( y ) = g mv + i ( g m − i ( x )) = ( g m ) v ( g i ( g m − i ( x )))= ( g m ) v ( g m ( x )) = ( g m ) v ( x ) = x ∈ U, and, for each k ∈ N , we have g u ( y k ) = g mv + i ( g m − i ( x k )) = ( g m ) v ( g m ( x k )) = g mv ( x k − ) . By definition of ultrafilter-limits, g mv ( x k − ) must be an accumulation point of the set { g mn ( x k − ) (cid:12)(cid:12) n ∈ N } .However, for n > k − g mn ( x k − ) = g m ( n − k +1) ( x ) / ∈ U , so g u ( y k ) / ∈ U for every k ∈ N , andtherefore the sequence ( g u ( y k )) k ∈ N does not converge to x = g u ( y ), showing that the function g u isdiscontinuous at y , and we are done. (cid:3) The previous lemma works for every compact metric space. For certain dendrites, there is a relationbetween g -backwards sequences and g -divergent sequences. Lemma 30.
Let X be a dendrite with only finitely many branching points, and let g : X −→ X be acontinuous function. If there is an arc I ⊆ X such that I contains a g -backwards sequence, then thereexists an m ∈ N such that X has a g m -divergent sequence.Proof. Let ( x n ) n ∈ N ∪{ } be a g -backwards sequence in the arc I , and let x = lim n →∞ x n . Notice that x is afixed point of g , and therefore lim k →∞ g k ( x ) = x .Now let us fix some notation. First of all, since X has only finitely many branching points, we may shrink I and drop finitely many terms of the sequence (and shift indices afterwards so that our sequence indexingstill starts at 0) ( x n ) n ∈ N { } to ensure that the arc I is free (that is, it contains no branching points otherthan possibly the endpoints). Now linearly order the arc I via a homeomorphism with [0 ,
1] in such a waythat x < x . Then the monotonicity of the g -backwards sequence ( x n ) n ∈ N ∪{ } means in this case that thesequence is increasing. Now let r I : X −→ I be the first point function from X onto the subcontinuum I of X . We will analyze the g -orbit of x . There are two cases to consider.
Case 1:
For every m ∈ N , r I ( g m ( x )) ≤ x . In this case, for each fixed n ∈ N ∪ { } we have that g m + n ( x n ) = g m ( x ), and so r I ( g m + n ( x n )) ≤ x < x for every m ∈ N . Since X is a dendrite and henceuniformly locally arcwise connected, there must be a δ > d ( x, z ) < δ and x (cid:54) = z ,then the arc xz must have diameter smaller than that of the arc x x . In particular, if r I ( z ) ≤ x , then QUICONTINUOUS MAPPINGS ON TREES 15 d ( x, z ) ≥ δ . So if we let U be the ball centred at x with radius δ , then for every n ∈ N it is the case that g n ( x ) / ∈ U , and consequently the sequence ( x n ) n ∈ N itself is already g -divergent. Case 2:
There exists an m ∈ N such that x ≤ r I ( g m ( x )). Fix one such m , and notice that the function r I ◦ ( g m (cid:22) I ) : I −→ I satisfies r I ( g m ( x m )) = r I ( x ) = x ≤ x m and x ≤ r I ( g m ( x )) . Therefore (by a standard result for continuous functions in the unit interval) this function must have afixed point in x x m , that is, there is a z ∈ x x m with z = r I ( g m ( z )). Since the arc I is free in X , wehave that r I ( w ) is one of the endpoints of I whenever w / ∈ I . Since z ∈ x x m \ { x , x m } (so z is aninterior point of I ), from z = r I ( g m ( z )) it follows that z = g m ( z ) and so z is actually a fixed point ofthe function g m .Now x x m = g m ( x m ) g m ( x m ) ⊆ g m [ x m x m ], so there must exist a z ∈ x m x m such that g m ( z ) = z . Wecontinue this process by induction: given a z n ∈ x nm x ( n +1) m = g m ( x ( n +1) m ) g m ( x ( n +2) m ) ⊆ g m [ x ( n +1) m x ( n +2) m ] , we find a z n +1 ∈ x ( n +1) m x ( n +2) m such that g m ( z n +1 ) = z n . This way we obtain a monotone sequence( z n ) n ∈ N ∪{ } , with limit x , which is g m -backwards and where z ∈ Fix( g m ). Since z (cid:54) = x , any open set U containing x and not containing z will satisfy ( g m ) n ( z ) = z / ∈ U , for every n ∈ N . Therefore thesequence ( z n ) n ∈ N ∪{ } is g m -divergent. (cid:3) We are ready to prove the Main Theorem of this paper.
Proof of Theorem 8 ( and of clause (2) of Remark 9 ) . The equivalence of (a), (b), (c) and (d) is establishedin Proposition 17. The equivalence of (e) and (f) is Lemma 18, and that of (e) and (g) is Corollary 27; inboth cases this equivalence works for arbitrary dendrites and so this establishes clause (2) of Remark 9.Finally, (e) implies (a) by Theorem 22; (a) implies (h) easily (by the remark in the Introduction right afterDefinition 2), and it is obvious that (h) implies (i) and that (d) implies (g). We also have that (i) implies(e): by contrapositive, if there exists an f -expanding arc in X then there is an f m -backward sequence forsome m , by Lemma 26; this yields an n ∈ N such that there is an f mn -divergent sequence by Lemma 30,and this in turn implies that there is no u ∈ N ∗ such that f u is continuous, by Theorem 29. The last chainof implications establishes the equivalence of (a) with (e), (h) and (i), which finishes the proof. (cid:3) Examples and open problems
This section contains examples showing that the previous results cannot be extended to other kinds ofdendrites. Theorem 8 holds for finite trees, and trees are dendrites satisfying two additional conditions:that they have finitely many branching points, and that each branching point has finite order. We showexamples of dendrites where one of these two conditions fails. Afterwards, we finish the paper by makinga few observations about functions defined on finite graphs.4.1.
Dendrites with finitely many branching points.
Recall that from clause (1) of Remark 9,conditions (a) and (d) from Theorem 8 are still equivalent if X is merely a dendrite with finitely manybranching points. In this subsection we proceed to exhibit an example of a dendrite, and two continuousfunctions defined on it, which together show that none of the other items from Theorem 8 are generalizableto dendrites with finitely many branching points (meaning that the hypothesis that all branching pointsare of finite order is really necessary in Theorem 8). Example 31.
A dendrite X with a unique branching point, which has infinite order, and continuousfunctions f, g : X −→ X such that f satisfies all conditions from (e) through (i) of Theorem 8 but fails tobe equicontinuous, while g is equicontinuous but fails to satisfy conditions (b) and (c) of Theorem 8 . I I I I I − I − I − v Figure 1.
The dendrite X that has v as its only branching point of infinite order.For other purposes, the dendrite X together with the function f , appear in [5, Example 5.1]. We reproducetheir description here for three reasons: for the reader’s convenience, to point out a few observationsabout the function f that are not made in [5], and in order to be able to also describe the function g . Webuild X by taking infinitely many disjoint arcs indexed by Z , { I n (cid:12)(cid:12) n ∈ Z } , with each I n of length | n | , andidentifying in a single point v (the vertex ) one end of each I n . The result X = (cid:83) n ∈ Z I n is a dendrite witha single infinite-order branching point v , as in Figure 1.We now describe the basic building blocks that will be used in the construction of f and g . For each n ∈ Z ,let us consider a function h n : I n −→ I n +1 defined by fixing v , and, for each e ∈ (0 , | n | ], if x is the uniqueelement of I n \ { v } at distance e from v , then h n ( x ) is the unique element of I n +1 \ { v } at distance 2 − n | n | e (at distance e in the case n = 0) from v . Note that h n maps I n homeomorphically onto I n +1 .We let f : X −→ X be defined by f = (cid:83) n ∈ Z h n , that is, by f ( v ) = v and f ( x ) = h n ( x ) whenever n is theunique element in Z so that x ∈ I n \ { v } . The sequence ( x k ) k ∈ N , where x k is the endpoint of I − k that isdistinct from v (this sequence converges to v ), together with ε = and the sequence of indices ( n k ) k ∈ N given by n k = k , witness the failure of the equicontinuity of f at v (since f n k ( x k ) = x , where x is theendpoint of I that is distinct from v ). For each n ∈ N , we have(2) Fix( f n ) = Per( f ) = { v } , which is a connected set. Using (2) it is straightforward to see that property ( j (cid:48) ) of Lemma 18 is notsatisfied. Hence, by the same lemma, X has no f -expanding arcs. Note that ω ( v, f ) = { v } (cid:40) X = ∞ (cid:92) m =1 f m [ X ] . Moreover, for every x ∈ X we have lim n →∞ f n ( x ) = v , which implies that, for each nonprincipal ultrafilter u ∈ N ∗ , it follows that f u : X −→ X is the function with constant value v , which is continuous.We now proceed to describe the function g . We stipulate that g (cid:22) (cid:83) n ≤ I n is the identity function. Foreach positive m ∈ N \ { n (cid:12)(cid:12) n ∈ N ∪ { }} , we let g (cid:22) I m = h m ; finally, we let g (cid:22) I m = h − m − (2 n − − ◦ h − m − (2 n − − ◦ · · · ◦ h − m − , whenever m = 2 n with n ≥ . Hence we have g (cid:22) I n : I n −→ I n − +1 . Therefore, for every n ≥
2, the function g will cyclicallypermute the finite sequence of arcs ( I n − +1 , I n − +2 , . . . , I n ), in such a way that f n − (cid:22) (cid:83) n i =2 n − +1 I i isthe identity function (and g will fix every point in each of the I m for m ∈ Z with m ≤ g ) = X , and so g will be equicontinuous by [5, Theorem 4.14]; at thesame time, although f is pointwise-periodic, X contains poins of arbitrarily high period (if x ∈ I m for2 n − + 1 ≤ m ≤ n , n ∈ N \ { } , then the period of x is equal to 2 n − ) and therefore, for every n ∈ N , wehave Fix( g n ) (cid:54) = (cid:84) ∞ m =1 g m [ X ] and g n (cid:22) (cid:84) ∞ m =1 g m [ X ] is not the identity function. QUICONTINUOUS MAPPINGS ON TREES 17
Dendrites with branching points of finite order.
We now show that, if we drop the requirementthat the dendrite X has finitely many branching points, then none of the equivalences of equicontinuityfrom Theorem 8 holds. The first few equivalences can be seen to fail by looking at [5, Example 5.4],which is the Gehman dendrite X (as a matter of fact, this dendrite is described and pictured in [13,Example 10.39]) with all branching points of finite order (with infinitely many branching points), and asurjective equicontinuous function f : X −→ X such that Per( f ) (cid:54) = X (consequently, X = (cid:84) ∞ m =1 f m [ X ]and Per( f ) (cid:54) = (cid:84) ∞ m =1 f m [ X ]). Therefore f is equicontinuous but fails to satisfy conditions (b), (c) and (d)of Theorem 8.Now for the remaining equivalences, the following example finishes our analysis. Example 32.
A dendrite X with infinitely many branching points ( each of which has finite order ) and acontinuous function f : X −→ X that fails to be equicontinuous, but satisfies (e) through (i) of Theorem 8.We build X as a subset of R as follows. We let K = [0 , × { } and, for each n ∈ N ∪ { } , we let I n = (cid:8) n (cid:9) × (cid:2) , n (cid:3) and J n = (cid:8) n (cid:9) × (cid:2) − n , (cid:3) . Define X = K ∪ (cid:32) ∞ (cid:91) n =0 I n (cid:33) ∪ (cid:32) ∞ (cid:91) n =0 J n (cid:33) . For notational convenience, we write K = (cid:83) ∞ n =1 K n where K n = (cid:2) n , n − (cid:3) × { } for each n ∈ N . Now wedefine the continuous function f : X −→ X as follows. First make f (cid:22) K the identity function. Now, foreach n ∈ N , f (cid:22) I n is given as follows: for (cid:0) n , y (cid:1) ∈ I n (0 ≤ y ≤ n ), we let f (cid:18) n , y (cid:19) = (cid:40) (cid:0) n + 2 y, (cid:1) , if 0 ≤ y ≤ n +1 ; (cid:0) n − , (cid:0) y − n +1 (cid:1)(cid:1) , if n +1 ≤ y ≤ n , so that f maps I n homeomorphically onto K n ∪ I n − . Furthermore, f (cid:22) I is defined by letting f (0 , y ) =(0 , − y ) so that f maps I homeomorphically onto J . Finally, for each n ∈ N ∪ { } , we define f (cid:22) J n byletting f (cid:18) n , y (cid:19) = (cid:40) (cid:0) n + y, (cid:1) , if − n +1 ≤ y ≤ (cid:0) n +1 , y + n +1 (cid:1) , if − n ≤ y ≤ − n +1 , whenever (cid:0) n , y (cid:1) ∈ J n ( − n ≤ y ≤ f maps J n homeomorphically onto K n +1 ∪ J n +1 . Thedendrite X , as well as the function f : X −→ X , are depicted in Figure 2.We will denote by v the point (0 , I n , (cid:0) n , n (cid:1) n ∈ N (whichconverges to v ), along with the increasing sequence of indices ( n ) n ∈ N and ε = 1, witness the failure ofequicontinuity of f at v (since f n (cid:0) n , n (cid:1) = (1 , ∈ I , which is at distance > f n ( v ) = v ). So f isnot equicontinuous.For every n ∈ N we have(3) Fix( f n ) = Per( f ) = K. Thus the sets Fix( f n ), as well as Per( f ), are all connected. Using (3) it is straightforward to see thatproperty ( j (cid:48) ) of Lemma 18 is not satisfied. Hence, by the same lemma, X has no f -expanding arcsIt remains to show that the function f u is continuous, whenever u is a nonprincipal ultrafilter. To do this,we define an auxiliary (continuous) function g : X −→ X as follows. First of all, g (cid:22) K will be the identityfunction. For every n ∈ N ∪ { } , we have g (cid:18) n , n (cid:19) = v = g (cid:18) n , − n (cid:19) . I f [ I ] J f [ J ] I f [ I ] J f [ J ] I f [ I ](0 ,
0) = v f ( v ) = v f · · · · · ·· · ·· · · K K Figure 2.
The dendrite X that has infinitely many branching points of finite order. Thecontinuous function f : X −→ X is not equicontinuous.Next, if (cid:0) n , y (cid:1) ∈ I n is not an endpoint (that is, if 0 < y < n ) then we let m ∈ N ∪ { } be unique suchthat 12 n − n + m ≤ y < n − n + m +1 , and define g (cid:18) n , y (cid:19) = (cid:18) n − m + 2 m +1 (cid:18) y − n − n + m (cid:19) , (cid:19) if m < n, and g (cid:18) n , y (cid:19) = (cid:18) − n (cid:18) y − (cid:18) n − n (cid:19)(cid:19) , (cid:19) if n ≤ m. Finally, if (cid:0) n , y (cid:1) ∈ J n is not an endpoint (i.e., − n < y < g (cid:18) n , y (cid:19) = (cid:18) n + y, (cid:19) . The function g is continuous; furthermore, for each x ∈ X we have lim n →∞ f n ( x ) = g ( x ) and therefore,for every nonprincipal ultrafilter u , it must be the case that f u = g . Thus the function f u is continuousfor every nonprincipal ultrafilter u .4.3. Finite graphs.
The case of finite graphs (compact connected polyhedra) might be harder to analyzethan the case of finite trees. The first difficulty that arises is the fact that the unit circle S is a finitegraph (any cyclic graph is represented by this space), and there are continuous functions f : S −→ S (such as, e.g., rotations by an irrational angle), which, though equicontinuous and surjective, lack anyperiodic points. Thus, items (a), (b), (c) and (d) from Theorem 8 are no longer equivalent if one attemptsto replace “finite tree” with “finite graph” in its statement. For finite graphs with at least one branchingpoint or at least one endpoint, however, the equivalence between items (a), (b) and (c) can be establishedby adapting the argument in the proof of Proposition 17. The following example shows that the equivalencebetween statements (a) and (f) from Theorem 8 does not hold on finite graphs, even if one demands thatthe graphs have branching points or endpoints. Example 33.
A finite graph X ( with two branching points and two endpoints ) and a continuous function f : X −→ X such that f is equicontinuous but the set Fix( f ) is disconnected. The graph is defined as a subset of R by X = (cid:8) ( x, y ) ∈ R (cid:12)(cid:12) x ∈ [ − , − ∪ [1 ,
2] and y = 0 , or x ∈ [ − ,
1] and y = ± x (cid:9) , QUICONTINUOUS MAPPINGS ON TREES 19
X f X
Figure 3.
The finite graph X from Example 33, along with a continuous function f : X −→ X that is equicontinuous even though Fix( f ) is disconnected.and we let f : X −→ X be given by f ( x, y ) = ( x, − y ). The graph X , as well as the function f : X −→ X ,are depicted in Figure 3. Notice that f is the identity function and so f is equicontinuous. However,Fix( f ) = { ( x, ∈ R (cid:12)(cid:12) x ∈ [ − , − ∪ [1 , } is a disconnected set.The observations, along with the example, from this subsection, suggest that the following might be aworthwhile question (a subset of the following question appears as [18, Question 3.10]). Question 34.
Let ( X, f ) be a discrete dynamical system. Which of the equivalences from Theorem 8 holdif we assume that X is an arbitrary finite graph? Which of them hold if we furthermore assume that X has at least one branching point or at least one endpoint? A subset the next question appears as [18, Question 3.9]. First recall that the cone over the harmonicsequence { , , , , . . . } is called the harmonic fan . Attempting to generalize some clauses of Theorem 8from finite trees to non-locally connected continua, we ask the following question. Question 35.
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Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, ´Area de la Investigaci´on Cient´ıfica,Circuito Exterior, Ciudad Universitaria, Coyoac´an, 04510, CDMX, Mexico.
E-mail address : [email protected] Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, ´Area de la Investigaci´on Cient´ıfica,Circuito Exterior, Ciudad Universitaria, Coyoac´an, 04510, CDMX, Mexico.
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