IMFM
Institute of Mathematics, Physics and Mechanics Jadranska 19, 1000 Ljubljana, Slovenia
Preprint series Vol. 51 (2013), 1187
ISSN 2232-2094
INVERSE LIMITS WITH GENERALIZED MARKOV INTERVAL FUNCTIONS
Iztok Banic Tjasa Lunder
Ljubljana, May 17, 2013
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Inverse limits with generalized Markov interval
functions
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Iztok Banic and Tjasa Lunder
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Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, Maribor 2000, Slovenia E-mail: iztok.banic@uni-mb.si E-mail: tjasa.lunder@uni-mb.si
May 8, 2013
CM	Abstract
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In [8] Markov interval maps were introduced and it was shown that any two inverse limits with Markov interval bonding maps with the same pattern were homeomorphic.
In this article we introduce generalized Markov interval functions, which are a generalization of Markov interval maps to set-valued functions, and show that any two generalized inverse limits with generalized Markov interval bonding functions with the same pattern are homeo-morphic.
1 Introduction
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In [8] Markov interval maps are defined as follows. Interval self-maps on I = [a0, am] are Markov with respect to A = {a0, a1,... , am}, if
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1.	a0 < ai < ... < am,
2.	f (A) C A,
3.	f is injective on every component of I\A. CD -
2010 Mathematics Subject Classification: Primary 54C60; Secondary 54F15. Key words and phrases: Inverse limits, Upper semicontinuous functions, Generalized Markov interval functions
Two interval self-maps, f and g, are Markov with the same pattern if f is Markov with respect to A = {a0,ai,... ,am}, g is Markov with respect to B = {b0, b\,..., bm}, and f (a j) = ak if and only if g(bj) = bk.
The main theorem in [8] says that any two Markov interval maps with the
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same pattern have homeomorphic inverse limits:
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Theorem 1.1. Let {/n}n=o be a sequence of surjective maps from I = [ao, am] to I, which are all Markov interval maps with respect to A = {a0, a1,..., am} and let {gn}n=0 be a sequence of surjective maps from J = [b0, bm] to J, which are all Markov interval maps with respect to B = {b0,b1,..., bm}. If for each n, /n and gn are Markov interval maps with the same pattern, then (I, /n) is homeomorphic to (J,gn).
In this paper we introduce generalized Markov interval functions, which generalize Markov interval maps from [8] (in such a way that every Markov interval map is naturally interpreted as a generalized Markov interval function). In this generalization we allow a generalized Markov interval function to be non single-valued only on points in A, and include a condition that provides the
injectivity of / on every component of I\A. The definition of two generalized Markov interval functions with the same pattern will generalize the definition of two Markov interval maps with the same pattern (as it is defined in [8]). We
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prove the following theorem, which is a generalization of Theorem 1.1, as the main result of the paper:
Theorem 1.2. Let {/n}n=0 be a sequence of u.s.c. functions from I = [a0, am] to 21 with surjective graphs, which are all generalized Markov interval functions with respect to A = {a0,ai,... ,am} and let {gn}n=0 be a sequence of u.s.c. functions from J = [b0,bm] to 2J with surjective graphs, which are all generalized Markov interval functions with respect to B = {b0,b1,..., bm}. If for each n, /n and gn are generalized Markov interval functions with the same pattern, then (I,/n) is homeomorphic to (J,gn).
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Since techniques we used in the proof of Theorem 1.2 are quite different from the ones used in [8], our proof can serve as an alternative proof of Holte's
result.
2 Definitions and notation
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A map is a continuous function. In the case where X = Y = R, a E R, and
/ : X ^ Y a map, we use lim /(x) to denote the right-hand limit and lim /(x)
x-la	xf a
to denote the left-hand limit of a function f at the point a G r. A detailed introduction of such limits can bee found in [21, p. 83-95].
Let X be a compact metric space, then 2X denotes the set of all nonempty closed subsets of X.
If f : X ^ 2 is a function, then the graph of f, r(f), is defined as r(f) = {(x,y) G X x Y| y G f (x)}.
A function f : X ^ 2Y has a surjective graph, if for each y G Y there is an x G X, such that y G f (x).
Let f : X ^ 2y be a function. If for each open set V C Y, the set {x G X | f (x) C V} is open in X, then f is an upper semicontinuous function
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(abbreviated u.s.c.) from X to 2Y.
The following is a well-known characterization of u.s.c. functions between metric compacta (for example, see [10, p. 120, Theorem 2.1]).
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Theorem 2.1. Let X and Y be compact metric spaces and f : X ^ 2Y a function. Then f is u.s.c. if and only if its graph r(f) is closed in X x Y.
the function F : X ^ 2Y, defined by F(x) = {f (x)}, is an u.s.c. function, since r(F) = r(f). Also if F : X ^ 2y is an u.s.c. function such that F(x) = {yj for each x G X, then the function f : X ^ Y, defined by f (x) = yx, is continuous. Such functions F will be addressed as single-valued functions. In the paper we frequently deal with such u.s.c. functions. Understanding them as
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Note that for any continuous function f : X ^ Y, where X and Y are
compact metric spaces, the graph of f is a closed subset of X x Y. Therefore
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mappings will simplify the notation and make the proof more reader-friendly. That is why in this case we write y = F(x) instead of y G F(x).
Let A be a subset of X and let f : X ^ 2Y be a function. The restriction of f on the set A, f |A, is the function from A to 2Y such that f |A(x) = f (x) for every x G A.
A sequence {Xk, fk}£=0 of compact metric spaces Xk and u.s.c. functions fk : Xk+1 ^ 2Xk, is an inverse sequence with u.s.c. bonding functions.
The inverse limit of an inverse sequence {Xk,fk}£=0 with u.s.c. bonding functions is defined as the subspace of c^=0 Xk of all points (x0, x1, x2,...), such that xk G fk (xk+1) for each k. The inverse limit of an inverse sequence {Xk, fk}^=0 is denoted by (Xk, fk).
In this paper we deal only with the case when for each k, Xk is a closed interval I and fk : I ^ 21. So, we denote the inverse limit simply by (I, fk). The notion of inverse limits of inverse sequences with upper semi-continuous
cu	bonding functions (also known as generalized inverse limits) was introduced by
Mahavier in [15] and later by Ingram and Mahavier in [10]. Since then, inverse limits have appeared in many papers, such as [1, 2, 3, 4, 6, 9, 11, 12, 13, 17, 18, 19, 20, 22].
3 Proof of Theorem 1.2
In this section we introduce the notion of generalized Markov interval functions and prove Theorem 1.2.
Definition 3.1. Let a0,am G r, a0 < am. We say that an u.s.c. function f from I = [a0,am] to 21 is a generalized Markov interval function with respect to A, where A = |a0,ai,... , am} is a subset of r, if
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1.	a0 < a1 < ... < am,
2.	the restriction of f on every component of I\A is an injective single-valued function,
3.	for each j = 0,1,..., m, the image f (aj) is an interval (possibly degenerated) [arij), ar2(j)] where ari(j),ar2 (j) G A (ari(j) < ar2(j)),
4.	for each j = 0,1,... , m — 1: lim f (x), lim f (x) G A.
Obviously, f can be single-valued on some points aj in A. In this case r1(j) = r2(j) for some 0 < j < m and f (aj) = {ari(j)}. Additionally, taking
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into account property 4. above, we see that:
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1.	if 0 < j < m, then lim f (x) = liin f (x) = ari(j) = ar2(j),
2.	if j = 0, lim f (x) = ari(j) = ar2(j),
3.	if j = m, lim f (x) = ari(j) = ar2(j).
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Definition 3.2. Let A = {a0, a1,..., am} and B = {b0, b1,..., bm}, where
a0 < a1 < ... < am and b0 < b1 < ... < Then we say that (a, b) G A x B is a pair of similar points (with respect to A and B), if a = a^ and b = hi for some i = 0,1,..., m.
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In the following definition we define when two generalized Markov interval functions follow the same pattern.
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Figure 1: A generalized Markov function
Definition 3.3. Let f : I = [a0,am] ^ 21 be a generalized Markov interval function with respect to A = {a0, a^..., am} and let g : J = [b0, bm] ^ 2J be a generalized Markov interval function with respect to B = {b0, b1,..., bm}.
We say that f and g are generalized Markov interval function with the same pattern if i) and ii) hold true:
i) for every j = 0,1,...,m: f (aj) = [ari(j), ar2j)] if and only if g(bj) =
[&ri(j),br2(j)] ,
ii) for every j = 0,1,... ,m — 1: ( lim f (x), lim g(y)) and
xfaj+1	yfbj+1
(lim f (x),limg(y)) are pairs of similar points.
x-laj	y\bj
Finally we prove Theorem 1.2.
Proof. Since we have different functions fk, gk, we introduce functions r\, rf : {0,1,...,m} ^ {0,1,...,m} serving as n,r2 from Definition 3.1, i.e. such that fk(aj) = [arfcj), arfcj)] for each j = 0,1,..., m and each k = 0,1, 2,.... According to Definition 3.3 the same functions r\, rf are also used for gk, i.e.
gk (bj ) = [brk(j),brk (j)].
For each j = 0,1,... m — 1 we define the subinterval Ij = [aj, aj+1] C I = [a0,am], and the subinterval Jj = [bj, bj+1] C J = [b0,bm]. We also define a
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piecewise linear mapping h : I ^ J such that h(aj) = bj for all j = 0,1,..., m by
ai—(x - ao) + bo;	if x E lo,
(x - ai) + bi;	if x e h,

bm ~bm- 1 am am-1
(X - am~l) + bm~i; if X G /m~i.
The mapping h : I ^ J is obviously continuous, monotone and surjective,
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therefore it is a homeomorphism.
Let x = (x0,x1,x2,...) be any element of (I, /n). We show first that there is a uniquely determined point y = (y0, y1, y2,...) in (J, gn), where y0 = h(x0), such that for all i = 0,1, 2,..., I(i) and II(i) hold true. Here for each i, I(i) and II(i) are defined as the following statements:
I(i) ... xj E Int(Ij) if and only if yj E Int(Jj), for each j = 0,1,... ,m — 1,
II(i) ... xj = aj if and only if yj = bj, for each j = 0,1,..., m.
To determine the point y we construct inductively the coordinates yj of y as follows.
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First we construct y0 as y0 = h(x0). It follows from the definition of h that I(0) and II(0) hold true.
Suppose we have already constructed y0, y1,y2,..., yk such that I(i) and II(i) hold true for each i = 0,1,..., k, and yj—1 E gj—1 (yj) hold true for each i = 1, 2,..., k.
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Now we construct yk+1 such that I(k + 1), II(k + 1), and yk E (yk+1). We consider the following two possible cases.
1.	xk+1 = aj for some j = 0,1,..., m. In this case we define yk+1 = bj. Obviously, I(k + 1) and II(k + 1) hold true. Next we show that yk E (yk+1). Since xfc E / (xfc+1) = / (aj) = K-kj), a^j for some ark (j),arfc (j) E A, and since and /k have the same pattern, it follows that (yk+1) =
gfc (bj ) = [brk(j),brk (j)].
If ar fc(j) = arfc(j), then fix an integer £0 such that xk E C [ar k(j), ark(j)].
Then yfc E Jfo C [brk(j) ,brk(j)] = gfc(yfc+1). If ar k(j) = ark(j), then xfc = arfc(j). It follows from the induction assumption II(k) that yk = br k(j) and therefore yfc E [b^Afe(j)] = gfc(yfc+1).
2.	xk+1 E Int(Ij) for some j = 0,1,... , m — 1. In this case, since /k|int(/j) is single-valued,
xfc = /(xfc+1) = /|int(/j)(xfc+1) E /fc(IntIj) = (a^, a^),
for some , a^2 G A (where {a^, a^2} = {lim f (x), lim f (x)}). There-
x^aj	xfaj+i
fore yk G , ) = gk (IntJj) since /k and gk follow the same pattern. We choose yk+i G Int(Jj) such that yk = gk|intj)(yfc+i). Such a point yk+i exists and is uniquely determined since gk|Int(j) : Int( Jj) ^ , ) IN	is bijective.
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Next we show, that if we fix y0 = h(x0), there is exactly one point y = (y0, y1, y2, • • •) in (J, gn), such that for each nonnegative integer i, I(i) and II(i) hold true. Suppose that y = (yo, yi, y2, • • •) and y' = (yo,y1 ,y2, • • •) G (J,g„) are two such points. We show using the induction on i that yi = y' for any i, hence it follows that y = y'. Suppose that for each k = 0,1, 2, • • •, i — 1, yk = yk. We prove that yi = yi. We examine the following two cases.
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1. For some j = 0,1, 2,... m — 1, x G Int(1j). Then yj are both in Int( Jj)
by I(i).
Since yi-i = yi_i, it follows that gi-ilintj)(yi) = gi-i(yi) = yi-i = y'-i = gi_i(y') = gi_i|int(jj)(yi). Since gi_i|int(jj) is one-to-one, it follows that yi = yi.
2. For each j = 0,1, 2, • • • m — 1, xi / Int(Ij). This means that xi = aj for some j = 0,1, • • •, m. In this case yi = bj = yi by II(i).
Next we define a function H : (I,/n) ^ (J, gn) and prove that it is a homeomorphism.
For each (x0, xi, x2, • • •) G (I,/n) we define H(x0, xi, x2, • • •) to be the unique point (y0, yi, y2, • • •) in (J, gn) such that y0 = h(x0) and for each i = 0,1, 2, • • •, I(i) and II(i) hold true.
We have already seen that H is well defined. Next we show that H is continuous, by proving that for any sequence {xi}°=0 in (I,/n) converging to x G (I,/n), the sequence {yi}i=0, where yi = H(xi) for each i, is convergent and its limit equals H(x).
(I, /n) and (J, gn) are both compact metric spaces since I and J are compact (for details see [10]).
Let {xi}i=0 be a convergent sequence of elements in (I,/n), where xi = (x0, xi, x2, • • •) for all i = 0,1, • • •. Let x = (x0, xi, x2, • • •) be the limit of this sequence. This means that xj is the limit of the sequence {xj}°=0 for each j. Let s = (s0, si, s2, • • •) G (J, gn) be any accumulation point of the sequence
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{yj }j=0. Let k be a strictly increasing sequence of nonnegative integers such that lim yki = s.
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(yk0 (yk1, (yk2
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(xo,xi,x2, • • •) G (/,/„) h
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(So,Si ,S2,...) G (J,gn)
(yo,yi,y2,---) G (J,gra)
Let y = (y0, ...) = H(x). We prove that s = y. One can easily see, that
s0 = lim y^ = lim h(xk) = h(lim xk) = h(x0) = y0. i—>■<< i—>■<< i—>■<<
Suppose we have already shown that yk = sk for each k = 0,1, 2,..., t — 1. We show that ye = se by distinguishing the following cases.
1. se G Int(Jj) for some 0 < j < m — 1. The point se is the limit of the sequence {yk }<=0. This means that there exists a nonnegative integer i0 such that yk G Int(Jj), for all i > i0. Therefore xk G Int(Ij), for all i > i0 by I(t). Since xe = lim xk, it follows that xe G Ij. We consider the following two subcases.
(a)	If xe G Int(/j), then ye G Int(Jj) by I(t). Then
ge-i|int(jj )(ye) = ye-i = se-i = ge-i|int(jj )(se)
and since ge-i|Int(Jj) is single-valued and one-to-one, ye = se follows.
(b)	If xe G A, then xe = aj or xe = aj+i (recall that xe G Ij). Without loss of generality, assume that xe = aj. It follows from
i.	lim
and aj is the left-hand endpoint of Ij,
ii.	for all i > i0, xk G Int(Ij) and fe-i is single-valued on Int(Ij), that
xe-i = lim x£-1 = lim /^((x^)) = lim fe-i(t) = ar where ar G [a/-ij), a/-i( J = fe-i(aj) (recall that fe-i is a gen-
eralized Markov function with respect to A). Therefore, xe-i = ar and by definition of H, it follows that ye-i = br. We also know that (lim fe-i(t), lim ge-i(t)) is a pair of similar points since fe-i and ge-i
t\.aj	t\.bj
follow the same pattern and therefore limge-i(t) = br.
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Using the fact that gg-1 is one-to-one on Int(Jj) and that sg G Int(Jj), we conclude that
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s¿_i = #g_i(sg) = lim gg_i(í) = br = yg_i. í^bj
Therefore yg-i = sg-i which contradicts the inductive assumption.
2. sg = bj for some 0 < j < m.
If there exists a nonnegative integer i1 such that yk = bj, for all i > i1;
then by II(£), x/ = aj holds true for all i > i1. This means that xg = aj, since it is the limit of the sequence {xk}i=0. Therefore, yg = bj = sg. If such an integer i1 does not exist, then we consider the following two possible cases:
0 < j < m. We chose a strictly increasing sequence of positive
integers n^ such that
-	{ykni }i=0 is a subsequence of {yk }°=0,
-	ygni = bj for all i,
-	yg^ G Int(Jj-1) for all i or yg^ G Int(Jj) for all i.
kn
Assume without any loss of generality that yg"4 G Int(Jj) for all 00	i. Recall that lim ygni = sg = bj and that by I(£), xgni G Int(Ij)
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for all i. This means that xg G Ij and we distinguish the following
i.	If xg G A, then either xg = aj or xg = aj+1. One can see, using
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similar arguments as in 1.(b), that
sg_1 = lim yg_1 = lim gg-^yg^) = lim gg_1(t) = br
i^x	i^x	t\bj
where br G [b^-ij), b^-ij ] = gg-1(bj). By inductive assumption yg-1 = sg-1 = br. By definition of H, it follows that xg-1 = ar. Since /g-1 and gg-1 follow the same pattern, it follows that xg-1 = ar = lim /g-1(t) = lim /g-1(t) since /g-1 is one-to-one
t^ttj	tfaj+i
on Int(Ij). Therefore, xg cannot equal to aj+1, hence xg = aj. By definition of H, it follows that yg = bj = sg.
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ii.	If xg G Int(Ij), then
axg-i = lim xg-\ = lim /g-i(xgn ) = lim /g-i(t)
i^x	i^x	tia,
since /g-i is single-valued and injective on Int(1j). Using the same arguments as in i. we conclude that also in this case
s— = br where br e [b^-jA^-j = g—(bj). Since fi-1 and gt-1 follow the same pattern, lim ft-1(t) = ar follows from
t\.aj
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lim gt-1(t) = br. Therefore xi-1 = ar. It follows that yt-1 =
t\bj
br = sl-1, which contradicts our inductive assumption.
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(b) j = 0 or j = m. Assume without any loss of generality that j = 0. We chose a strictly increasing sequence of positive integers ui, such that
-	{y^ni }i=0 is a subsequence of [y^ }°=0,
kn
-	Vt i = bj for all i,
-	ytTH e Int(J0) for all i.
The rest of the proof is similar to the proof of (a), replacing j with
We can define H 1 : (J, gn) ^ (I,fn) in the same fashion as we did with H. Every element (y0,y1,...) of (J,gn) has the unique image (x0 ,x1,...) in (I, fn), such that x0 = h-1(y0) and for each i = 0,1, 2,..., I(i) and II(i) hold true. Therefore H is a homeomorphism.	□
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Theorem 1.2.
We conclude the paper with the following corollary that easily follows from
Corollary 3.4. Let f : I = [a0,am] ^ 21 be a generalized Markov interval function with respect to A = [a0, a1,..., am} with a surjective graph and g : J = [b0,bm] ^ 2J be a generalized Markov interval function with respect to B = [b0,b1,... ,bm} with a surjective graph. If f and g are generalized Markov interval functions with the same pattern, then (I, f) is homeomorphic to (J,g).	□
Acknowledgements
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This work was supported in part by the Slovenian Research Agency, under Grant P1-0285.
We thank Uros Milutinovic for constructive suggestions.
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