Zero entropy continuous interval maps and MMLS-MMA property
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Nonlinearity PAPER Zero entropy continuous interval maps and MMLS- MMA property To cite this article: Yunping Jiang 2018 Nonlinearity 31 2689 View the article online for updates and enhancements. Related content On dynamics of graph maps with zero topological entropy Jian Li, Piotr Oprocha, Yini Yang et al. - On various definitions of shadowing with average error in tracing Xinxing Wu, Piotr Oprocha and Guanrong Chen - Stability of the Markov operator and synchronization of Markovian random products Lorenzo J Díaz and Edgar Matias - This content was downloaded from IP address 89.202.245.164 on 09/05/2018 at 14:20
Transcript of Zero entropy continuous interval maps and MMLS-MMA property
Zero entropy continuous interval maps and MMLS-MMA
propertyNonlinearity
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Yunping Jiang
Department of Mathematics, Queens College of the City University of New York, Flushing, NY 11367-1597, United States of America Department of Mathematics, Graduate School of the City University of New York, 365 Fifth Avenue, New York, NY 10016, United States of America
E-mail: [email protected]
Received 25 September 2017, revised 7 February 2018 Accepted for publication 9 March 2018 Published 2 May 2018
Recommended by Professor Masato Tsujii
Abstract We prove that the flow generated by any continuous interval map with zero topological entropy is minimally mean-attractable and minimally mean-L- stable. One of the consequences is that any oscillating sequence is linearly disjoint from all flows generated by all continuous interval maps with zero topological entropy. In particular, the Möbius function is linearly disjoint from all flows generated by all continuous interval maps with zero topological entropy (Sarnak’s conjecture for continuous interval maps). Another consequence is a non-trivial example of a flow having discrete spectrum. We also define a log-uniform oscillating sequence and show a result in ergodic theory for comparison.
Keywords: minimally mean-attractable (MMA), minimally mean-L-stable (MMLS), linear disjointness, Mobius function, oscillating sequence, Sarnak’s conjecture, continuous interval map with zero topological entropy Mathematics Subject Classification numbers: Primary; 11K65, 37A35, Secondary; 37A25, 11N05
Y Jiang
Printed in the UK
31
Nonlinearity
NON
10.1088/1361-6544/aab593
Paper
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2689
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Nonlinearity
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* This material is based upon work supported by the National Science Foundation. It is also partially supported by a collaboration grant from the Simons Foundation (grant number 523341) and PSC-CUNY awards and a grant from NSFC (grant number 11571122).
2018
1361-6544
Nonlinearity 31 (2018) 2689–2702 https://doi.org/10.1088/1361-6544/aab593
1. Introduction
Let C and R denote the complex plane and the real line. Let N = {1, 2, 3, · · · , n · · · } be the set of natural numbers. Suppose X is a compact metric space with metric d(·, ·). Let C(X,C) be the space of all continuous functions : X → C. Consider a continuous map f : X → X . Then f generates a flow (i.e. dynamical system)
X = { f n : X → X}n∈{0}∪N
where f n means the nth iteration
f f · · · f n
and f 0 means the idenity. The Möbius function is the arithmetic function µ(n) : N → {−1, 0, 1} defined as
µ(n) =
1 if n = 1; (−1)r if n = p1 · · · pr for r distinct prime numbers pi; 0 if p2|n for a prime number p.
Sarnak proposed a conjecture which makes a connection between number theory and ergodic theory. The conjecture can be stated as
Conjecture 1 (Sarnak [14, 15]). The Möbius function is linearly disjoint from (i.e. or- thogonal to) all flows with zero topological entropy.
The linear disjointness can be defined more generally. Suppose c = (cn)n∈N− is a sequence of complex numbers.
Definition 1 (Linear disjointness). We say c is linearly disjoint from X if
lim N→∞
1 N
for any ∈ C(X,C) and any x ∈ X .
In order to understand conjecture 1 in ergodic theory, we gave the following definition in [5].
Definition 2 (Oscillating Sequence). The sequence c is said to be an oscillating sequence if
lim N→∞
1 N
with a technical condition
for some constant λ > 1.
An important example of an oscillating sequence is the Möbius sequence u = (µ(n))n∈N due to Davenport’s theorem [2]. However, there are many other oscillating sequences generated
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by arithmetic functions, for example, the Liouville function defined as l(n) = (−1)(n), where (n) is the number of prime factors of n, counted with multiplicity (see [1, 4]). There are
also many oscillating sequences from complex analysis and harmonic analysis, for examples,
c = ( e2πin2α
( e2πicn log n
) for any fixed constant
c > 0 are oscillating sequences. One can find other examples of oscillating sequences in [5]. Therefore, our new result theorem 2 and corollary 1 applies not only to Sarnak’s conjecture in number theory corollary 2 but also to many other branches in mathematics like ergodic theory, probability, and harmonic analysis.
The key point of conjecture 1 in the view of definition 1 is that the convergence is for every point in the space. However, in ergodic theory, the convergence is for almost all points for a reference measure. To distinct our main result theorem 2 and corollary 1 to those results in ergodic theory, we give a definition of a log-uniform oscillating sequence and prove a result in ergodic theory. This result in ergodic theory indicates why our study of the MMA (minimally mean-attractable)-MMLS (minimally mean-L-stable) property for flows is important.
Definition 3 (Log-uniform oscillating sequence). A sequence c is said to be a log- uniform oscillating sequence if there are constants A > 1 and B > 0 such that
max 0t<1
with also the technical condition (3).
An example of a log-uniform oscillating sequence is the Möbius sequence u due to Davenport’s theorem [2].
Suppose B is the standard Borel σ-algebra on X (here we can relax the compactness con- dition on X). Suppose ν is a Borel measure on X. Then (X,B, ν) is a measurable space. A measurable and bijective map f : (X,B, ν) → (X,B, ν) is called an automorphism if it is measure-preserving, that is, ν( f−1(A)) = ν(A) for every A ∈ B (in other words, we also called ν f-invariant). Let L1(X,B, ν) be the space of all L1-functions on X.
Theorem 1. Suppose f : (X,B, ν) → (X,B, ν) is an automorphism. Suppose c = (cn) is a log-uniform oscillating sequence. Then for any φ ∈ L1(X,B, ν), we have that
lim N→∞
1 N
cn( f nx) = 0, for ν − a.e. x ∈ X. (5)
Since this theorem is not our main topic, just for a comparison, we will prove it in the last section 3. In general, it is difficult to get the convergence for every point in the above theorem. That is a reason we need the following definitions.
Suppose E ⊂ N. The upper density D(E) of E in N is
D(E) = lim sup N→∞
N .
Definition 4 (MLS). A flow X is said to be mean-L-stable (MLS) if for every ε > 0, there is a δ > 0 such that d(x, y) < δ implies d( f nx, f ny) < ε for all n ∈ N except a subset E of natural numbers with upper density D(E) less than ε.
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Mean-L-stable flows are an important class of dynamical systems in ergodic theory. An example of a MLS flow is an equicontinuous flow.
Definition 5 (Equicontinuity). A flow X is said to be equicontinuous if for every ε > 0, there is a δ > 0 such that d(x, y) < δ implies d( f nx, f ny) < ε for all n ∈ N.
We says a closed subset K ⊂ X is minimal (with respect to f) if f (K) ⊂ K and if { f nx | n ∈ N} = K for every x ∈ K. We use K to denote the sub-flow {( f |K)n : K → K}n∈{0}∪N. Again, for the purpose of understanding conjecture 1 in ergodic theory, we gave the following definitions in [5].
Definition 6 (MMLS). We say that a flow X is minimally MLS (MMLS) if for every mini- mal subset K ⊆ X , the sub-flow K is MLS.
Definition 7 (MMA). Suppose X is a flow. Suppose K ⊂ X is minimal. We say x ∈ X is mean-attracted to K if for any ε > 0 there is a point z = zε,x ∈ K (depending on x and ε) such that
lim sup N→∞
d( f nx, f nz) < ε. (6)
The basin of mean-attraction of K, denoted Basin(K), is defined to be the set of all points x which are mean-attracted to K. It is trivial that K ⊂ Basin(K). We call X minimally mean- attractable (MMA) if
X = K
Basin(K) (7)
where K varies among all minimal subsets of X.
Recall that a point x ∈ X is attracted to K if
lim n→∞
d( f nx, K) = 0.
In general, this does not imply to that x is mean-attracted to K. We proved the following theorem in [5]. The reader is encouraged to compare this result
with theorem 1.
Theorem 1 (MMA/MMLS and disjointness). Any oscillating sequence c is linearly disjoint from all MMA and MMLS flows X . Moreover, the limit in (1) is uniform on every minimal subset K.
Since a MMA and MMLS flow has zero topological entropy (for example, this follows from [9]), this theorem confirms Sarnak’s conjecture for all MMA and MMLS flows. We provided some examples of flows which are MMA and MMLS in [5]. However, these exam- ples except for Denjoy counter-examples are equicontinuous when they are restricted on their minimal subsets. It is an interesting problem to find further examples which are MMA and MMLS but not equicontinuous when they are restricted on their minimal subsets. We give a complete answer to this problem for continuous interval maps with zero topological entropy in this paper. The main result in this paper is that
Theorem 2 (Main theorem). Suppose I = [a, b] ⊂ R is a closed interval. Suppose f : I → I is a continuous map with zero topological entropy. Then the flow X generated by f is a MMA and MMLS flow.
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We will prove this theorem in section 2. Now let us discuss some very important conse- quences from our main result theorem 2. The first one is our main goal in this area.
Corollary 1 (Continuous interval map and disjointness). Suppose I = [a, b] ⊂ R is a closed interval. Suppose f : I → I is a continuous map with zero topological entropy. Then any oscillating sequence c is linearly disjoint from the flow X generated by f.
The proof of corollary 1 is given by combining the proof of theorem 2 and the proof of theorem 1. This proof provides a completely new proof of Sarnak’s conjecture for all flows generated by continuous interval maps with zero topological entropy (see [8] for a different proof and remark 1 for another approach) as follows.
Corollary 2 (Continuous interval map and Möbius function). Suppose I = [a, b] ⊂ R is a closed interval. Suppose f : I → I is a continuous map with zero topologi- cal entropy. The Möbius function is linearly disjoint from the flow X generated by f.
Let us also discuss some consequence of our main result theorem 2 in ergodic theory. Suppose K ⊂ X is a subspace. Let BK be the standard Borel σ-algebra on K and let ν be a Borel probability measure on K. Let L2(K,BK , ν) be the space of all L2 functions on K.
Suppose f is a measure-preserving map from (K,BK , ν) into itself. A complex number λ is called an eigenvalue if there is a function φ ∈ L2(K,BK , ν) such that φ( fx) = λφ(x) for ν-a.e. x ∈ K. Here φ is called an eigenfunction with respect to λ. We say f : (K,BK , ν) → (K,BK , ν) has discrete spectrum or pure point spectrum if there exists an orthonormal L2(K,BK , ν)-basis consisting of eigenfunctions. Theorem 2 provides a non-trivial example of flows having dis- crete spectrum (see [9, theorem 3.8]).
Corollary 3 (Continuous interval map and discrete spectrum). Suppose I = [a, b] ⊂ R is a closed interval. Suppose f : I → I is a continuous map with zero topo- logical entropy. Suppose X is the flow generated by f. Suppose K ⊆ X is a minimal subset with respect to f. Then for every ergodic f-invariant measure ν on K, f : (K,B, ν) → (K,B, ν) is a measure-preserving map having discrete spectrum.
Remark 1. Corollary 3 combining with the work in [11] (see also [6]) gives another proof of Sarnak’s conjecture for all flows generated by continuous interval maps with zero topologi- cal entropy as present in corollary 2.
Remark 2. I would like to point out that there are flows generated by higher-dimensional maps with zero topological entropy which are not MLS when they are restricted on their mini- mal subsets. Thus in the higher-dimensional case, the oscillation property alone is not enough. We need instead the higher-order oscillation property. The reader who is interested in the higher-order oscillation property for a sequence c and the role it plays in Sarnak’s conjecture for higher-dimensional dynamical systems is encouraged to read my paper [7].
2. Proof of the main result theorem 2
Suppose I = [a, b] ⊂ R is a closed interval and f : I → I is a continuous map. Suppose the topological entropy ent( f ) = 0. In other words, this means that f cannot have any horseshoe- type invariant subset in I. More precisely, this means that f can not have a forward f-invariant closed subset X ⊂ I such that f : X → X is topologically conjugate (or semi-conjugate) to a sub-shift of finite type σA : ΣA → ΣA with the maximal Lyapunov exponent greater than 0.
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For the sub-shift of finite type σA : ΣA → ΣA, we mean that A = (aij)d×d is a d × d-square matrix of 0’s and 1’s,
ΣA = {w = i0i1 · · · in−1in · · · | in−1 ∈ {1, 2, · · · , d}, ain−1in = 1, n ∈ N}
and
σA(w) = i1 · · · in−1in · · · .
By abuse of notation, we use ΣA to mean a horseshoe-type forward f-invariant subset X ⊂ I . Since A is a nonnegative square matrix, from the Perron–Frobenius theorem, it has a maximal nonnegative eigenvalue eA. If A is also irreducible, eA is simple, unique, and positive. The statement that maximal Lyapunov exponent of σA is greater than 0 is equivalent to saying that eA > 1.
A point p ∈ X is called a periodic point of period n 1 if fn( p ) = p. The minimal such n is called the primitive period. Following the proof of Sharkovskii’s theorem (for example, the reader may refer to book [3] or book [13, chapter 4]), we see that if f has a periodic point whose period is different from all powers of 2, then f has a horseshoe-type invariant subset. For example, if f has a periodic point of period 3, then f has a horseshoe-type invariant subset σA : ΣA → ΣA for
A =
5)/2. Thus we have that
Lemma 1. The period of any periodic point of f is 2n for some n 0.
For any x ∈ I , let
ω(x) = ∩∞ n=0{ f kx | k n},
be the ω-limit set of x under f. Let K = ω(x). Consider the sub-flow
K = {( f |K)n : K → K}n∈{0}∪N. (8)
It is a minimal flow (refer to [13] or the argument after lemma 2). We have two cases:
(I) the set K is finite and (II) the set K is infinite.
Case (I) is easier. In this case,
K = { p0, p1, · · · , p2n−1}
is a periodic orbit of period 2n for some n 0. That is, f ( pi) = pi+1 (mod 2n) for i = 0, 1, · · · , 2n − 1. Therefore, the following lemma is easily verified.
Lemma 2. In Case (I), the sub-flow (8) is equicontinuous, thus, MLS. Moreover, x is mean-attractable to K.
Case (II) is more complicated. Since the topological entropy of f is zero, in this case, K con- tains no periodic point (see, for example, [13, chapter 5]). Since f cannot have any horseshoe- type forward f-invariant subset in I, we can find a sequence of sets
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of closed intervals In,k such that
(1) for each n 1, intervals in ηn are pairwise disjoint; (2) f (In,k) = In,k+1 (mod 2n) for any n 1 and 0 k 2n − 1; (3) for each n 1 and In,k ∈ ηn, two and only two intervals In+1,k and In+1,k+2n in ηn+1 are
sub-intervals of In,k, that is,
In+1,k ∪ In+1,k+2n ⊆ In,k;
(4) K ⊆ ∩∞ n=1 ∪
2n−1 k=0 In,k .
This is a well-known fact about continuous interval maps with zero topological entropy among all experts in this field (for example, refer to [13, Chapter 5]).
We put a symbolic coding on each interval In,k in ηn for n 1 and 0 k 2n − 1. Consider the binary expansion of k as
k = i0 + i12 + · · · im2m + · · ·+ in−12n−1
where i0, i1, · · · , im, · · · , in−1 = 0 or 1. Let us only remember the coding wn = i0i1 · · · im · · · in−1 and label In,k by wn, that is, In,k = Iwn.
Define the space of codings
Σn = {wn = i0i1 · · · im · · · in−1 | im ∈ {0, 1}, m = 0, 1, · · · , n − 1}
with the product topology. We have a natural shift map
σn(wn) = i1 · · · im · · · in−1 : Σn → Σn−1.
It is a continuous 2-to-1 map. Then we have an inverse limit system
{(σn : Σn → Σn−1); n ∈ N}.
Let
(σ : Σ → Σ) = lim ←−
(σn : Σn → Σn−1).
Explicitly, we have that
Σ = {w = i0i1 · · · in−1 · · · | in−1 ∈ {0, 1}, n ∈ N}
and
Consider the metric d(·, ·) on Σ defined as
d(w, w′) =
∞∑ n=1
|in−1 − i′n−1| 2n
for w = i0i1 · · · in−1 · · · and w′ = i′0i′1 · · · i′n−1 · · ·. It induces the same topology as the topology from the inverse limit and makes Σ a compact metric space.
On Σ, we have a map called the adding machine denoted as add and defined as follows: for any w = i0i1 · · · ∈ Σ, consider the formal binary expansion a =
∑∞ n=1 in−12n−1, then
a + 1 = ∑∞
n=1 i′n−12n−1 has a unique formal binary expansion. Here ‘a + 1’ is just a notation. It means if i0 = 0, then i0 = 1 and all other i′n = in and if i0 = 1, then i0 = 0 and then consider i1 + 1, and so on. The adding machine is
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add(w) = i′0i′1 · · · i′n−1 · · · : Σ → Σ.
It is a homeomorphism of Σ to itself. Moreover, we have
Lemma 3. The flow
is minimal and equicontinuous.
We leave the proof for the reader since it is not difficult. For a point w = i0i1 · · · in−1 · · · ∈ Σ, let wn = i0i1 · · · in−1. We use
[w]n = {w′ ∈ Σ | w′ n = wn}
to denote the n-cylinder containing w. One can check that for every w ∈ Σ,
· · · ⊂ Iwn ⊂ Iwn−1 ⊂ · · · ⊂ Iw2 ⊂ Iw1 ,
is a nested sequence of closed intervals. Therefore
Iw = ∩∞ n=1Iwn
is a non-empty, connected, and compact subset of I. Moreover, we have that f (Iw) = Iadd(w) and
K ⊂ ∪w∈ΣIw.
The set C = {[w]n | w ∈ Σ, n ∈ N} of all n-cylinders forms a topological basis for Σ. Let B be the σ-algebra generated by C. We have a standard probability measure ν defined on B such that ν([w]n) = 1/2n for all w ∈ Σ. We say ν is add-invariant if ν(add−1(A)) = ν(A) for all A ∈ B. We say ν is ergodic if add−1(A) = A for A ∈ B implies either ν(A) = 0 or 1. The following lemma is also well-known among experts in this field and not difficult to be verified. We also leave it to the reader.
Lemma 4. The measure ν is add-invariant and ergodic.
In Case (II), we consider two different cases:
(a) there is a point w0 ∈ Σ such that Iw0 = {xw0} contains only one point or (b) all Iw for w ∈ Σ are closed intervals.
In Case (II)(a), we have that ω(x) = ω(xw0). Without loss of generality, we assume x = xw0 and every Iw = {xw} contains only one point. Let
τn = max wn∈Σn
|Iwn |.
We claim that τn → 0 as n → ∞. We prove it by contradiction. If there is a subsequence {nk}∞k=1 of the natural numbers and a number ε0 > 0 such that τnk ε0, then we have an inter- val Iwnk
∈ ηnk such that |Iwnk | ε0. Since Σ is a compact metric space, we have a convergent
subsequence of {wnk}∞k=1 in Σ. Suppose {wnk}∞k=1 is itself a convergent sequence with the limit point w ∈ Σ, then the interval Iwnk
converges to an interval Iw as k → ∞. The length |Iw| is greater than ε0. This contradicts the fact Iw contains only one point. This proves the claim.
Lemma 5. In Case (II)(a), the sub-flow K in (8) is equicontinuous, thus, MLS.
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Proof. We can set a one-to-one and onto correspondence h(w) = xw from Σ to K. Since τn → 0 as n → ∞, for any ε > 0, there is a N > 0 such that τN < ε. For the given N, we can find a δ > 0 such that for any w, w′ ∈ Σ with d(w, w′) < δ , then wN = w′
N . This implies that both points xw and xw are in the same interval IwN. So |h(w)− h(w′)| τN < ε. This proves that h is continuous. Since both Σ and K are compact, we have that h−1 is also continuous. Thus h is a homeomorphism. Since f |K = h add h−1 and since the flow A is equicontinu- ous, the sub-flow K is equicontinuous.
Lemma 6. In Case (II)(a), x is mean-attracted to K.
Proof. Since K ⊂ ∩∞ n=1 ∪wn∈Σn Iwn and since τn → 0 as n → ∞, for any ε > 0, there is
an integer n0 > 0 such that τn0 < ε. Then we have an integer m0 > 0 such that f m0 x ∈ Iwn0
for some w ∈ Σ. Since K ∩ Iwn0 = ∅ and f |K = h add h−1 is a homeomorphism, we have
z ∈ K such that f m0 z ∈ K ∩ Iwn0 . This implies that
| f m0 x − f m0 z| τn0 < ε.
Since f m0+n(Iwn0 ) ∈ ηn0 for all n 0, we have that
| f m0+nx − f m0+nz| τn0 < ε
for all n 0. This implies that
lim sup N→∞
We proved that x is mean-attracted to K.
In Case (II)(b), we consider a fixed interval Iw0. Let
ε0 = |Iw0 | > 0.
The orbit {addn(w0) | n ∈ Z} is dense in Σ. Then we can find two points u, v ∈ K such that the distance |u − v| is arbitrary small and such that there are infinitely many positive integers n such that Iadd−n(w0) ⊂ Iuv. Then we have
f n(Iuv) ⊃ f n(Iadd−n(w0)) = Iw0 .
Thus | f nu − f nv| ε0 for infinitely many positive integers n. This proves that
Lemma 7. In Case (II)(b), the sub-flow K in (8) is not equicontinuous.
The main work in this paper is to prove the sub-flow K in (8) is still MLS in Case (II)(b).
Lemma 8. In Case (II)(b), the sub-flow (8) is MLS.
Proof. Since the probability measure ν is add-invariant and ergodic, the Birkhoff ergodic theorem implies that
lim N→∞
= lim N→∞
= ν(A),
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for any A ∈ B and ν-a.e. w ∈ Σ. For any ε > 0, since Iw ∩ Iw′ = ∅ for all w = w′ ∈ Σ and since
∑ w∈Σ |Iw| |I| < ∞,
there are an integer N0 > |I|/ε+ 1 and 0 m0 |I|/ε+ 1 points, denoted as wj, 1 j m0, in Σ such that for any w = w j ∈ Σ for all 1 j m0 and n N0, |Iwn | < ε.
Let N1 > N0 be an integer such that ν([w]N1) < ε/m0 for all w ∈ Σ. We have a number δ > 0 such that for any u, v ∈ K with |u − v| < δ, the interval [u, v] ⊂ Iwu,v
N1 for some wu,v ∈ Σ.
For any given |u − v| < δ and for each 1 j m0, define
Ej = {k ∈ N | add−k([w j]N1) = [wu,v]N1}.
From the Birkhoff ergodic theorem (see the formula in the beginning of the proof), we have that
lim N→∞
lim N→∞
m0∑ j=1
#(Ej ∩ [1, N])
N < ε. (10)
Thus we have that the upper density D(E) is less than ε. For any n ∈ N \ E , addn([wu,v]N1) = [w j]N1 for all 1 j m0. This implies f n(u),
f n(v) ∈ IwN1 for w = w j for all 1 j m0. Thus,
| f n(u)− f n(v)| < ε, ∀ n ∈ N \ E. (11)
Both (10) and (11) imply that f : K → K is MLS. We completed the proof.
Lemma 9. In Case (II)(b), x is mean-attracted to K.
Proof. We use the same notation as that in the proof of lemma 8. For any ε > 0, since Iw ∩ Iw′ = ∅ for all w = w′ ∈ Σ and since
∑ w∈Σ |Iw| |I| < ∞, there are an integer
N0 > |I|/ε+ 1 and 0 m0 |I|/ε+ 1 points, denoted as wj, 1 j m0, in Σ such that for any w = w j ∈ Σ for all 1 j m0 and n N0, |Iwn | < ε/3.
Let N1 > N0 be an integer such that ν([w]N1) < ε/(3|I|m0) for all w ∈ Σ. Fix an inter- val Iw′
N1 with length less than ε. There is an integer k 0 such that f kx ∈ Iw′
N1 . Take
z0 ∈ Iw′ N1 ∩ K . Then we have a point z = zx,ε ∈ I(add−k(w′))N1
∩ K such that f kz = z0. Thus we have that | f kx − f kz| < ε.
Define
Ej = {k ∈ N | add−k([w j]N1) = [w′]N1}.
From the Birkhoff ergodic theorem (see the formula in the beginning of the proof of lemma 8), we have that
lim N→∞
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lim N→∞
k∑ n=1
| f k+nx − f k+nz|+ 1 N
∑ n∈N,k+n∈E
= I + II + III.
First for N large, we have I < ε/3. Secondly, for any n ∈ N such that k + n ∈ N \ E,
[w]N1 = addk+n([w′]N1) = [w j]N1 .
This implies f k+nx, f k+nz ∈ IwN1 for w = w j for all 1 j m0. Thus,
| f k+nx − f k+nz| < ε
3 , ∀ n ∈ N and k + n ∈ N \ E.
This implies that II < ε/3.
Finally, since | f k+nx − f k+nz| |I| is always true, we have
III #(E ∩ [1, N])|I|
N .
The upper density of E given in (13) implies that
lim sup N→∞
This proves that x is mean-attracted to K.
Lemmas 2, 5, 6, 8 and 9 complete the proof of theorem 2.
Remark 3. The proof given in this section not only works for continuous interval maps with zero topological entropy but also works for any dynamical system f : X → X of a com- pact space X satisfying that (i) it is MMA and (ii) when f is restricted to any minimal subset K ⊆ X , f : K → K is semi-conjugate to the adding machine add : Σ → Σ, that is, there is a continuous surjective map h : K → Σ such that h f = add h. Jian Li sent me a note after the first version of this paper was appeared in ArXiv and mentioned that they are working
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on a similar result as stated in corollary 2 for graph maps with zero topological entropy by using a different approach [10]. Graph maps with zero topological entropy and circle ho- meomorphisms and continuous interval maps with zero topological entropy are interesting examples satisfying (i) and (ii). Thus our proof in this section could be used to prove not only continuous interval maps with zero topological entropy but also graph maps with zero topo- logical entropy and circle homeomorphisms. It is still not clear to me if there is an example of a dynamical system f : X → X of a compact space X which satisfies only (i) but not (ii), however, it is still MMLS.
3. Proof of theorem 1
Without loss of generality, we assume that ν is ergodic. Otherwise, we consider on every ergodic component one by one. For any φ ∈ L1(X,B, ν), we define
σφ(n) = ∫
∫
These numbers character a measure σφ on the unit circle
T = {e2πit | 0 t < 1}
∫
SN(x) = 1 N
N∑ n=1
cnφ( f nx).
Since c = (cn) is a log-uniform oscillating sequence, we have two constants A > 1 and B > 0 such that
SN2 = (∫
, ∀ N 2.
For any r > 1, let [rm] be the largest integer less than or equal to rm. Then we have that
S[rm]2 B
This implies that
S[rm]2 B
S[rm](x) = 1
cnφ( f nx) → 0 as m → ∞ for ν − a.e.x ∈ X.
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For any integer N > 0, we have a unique integer m > 0 such that [rm] N < [rm] + 1 [rm+1], then we have that
|SN(x)| = 1 N
N∑ n=1
1 N
N∑ [rm]+1
cnφ( f nx) .
Let us first prove the theorem under the assumption that φ ∈ L∞(X,B, ν). Under this assumption, let λ′ > 1 be the dual number of λ in the technical condition (3), that is, 1/λ+ 1/λ′ = 1. The Hölder inequality implies that
1 N
λ′ .
1 λ′ as m → ∞.
Therefore, for any ε > 0, we have r > 1 such that Cφ∞(r − 1) < ε/4. For this fixed r, we have a m0 > 0 such that
C 1 λ φ∞
1 λ′
4 , ∀ m > m0,
and |S[rm](x)| < ε/2 for ν-a.e. x ∈ X for all m > m0. Thus we get that
|SN(x)| < ε for ν − a.e. x ∈ X and m > m0.
We proved that
SN(x) → 0 as m → ∞ for ν − a.e. x ∈ X
when φ ∈ L∞(X,B, ν). In general, we find a sequence {φm}∞m=1 ⊂ L∞(X,B, ν) such that
φ− φm1 =
Note that
lim N→∞
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For any ε > 0, we have m > 0 such that φ− φm1 < ε/2. For this fixed m, from the first part of the proof, we have a N0 > 0 such that
IN,m(x), IIN,m(x) < ε
Thus we get SN(x)
≤ ε, for ν-a.e. x ∈ X and any N > N0.
Therefore, SN(x) → 0 as N → ∞ for ν-a.e. x ∈ X . This completes the proof.
Acknowledgment
This paper was completed during my visit to the National Center for Theoretical Sciences (NCTS) at National Taiwan University in 2016. I would like to thank NCTS for its hospitality. I would like to thank my student John Adamski for his help on revising this paper.
References
[1] Daboussi H and Delange H 1982 On multiplicative arithmetical functions whose modulus does not exceed one J. London Math. Soc. 26 245–64
[2] Davenport H 1937 On some infinite series involving arithmetical functions (II) Quart. J. Math. Oxford 8 313–20
[3] Devaney R 2003 Introduction to Chaotic Dynamical Systems (Boulder, CO: Westview Press) [4] Delange H 1994 Sur les fonctions multiplicatives complexes de module Ann. Inst. Fourier
44 1323–49 [5] Fan A and Jiang Y 2017 Oscillating sequences, minimal mean attractability and minimal mean-
lyapunov-stability Ergod. Theor. Dynam. Syst. (https://doi.org/10.1017/etds.2016.121) [6] Huang W, Wang Z and Zhang G 2016 Möbius disjointness for topological models of ergodic
systems with discrete spectrum (arXiv:1608.08289v2 [math.DS]) [7] Jiang Y 2016 Higher order oscillating sequences, affine distal flows on the d-torus, and Sarnak’s
conjecture (arXiv:1612.04306v2 [math.DS]) [8] Krengel U 1982 Ergodic Theorems (New York: Walter de Gruyter) [9] Li J, Tu S and Ye X D 2015 Mean-equicontinuity and mean sensitivity Ergod. Theor. Dynam. Syst.
35 2587–612 [10] Li J, Oprocha P, Yang Y and Zeng T 2017 On dynamics of graph maps with zero topological
entropy (arXiv:1711.01101v2 [math.DS]) [11] Matomãki K, Radziwill M and Tao T 2015 An averaged form of Chowla’s conjecture
(arXiv:1503.05121v1 [math.NT]) [12] Karagulyan D 2015 On Möbius orthogonality for interval maps of zero entropy and orientation-
preserving circle homeomorphisms Ark. Mat. 53 317–27 [13] Ruette S 2017 Chaos on the Interval (University Lecture Series vol 67) (Providence, RI: American
Mathematical Society) [14] Sarnak P 2009 Three lectures on the Möbius function, randomness and dynamics IAS Lecture
Notes (Princeton, NJ: Princeton University Press) (http://publications.ias.edu/sites/default/files/ MobiusFunctionsLectures(2).pdf)
[15] Sarnak P 2012 Möbius randomness and dynamics Not. S. Afr. Math. Soc. 43 89–97
Y Jiang Nonlinearity 31 (2018) 2689
Abstract
3. Proof of theorem 1
Acknowledgment
References
PAPER
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This content was downloaded from IP address 89.202.245.164 on 09/05/2018 at 14:20
Yunping Jiang
Department of Mathematics, Queens College of the City University of New York, Flushing, NY 11367-1597, United States of America Department of Mathematics, Graduate School of the City University of New York, 365 Fifth Avenue, New York, NY 10016, United States of America
E-mail: [email protected]
Received 25 September 2017, revised 7 February 2018 Accepted for publication 9 March 2018 Published 2 May 2018
Recommended by Professor Masato Tsujii
Abstract We prove that the flow generated by any continuous interval map with zero topological entropy is minimally mean-attractable and minimally mean-L- stable. One of the consequences is that any oscillating sequence is linearly disjoint from all flows generated by all continuous interval maps with zero topological entropy. In particular, the Möbius function is linearly disjoint from all flows generated by all continuous interval maps with zero topological entropy (Sarnak’s conjecture for continuous interval maps). Another consequence is a non-trivial example of a flow having discrete spectrum. We also define a log-uniform oscillating sequence and show a result in ergodic theory for comparison.
Keywords: minimally mean-attractable (MMA), minimally mean-L-stable (MMLS), linear disjointness, Mobius function, oscillating sequence, Sarnak’s conjecture, continuous interval map with zero topological entropy Mathematics Subject Classification numbers: Primary; 11K65, 37A35, Secondary; 37A25, 11N05
Y Jiang
Printed in the UK
31
Nonlinearity
NON
10.1088/1361-6544/aab593
Paper
6
2689
2702
Nonlinearity
IOP
* This material is based upon work supported by the National Science Foundation. It is also partially supported by a collaboration grant from the Simons Foundation (grant number 523341) and PSC-CUNY awards and a grant from NSFC (grant number 11571122).
2018
1361-6544
Nonlinearity 31 (2018) 2689–2702 https://doi.org/10.1088/1361-6544/aab593
1. Introduction
Let C and R denote the complex plane and the real line. Let N = {1, 2, 3, · · · , n · · · } be the set of natural numbers. Suppose X is a compact metric space with metric d(·, ·). Let C(X,C) be the space of all continuous functions : X → C. Consider a continuous map f : X → X . Then f generates a flow (i.e. dynamical system)
X = { f n : X → X}n∈{0}∪N
where f n means the nth iteration
f f · · · f n
and f 0 means the idenity. The Möbius function is the arithmetic function µ(n) : N → {−1, 0, 1} defined as
µ(n) =
1 if n = 1; (−1)r if n = p1 · · · pr for r distinct prime numbers pi; 0 if p2|n for a prime number p.
Sarnak proposed a conjecture which makes a connection between number theory and ergodic theory. The conjecture can be stated as
Conjecture 1 (Sarnak [14, 15]). The Möbius function is linearly disjoint from (i.e. or- thogonal to) all flows with zero topological entropy.
The linear disjointness can be defined more generally. Suppose c = (cn)n∈N− is a sequence of complex numbers.
Definition 1 (Linear disjointness). We say c is linearly disjoint from X if
lim N→∞
1 N
for any ∈ C(X,C) and any x ∈ X .
In order to understand conjecture 1 in ergodic theory, we gave the following definition in [5].
Definition 2 (Oscillating Sequence). The sequence c is said to be an oscillating sequence if
lim N→∞
1 N
with a technical condition
for some constant λ > 1.
An important example of an oscillating sequence is the Möbius sequence u = (µ(n))n∈N due to Davenport’s theorem [2]. However, there are many other oscillating sequences generated
Y Jiang Nonlinearity 31 (2018) 2689
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by arithmetic functions, for example, the Liouville function defined as l(n) = (−1)(n), where (n) is the number of prime factors of n, counted with multiplicity (see [1, 4]). There are
also many oscillating sequences from complex analysis and harmonic analysis, for examples,
c = ( e2πin2α
( e2πicn log n
) for any fixed constant
c > 0 are oscillating sequences. One can find other examples of oscillating sequences in [5]. Therefore, our new result theorem 2 and corollary 1 applies not only to Sarnak’s conjecture in number theory corollary 2 but also to many other branches in mathematics like ergodic theory, probability, and harmonic analysis.
The key point of conjecture 1 in the view of definition 1 is that the convergence is for every point in the space. However, in ergodic theory, the convergence is for almost all points for a reference measure. To distinct our main result theorem 2 and corollary 1 to those results in ergodic theory, we give a definition of a log-uniform oscillating sequence and prove a result in ergodic theory. This result in ergodic theory indicates why our study of the MMA (minimally mean-attractable)-MMLS (minimally mean-L-stable) property for flows is important.
Definition 3 (Log-uniform oscillating sequence). A sequence c is said to be a log- uniform oscillating sequence if there are constants A > 1 and B > 0 such that
max 0t<1
with also the technical condition (3).
An example of a log-uniform oscillating sequence is the Möbius sequence u due to Davenport’s theorem [2].
Suppose B is the standard Borel σ-algebra on X (here we can relax the compactness con- dition on X). Suppose ν is a Borel measure on X. Then (X,B, ν) is a measurable space. A measurable and bijective map f : (X,B, ν) → (X,B, ν) is called an automorphism if it is measure-preserving, that is, ν( f−1(A)) = ν(A) for every A ∈ B (in other words, we also called ν f-invariant). Let L1(X,B, ν) be the space of all L1-functions on X.
Theorem 1. Suppose f : (X,B, ν) → (X,B, ν) is an automorphism. Suppose c = (cn) is a log-uniform oscillating sequence. Then for any φ ∈ L1(X,B, ν), we have that
lim N→∞
1 N
cn( f nx) = 0, for ν − a.e. x ∈ X. (5)
Since this theorem is not our main topic, just for a comparison, we will prove it in the last section 3. In general, it is difficult to get the convergence for every point in the above theorem. That is a reason we need the following definitions.
Suppose E ⊂ N. The upper density D(E) of E in N is
D(E) = lim sup N→∞
N .
Definition 4 (MLS). A flow X is said to be mean-L-stable (MLS) if for every ε > 0, there is a δ > 0 such that d(x, y) < δ implies d( f nx, f ny) < ε for all n ∈ N except a subset E of natural numbers with upper density D(E) less than ε.
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Mean-L-stable flows are an important class of dynamical systems in ergodic theory. An example of a MLS flow is an equicontinuous flow.
Definition 5 (Equicontinuity). A flow X is said to be equicontinuous if for every ε > 0, there is a δ > 0 such that d(x, y) < δ implies d( f nx, f ny) < ε for all n ∈ N.
We says a closed subset K ⊂ X is minimal (with respect to f) if f (K) ⊂ K and if { f nx | n ∈ N} = K for every x ∈ K. We use K to denote the sub-flow {( f |K)n : K → K}n∈{0}∪N. Again, for the purpose of understanding conjecture 1 in ergodic theory, we gave the following definitions in [5].
Definition 6 (MMLS). We say that a flow X is minimally MLS (MMLS) if for every mini- mal subset K ⊆ X , the sub-flow K is MLS.
Definition 7 (MMA). Suppose X is a flow. Suppose K ⊂ X is minimal. We say x ∈ X is mean-attracted to K if for any ε > 0 there is a point z = zε,x ∈ K (depending on x and ε) such that
lim sup N→∞
d( f nx, f nz) < ε. (6)
The basin of mean-attraction of K, denoted Basin(K), is defined to be the set of all points x which are mean-attracted to K. It is trivial that K ⊂ Basin(K). We call X minimally mean- attractable (MMA) if
X = K
Basin(K) (7)
where K varies among all minimal subsets of X.
Recall that a point x ∈ X is attracted to K if
lim n→∞
d( f nx, K) = 0.
In general, this does not imply to that x is mean-attracted to K. We proved the following theorem in [5]. The reader is encouraged to compare this result
with theorem 1.
Theorem 1 (MMA/MMLS and disjointness). Any oscillating sequence c is linearly disjoint from all MMA and MMLS flows X . Moreover, the limit in (1) is uniform on every minimal subset K.
Since a MMA and MMLS flow has zero topological entropy (for example, this follows from [9]), this theorem confirms Sarnak’s conjecture for all MMA and MMLS flows. We provided some examples of flows which are MMA and MMLS in [5]. However, these exam- ples except for Denjoy counter-examples are equicontinuous when they are restricted on their minimal subsets. It is an interesting problem to find further examples which are MMA and MMLS but not equicontinuous when they are restricted on their minimal subsets. We give a complete answer to this problem for continuous interval maps with zero topological entropy in this paper. The main result in this paper is that
Theorem 2 (Main theorem). Suppose I = [a, b] ⊂ R is a closed interval. Suppose f : I → I is a continuous map with zero topological entropy. Then the flow X generated by f is a MMA and MMLS flow.
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We will prove this theorem in section 2. Now let us discuss some very important conse- quences from our main result theorem 2. The first one is our main goal in this area.
Corollary 1 (Continuous interval map and disjointness). Suppose I = [a, b] ⊂ R is a closed interval. Suppose f : I → I is a continuous map with zero topological entropy. Then any oscillating sequence c is linearly disjoint from the flow X generated by f.
The proof of corollary 1 is given by combining the proof of theorem 2 and the proof of theorem 1. This proof provides a completely new proof of Sarnak’s conjecture for all flows generated by continuous interval maps with zero topological entropy (see [8] for a different proof and remark 1 for another approach) as follows.
Corollary 2 (Continuous interval map and Möbius function). Suppose I = [a, b] ⊂ R is a closed interval. Suppose f : I → I is a continuous map with zero topologi- cal entropy. The Möbius function is linearly disjoint from the flow X generated by f.
Let us also discuss some consequence of our main result theorem 2 in ergodic theory. Suppose K ⊂ X is a subspace. Let BK be the standard Borel σ-algebra on K and let ν be a Borel probability measure on K. Let L2(K,BK , ν) be the space of all L2 functions on K.
Suppose f is a measure-preserving map from (K,BK , ν) into itself. A complex number λ is called an eigenvalue if there is a function φ ∈ L2(K,BK , ν) such that φ( fx) = λφ(x) for ν-a.e. x ∈ K. Here φ is called an eigenfunction with respect to λ. We say f : (K,BK , ν) → (K,BK , ν) has discrete spectrum or pure point spectrum if there exists an orthonormal L2(K,BK , ν)-basis consisting of eigenfunctions. Theorem 2 provides a non-trivial example of flows having dis- crete spectrum (see [9, theorem 3.8]).
Corollary 3 (Continuous interval map and discrete spectrum). Suppose I = [a, b] ⊂ R is a closed interval. Suppose f : I → I is a continuous map with zero topo- logical entropy. Suppose X is the flow generated by f. Suppose K ⊆ X is a minimal subset with respect to f. Then for every ergodic f-invariant measure ν on K, f : (K,B, ν) → (K,B, ν) is a measure-preserving map having discrete spectrum.
Remark 1. Corollary 3 combining with the work in [11] (see also [6]) gives another proof of Sarnak’s conjecture for all flows generated by continuous interval maps with zero topologi- cal entropy as present in corollary 2.
Remark 2. I would like to point out that there are flows generated by higher-dimensional maps with zero topological entropy which are not MLS when they are restricted on their mini- mal subsets. Thus in the higher-dimensional case, the oscillation property alone is not enough. We need instead the higher-order oscillation property. The reader who is interested in the higher-order oscillation property for a sequence c and the role it plays in Sarnak’s conjecture for higher-dimensional dynamical systems is encouraged to read my paper [7].
2. Proof of the main result theorem 2
Suppose I = [a, b] ⊂ R is a closed interval and f : I → I is a continuous map. Suppose the topological entropy ent( f ) = 0. In other words, this means that f cannot have any horseshoe- type invariant subset in I. More precisely, this means that f can not have a forward f-invariant closed subset X ⊂ I such that f : X → X is topologically conjugate (or semi-conjugate) to a sub-shift of finite type σA : ΣA → ΣA with the maximal Lyapunov exponent greater than 0.
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For the sub-shift of finite type σA : ΣA → ΣA, we mean that A = (aij)d×d is a d × d-square matrix of 0’s and 1’s,
ΣA = {w = i0i1 · · · in−1in · · · | in−1 ∈ {1, 2, · · · , d}, ain−1in = 1, n ∈ N}
and
σA(w) = i1 · · · in−1in · · · .
By abuse of notation, we use ΣA to mean a horseshoe-type forward f-invariant subset X ⊂ I . Since A is a nonnegative square matrix, from the Perron–Frobenius theorem, it has a maximal nonnegative eigenvalue eA. If A is also irreducible, eA is simple, unique, and positive. The statement that maximal Lyapunov exponent of σA is greater than 0 is equivalent to saying that eA > 1.
A point p ∈ X is called a periodic point of period n 1 if fn( p ) = p. The minimal such n is called the primitive period. Following the proof of Sharkovskii’s theorem (for example, the reader may refer to book [3] or book [13, chapter 4]), we see that if f has a periodic point whose period is different from all powers of 2, then f has a horseshoe-type invariant subset. For example, if f has a periodic point of period 3, then f has a horseshoe-type invariant subset σA : ΣA → ΣA for
A =
5)/2. Thus we have that
Lemma 1. The period of any periodic point of f is 2n for some n 0.
For any x ∈ I , let
ω(x) = ∩∞ n=0{ f kx | k n},
be the ω-limit set of x under f. Let K = ω(x). Consider the sub-flow
K = {( f |K)n : K → K}n∈{0}∪N. (8)
It is a minimal flow (refer to [13] or the argument after lemma 2). We have two cases:
(I) the set K is finite and (II) the set K is infinite.
Case (I) is easier. In this case,
K = { p0, p1, · · · , p2n−1}
is a periodic orbit of period 2n for some n 0. That is, f ( pi) = pi+1 (mod 2n) for i = 0, 1, · · · , 2n − 1. Therefore, the following lemma is easily verified.
Lemma 2. In Case (I), the sub-flow (8) is equicontinuous, thus, MLS. Moreover, x is mean-attractable to K.
Case (II) is more complicated. Since the topological entropy of f is zero, in this case, K con- tains no periodic point (see, for example, [13, chapter 5]). Since f cannot have any horseshoe- type forward f-invariant subset in I, we can find a sequence of sets
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of closed intervals In,k such that
(1) for each n 1, intervals in ηn are pairwise disjoint; (2) f (In,k) = In,k+1 (mod 2n) for any n 1 and 0 k 2n − 1; (3) for each n 1 and In,k ∈ ηn, two and only two intervals In+1,k and In+1,k+2n in ηn+1 are
sub-intervals of In,k, that is,
In+1,k ∪ In+1,k+2n ⊆ In,k;
(4) K ⊆ ∩∞ n=1 ∪
2n−1 k=0 In,k .
This is a well-known fact about continuous interval maps with zero topological entropy among all experts in this field (for example, refer to [13, Chapter 5]).
We put a symbolic coding on each interval In,k in ηn for n 1 and 0 k 2n − 1. Consider the binary expansion of k as
k = i0 + i12 + · · · im2m + · · ·+ in−12n−1
where i0, i1, · · · , im, · · · , in−1 = 0 or 1. Let us only remember the coding wn = i0i1 · · · im · · · in−1 and label In,k by wn, that is, In,k = Iwn.
Define the space of codings
Σn = {wn = i0i1 · · · im · · · in−1 | im ∈ {0, 1}, m = 0, 1, · · · , n − 1}
with the product topology. We have a natural shift map
σn(wn) = i1 · · · im · · · in−1 : Σn → Σn−1.
It is a continuous 2-to-1 map. Then we have an inverse limit system
{(σn : Σn → Σn−1); n ∈ N}.
Let
(σ : Σ → Σ) = lim ←−
(σn : Σn → Σn−1).
Explicitly, we have that
Σ = {w = i0i1 · · · in−1 · · · | in−1 ∈ {0, 1}, n ∈ N}
and
Consider the metric d(·, ·) on Σ defined as
d(w, w′) =
∞∑ n=1
|in−1 − i′n−1| 2n
for w = i0i1 · · · in−1 · · · and w′ = i′0i′1 · · · i′n−1 · · ·. It induces the same topology as the topology from the inverse limit and makes Σ a compact metric space.
On Σ, we have a map called the adding machine denoted as add and defined as follows: for any w = i0i1 · · · ∈ Σ, consider the formal binary expansion a =
∑∞ n=1 in−12n−1, then
a + 1 = ∑∞
n=1 i′n−12n−1 has a unique formal binary expansion. Here ‘a + 1’ is just a notation. It means if i0 = 0, then i0 = 1 and all other i′n = in and if i0 = 1, then i0 = 0 and then consider i1 + 1, and so on. The adding machine is
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add(w) = i′0i′1 · · · i′n−1 · · · : Σ → Σ.
It is a homeomorphism of Σ to itself. Moreover, we have
Lemma 3. The flow
is minimal and equicontinuous.
We leave the proof for the reader since it is not difficult. For a point w = i0i1 · · · in−1 · · · ∈ Σ, let wn = i0i1 · · · in−1. We use
[w]n = {w′ ∈ Σ | w′ n = wn}
to denote the n-cylinder containing w. One can check that for every w ∈ Σ,
· · · ⊂ Iwn ⊂ Iwn−1 ⊂ · · · ⊂ Iw2 ⊂ Iw1 ,
is a nested sequence of closed intervals. Therefore
Iw = ∩∞ n=1Iwn
is a non-empty, connected, and compact subset of I. Moreover, we have that f (Iw) = Iadd(w) and
K ⊂ ∪w∈ΣIw.
The set C = {[w]n | w ∈ Σ, n ∈ N} of all n-cylinders forms a topological basis for Σ. Let B be the σ-algebra generated by C. We have a standard probability measure ν defined on B such that ν([w]n) = 1/2n for all w ∈ Σ. We say ν is add-invariant if ν(add−1(A)) = ν(A) for all A ∈ B. We say ν is ergodic if add−1(A) = A for A ∈ B implies either ν(A) = 0 or 1. The following lemma is also well-known among experts in this field and not difficult to be verified. We also leave it to the reader.
Lemma 4. The measure ν is add-invariant and ergodic.
In Case (II), we consider two different cases:
(a) there is a point w0 ∈ Σ such that Iw0 = {xw0} contains only one point or (b) all Iw for w ∈ Σ are closed intervals.
In Case (II)(a), we have that ω(x) = ω(xw0). Without loss of generality, we assume x = xw0 and every Iw = {xw} contains only one point. Let
τn = max wn∈Σn
|Iwn |.
We claim that τn → 0 as n → ∞. We prove it by contradiction. If there is a subsequence {nk}∞k=1 of the natural numbers and a number ε0 > 0 such that τnk ε0, then we have an inter- val Iwnk
∈ ηnk such that |Iwnk | ε0. Since Σ is a compact metric space, we have a convergent
subsequence of {wnk}∞k=1 in Σ. Suppose {wnk}∞k=1 is itself a convergent sequence with the limit point w ∈ Σ, then the interval Iwnk
converges to an interval Iw as k → ∞. The length |Iw| is greater than ε0. This contradicts the fact Iw contains only one point. This proves the claim.
Lemma 5. In Case (II)(a), the sub-flow K in (8) is equicontinuous, thus, MLS.
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Proof. We can set a one-to-one and onto correspondence h(w) = xw from Σ to K. Since τn → 0 as n → ∞, for any ε > 0, there is a N > 0 such that τN < ε. For the given N, we can find a δ > 0 such that for any w, w′ ∈ Σ with d(w, w′) < δ , then wN = w′
N . This implies that both points xw and xw are in the same interval IwN. So |h(w)− h(w′)| τN < ε. This proves that h is continuous. Since both Σ and K are compact, we have that h−1 is also continuous. Thus h is a homeomorphism. Since f |K = h add h−1 and since the flow A is equicontinu- ous, the sub-flow K is equicontinuous.
Lemma 6. In Case (II)(a), x is mean-attracted to K.
Proof. Since K ⊂ ∩∞ n=1 ∪wn∈Σn Iwn and since τn → 0 as n → ∞, for any ε > 0, there is
an integer n0 > 0 such that τn0 < ε. Then we have an integer m0 > 0 such that f m0 x ∈ Iwn0
for some w ∈ Σ. Since K ∩ Iwn0 = ∅ and f |K = h add h−1 is a homeomorphism, we have
z ∈ K such that f m0 z ∈ K ∩ Iwn0 . This implies that
| f m0 x − f m0 z| τn0 < ε.
Since f m0+n(Iwn0 ) ∈ ηn0 for all n 0, we have that
| f m0+nx − f m0+nz| τn0 < ε
for all n 0. This implies that
lim sup N→∞
We proved that x is mean-attracted to K.
In Case (II)(b), we consider a fixed interval Iw0. Let
ε0 = |Iw0 | > 0.
The orbit {addn(w0) | n ∈ Z} is dense in Σ. Then we can find two points u, v ∈ K such that the distance |u − v| is arbitrary small and such that there are infinitely many positive integers n such that Iadd−n(w0) ⊂ Iuv. Then we have
f n(Iuv) ⊃ f n(Iadd−n(w0)) = Iw0 .
Thus | f nu − f nv| ε0 for infinitely many positive integers n. This proves that
Lemma 7. In Case (II)(b), the sub-flow K in (8) is not equicontinuous.
The main work in this paper is to prove the sub-flow K in (8) is still MLS in Case (II)(b).
Lemma 8. In Case (II)(b), the sub-flow (8) is MLS.
Proof. Since the probability measure ν is add-invariant and ergodic, the Birkhoff ergodic theorem implies that
lim N→∞
= lim N→∞
= ν(A),
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for any A ∈ B and ν-a.e. w ∈ Σ. For any ε > 0, since Iw ∩ Iw′ = ∅ for all w = w′ ∈ Σ and since
∑ w∈Σ |Iw| |I| < ∞,
there are an integer N0 > |I|/ε+ 1 and 0 m0 |I|/ε+ 1 points, denoted as wj, 1 j m0, in Σ such that for any w = w j ∈ Σ for all 1 j m0 and n N0, |Iwn | < ε.
Let N1 > N0 be an integer such that ν([w]N1) < ε/m0 for all w ∈ Σ. We have a number δ > 0 such that for any u, v ∈ K with |u − v| < δ, the interval [u, v] ⊂ Iwu,v
N1 for some wu,v ∈ Σ.
For any given |u − v| < δ and for each 1 j m0, define
Ej = {k ∈ N | add−k([w j]N1) = [wu,v]N1}.
From the Birkhoff ergodic theorem (see the formula in the beginning of the proof), we have that
lim N→∞
lim N→∞
m0∑ j=1
#(Ej ∩ [1, N])
N < ε. (10)
Thus we have that the upper density D(E) is less than ε. For any n ∈ N \ E , addn([wu,v]N1) = [w j]N1 for all 1 j m0. This implies f n(u),
f n(v) ∈ IwN1 for w = w j for all 1 j m0. Thus,
| f n(u)− f n(v)| < ε, ∀ n ∈ N \ E. (11)
Both (10) and (11) imply that f : K → K is MLS. We completed the proof.
Lemma 9. In Case (II)(b), x is mean-attracted to K.
Proof. We use the same notation as that in the proof of lemma 8. For any ε > 0, since Iw ∩ Iw′ = ∅ for all w = w′ ∈ Σ and since
∑ w∈Σ |Iw| |I| < ∞, there are an integer
N0 > |I|/ε+ 1 and 0 m0 |I|/ε+ 1 points, denoted as wj, 1 j m0, in Σ such that for any w = w j ∈ Σ for all 1 j m0 and n N0, |Iwn | < ε/3.
Let N1 > N0 be an integer such that ν([w]N1) < ε/(3|I|m0) for all w ∈ Σ. Fix an inter- val Iw′
N1 with length less than ε. There is an integer k 0 such that f kx ∈ Iw′
N1 . Take
z0 ∈ Iw′ N1 ∩ K . Then we have a point z = zx,ε ∈ I(add−k(w′))N1
∩ K such that f kz = z0. Thus we have that | f kx − f kz| < ε.
Define
Ej = {k ∈ N | add−k([w j]N1) = [w′]N1}.
From the Birkhoff ergodic theorem (see the formula in the beginning of the proof of lemma 8), we have that
lim N→∞
2699
lim N→∞
k∑ n=1
| f k+nx − f k+nz|+ 1 N
∑ n∈N,k+n∈E
= I + II + III.
First for N large, we have I < ε/3. Secondly, for any n ∈ N such that k + n ∈ N \ E,
[w]N1 = addk+n([w′]N1) = [w j]N1 .
This implies f k+nx, f k+nz ∈ IwN1 for w = w j for all 1 j m0. Thus,
| f k+nx − f k+nz| < ε
3 , ∀ n ∈ N and k + n ∈ N \ E.
This implies that II < ε/3.
Finally, since | f k+nx − f k+nz| |I| is always true, we have
III #(E ∩ [1, N])|I|
N .
The upper density of E given in (13) implies that
lim sup N→∞
This proves that x is mean-attracted to K.
Lemmas 2, 5, 6, 8 and 9 complete the proof of theorem 2.
Remark 3. The proof given in this section not only works for continuous interval maps with zero topological entropy but also works for any dynamical system f : X → X of a com- pact space X satisfying that (i) it is MMA and (ii) when f is restricted to any minimal subset K ⊆ X , f : K → K is semi-conjugate to the adding machine add : Σ → Σ, that is, there is a continuous surjective map h : K → Σ such that h f = add h. Jian Li sent me a note after the first version of this paper was appeared in ArXiv and mentioned that they are working
Y Jiang Nonlinearity 31 (2018) 2689
2700
on a similar result as stated in corollary 2 for graph maps with zero topological entropy by using a different approach [10]. Graph maps with zero topological entropy and circle ho- meomorphisms and continuous interval maps with zero topological entropy are interesting examples satisfying (i) and (ii). Thus our proof in this section could be used to prove not only continuous interval maps with zero topological entropy but also graph maps with zero topo- logical entropy and circle homeomorphisms. It is still not clear to me if there is an example of a dynamical system f : X → X of a compact space X which satisfies only (i) but not (ii), however, it is still MMLS.
3. Proof of theorem 1
Without loss of generality, we assume that ν is ergodic. Otherwise, we consider on every ergodic component one by one. For any φ ∈ L1(X,B, ν), we define
σφ(n) = ∫
∫
These numbers character a measure σφ on the unit circle
T = {e2πit | 0 t < 1}
∫
SN(x) = 1 N
N∑ n=1
cnφ( f nx).
Since c = (cn) is a log-uniform oscillating sequence, we have two constants A > 1 and B > 0 such that
SN2 = (∫
, ∀ N 2.
For any r > 1, let [rm] be the largest integer less than or equal to rm. Then we have that
S[rm]2 B
This implies that
S[rm]2 B
S[rm](x) = 1
cnφ( f nx) → 0 as m → ∞ for ν − a.e.x ∈ X.
Y Jiang Nonlinearity 31 (2018) 2689
2701
For any integer N > 0, we have a unique integer m > 0 such that [rm] N < [rm] + 1 [rm+1], then we have that
|SN(x)| = 1 N
N∑ n=1
1 N
N∑ [rm]+1
cnφ( f nx) .
Let us first prove the theorem under the assumption that φ ∈ L∞(X,B, ν). Under this assumption, let λ′ > 1 be the dual number of λ in the technical condition (3), that is, 1/λ+ 1/λ′ = 1. The Hölder inequality implies that
1 N
λ′ .
1 λ′ as m → ∞.
Therefore, for any ε > 0, we have r > 1 such that Cφ∞(r − 1) < ε/4. For this fixed r, we have a m0 > 0 such that
C 1 λ φ∞
1 λ′
4 , ∀ m > m0,
and |S[rm](x)| < ε/2 for ν-a.e. x ∈ X for all m > m0. Thus we get that
|SN(x)| < ε for ν − a.e. x ∈ X and m > m0.
We proved that
SN(x) → 0 as m → ∞ for ν − a.e. x ∈ X
when φ ∈ L∞(X,B, ν). In general, we find a sequence {φm}∞m=1 ⊂ L∞(X,B, ν) such that
φ− φm1 =
Note that
lim N→∞
Y Jiang Nonlinearity 31 (2018) 2689
2702
For any ε > 0, we have m > 0 such that φ− φm1 < ε/2. For this fixed m, from the first part of the proof, we have a N0 > 0 such that
IN,m(x), IIN,m(x) < ε
Thus we get SN(x)
≤ ε, for ν-a.e. x ∈ X and any N > N0.
Therefore, SN(x) → 0 as N → ∞ for ν-a.e. x ∈ X . This completes the proof.
Acknowledgment
This paper was completed during my visit to the National Center for Theoretical Sciences (NCTS) at National Taiwan University in 2016. I would like to thank NCTS for its hospitality. I would like to thank my student John Adamski for his help on revising this paper.
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Y Jiang Nonlinearity 31 (2018) 2689
Abstract
3. Proof of theorem 1
Acknowledgment
References
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