Some Results and Problems on Quasi Weakly Almost Periodic Points

Λ43ΨΛ3fi ffi Φ ο Vol.43, No.3 204ff5μ ADVANCES IN MATHEMATICS(CHINA) May, 204 doi: 0.845/sxjz.202002a Some Results and Problems on Quasi Weakly Almost Periodic Points YIN Jiandong, YANG Zhongxuan (Department of Mathematics, Nanchang University, Nanchang, Jiangxi, 33003, P. R. China) Abstract: In this note, we mainly present some results and problems on quasi weakly almost periodic points in discrete dynamical systems. Keywords: quasi weakly almost periodic point; chaos; topological entropy MR(200) Subject Classification: 54H20; 58F08 / CLC number: O89.3 Document code: A Article ID: 000-097(204)03-032-08 0 Introduction For a topological dynamical system (X, f) (a dynamical system for short), where X is a compact metric space with metric d and f : X X is continuous, it is well known that the central problem is the asymptotic behavior or topological structure of the orbits of points. Is the orbit of every point of equal importance? From the viewpoint of pure topology, we know that only the orbits of the points possessing certain recurrence are important. Classically, the recurrence has four layers: periodic points, almost periodic points, recurrent points and non-wandering points, denoted by P(f), A(f), R(f) andω(f) respectively. The latter possesses the mildest recurrence and thus only the orbits generated by the non-wandering points are important and it suffices to study the non-wandering set. In this sense, we may say that all important dynamical behaviors (including the structure of the orbits of points) of a dynamical system concentrate upon its non-wandering set. But the ergodic theory reveals that all the important behaviors of a dynamical system take place on a full measure set. Of course, for such a set, the smaller one is better (in the sense of set inclusion). We know that the non-wandering set is such a set, but is it the smallest one? The answer is negative since the recurrent points also form a full measure set which (and its closure) may be contained properly in the non-wandering set. We also know that the set of almost periodic points (and its closure) is not of full measure. In order to look for the smallest full measure set, Zhou introduced the notions of weakly almost periodic point and measure center, quasi weakly almost periodic point in [30] and [32], respectively. Let x X and U be a neighborhood of x, put P x (U) = n #{0 j<n: f j (x) U}, P x (U) = lim sup n #{0 j<n: f j (x) U}, Received date: 202-02-27. Revised date: 202-07-3. Foundation item: This work is supported by NSFC (No. 26039) and the Natural Science Foundation of Jiangxi Province (No. 2032BAB20009). E-mail: yjdaxf@63.com

322 fl ν 43Ψ where #( ) denotes the cardinality. Apointx X is called a weakly almost periodic point with respect to f or x W(f), if for every neighborhood U of x, P x (U) > 0. A point y X is called a quasi weakly almost periodic point with respect to f or y QW(f), if for every neighborhood V of y, P y (V ) > 0 (see [32] for more details). Clearly, any weakly almost periodic point with respect to f is quasi weakly almost periodic. Let M X. We call M is invariant under f if f(m) M. A closed subset M of X is called the measure center of f if f(m) M, μ(m) = for any μ M(X) andthereisnoproper subset of M with these properties, where M(X) denotes the set of invariant measures of f (see [23] for more details). We denote the measure center of f by M(f). Zhou et al., in the 990s, proved that the closure of all (quasi) weakly almost periodic points is just the measure center and pointed out that almost all the dynamical behaviors of a dynamical system concentrate on its measure center in the framework of ergodic theory. This result makes the intrinsic properties of a dynamical system much clearer not only in the sense of topological structure but also in the sense of ergodic theory. From the definitions of quasi weakly almost periodic point and weakly almost periodic point, we know that a dynamical system (X, f) should be complex if there exists a proper quasi weakly almost periodic point (a quasi weakly almost periodic point but not a weakly almost periodic point) or a proper weakly almost periodic point (a weakly almost periodic point but not a minimal point) with respect to f in X. For simplicity, we denote in this note the sets of all the proper quasi weakly almost periodic points and all the proper weakly almost periodic points with respect to f by PQW(f) andpw(f) respectively, that is, PQW(f) =QW(f) W(f) and PW(f) =W(f) A(f). In general, a dynamical system having positive topological entropy as well as being chaotic is deemed to be very complex. Naturally, whether f has positive entropy and some chaotic properties if PQW(f) or PW(f) is a problem worthy of attention. In the following sections, we fix our attention on this problem. Quasi Weakly Almost Periodic Point and Topological Entropy First, we consider the relationship between positive topological entropy and (quasi) weakly almost periodic point. Let (X, f) be a dynamical system. It is well known that f with positive topological entropy does not imply PQW(f) or PW(f) since there exists a minimal dynamical system with positive topological entropy [34]. Conversely, whether f has positive topological entropy if PQW(f) or PW(f) is just the open problem posed by Zhou in [35] as follows: Problem A Does a dynamical system possessing a proper quasi weakly almost periodic point or a proper weakly almost periodic point have positive topological entropy? Obadalová et al. proved the following result. Theorem. [8] Let F be the skew-product map of the space C I, wherec is the Cantor set and I =[0, ] the unit interval. Then

3fi ΠΞ,»ß : Some Results and Problems on Quasi Weakly Almost Periodic Points 323 () PQW(F ) ; (2) F has zero topological entropy. This result gives part of a negative answer to Problem A. However, whether there exists a dynamical system (X, f) with PQW(f) having positive topological entropy is a valuable problem. Next, we point out that interval self-maps and sub-shifts of finite type have positive topological entropy if they possess a proper quasi weakly almost periodic point or a proper weakly almost periodic point (in fact, such results are known, for the completeness of this note, we give the results but omit their proofs). Theorem.2 [35] Let f : I I be a continuous self map defined on unit interval I =[0, ]. Then f has positive topological entropy iff PQW(f) or PW(f). Theorem.3 [3] Let σ A be the sub-shift of finite type determined by some {0, } matrix A. Thenσ A has positive topological entropy iff PQW(σ A ) or PW(σ A ). According to the results above, Problem A can be reduced to Problem. Does f have positive topological entropy if PW(f)? 2 Quasi Weakly Almost Periodic Point and Chaos The chaotic properties are frequently used in the studies of the complexity of dynamical systems (see [, 3 5, 7 8,, 4, 6 8, 20, 24]). Devaney [8] gave a mathematical definition for a map to be chaotic on the whole space where a map is defined. f is said to be chaotic in the sense of Devaney if it satisfies the following three conditions: () f is topologically transitive, that is, for any nonempty open sets U, V X, there is a non-negative integer k such that f k (U) V ; (2)Thesetofperiodicpointsoff is dense in X; (3) f is sensitively dependent on initial conditions, that is, there is a constant ε>0such that for every point x X and every δ>0, there exist a point y X with d(x, y) <δand an integer n>0 such that d(f n (x),f n (y)) >ε. Since that time, there have been several different definitions of chaos which emphasize different aspects of the map. Some of these are more computable and others are more mathematical. See [7] and [2] for a comparison of many of these definitions. But it is worth noting that Banks et al. [5] showed that () and (2) imply (3) in the definition of chaos in the sense of Devaney. However, sensitivity is understood as being the central idea of chaos. Hence, when deleting one of the three conditions from the definition of chaos in the sense of Devaney, we should delete the second one, namely, the set of periodic points of f is dense in X, as it excludes many complex systems from the chaotic systems in the sense of Devaney. As luck would have it, Ruelle and Takens [20] called a map to be chaotic if it is both topologically transitive and sensitively dependent on initial conditions. From then on, we say that such a map is chaotic in the sense of Takens-Ruelle. In the past several years, some chaotic results were gotten for the systems with proper (quasi) weakly almost periodic points. Before introducing the results, we recall some basic notions. x X is called a transitive point of f if the orbit {x, f(x),f 2 (x), } of x under f is dense

324 fl ν 43Ψ in X. DenotebyTran(f) the set of transitive points of f. (x, y) X X is called a Li-Yorke pair of f if d(f n (x),f n (y)) = 0, lim sup d(f n (x),f n (y)) > 0. f is called to be countable Li-Yorke chaotic if there exist countable Li-Yorke pairs of f. f is called to be Li-Yorke chaotic if there exists an uncountable subset C of X such that any (x, y) C 2 (x y) is a Li-Yorke pair of f. Finally, we recall the notion of distributional chaos. For any (x, y) X X, andanyt with 0 <t diam(x), put ρ xy (j) =d(f j (x),f j (y)), and let Φ xy (t) = Φ xy(t) = lim sup n #{0 j<n: ρ xy(j) <t}, n #{0 j<n: ρ xy(j) <t}. Then Φ xy (t) andφ xy (t) are called the lower distribution functions and the upper distribution functions of x and y, respectively. Obviously, Φ xy (t) Φ xy(t) for any t (0, diam(x)]. If Φ xy (t) Φ xy (t) for all t in an interval, we write Φ xy Φ xy. There are three types of distributional chaos. DC and DC2 are given by the following conditions: DC Φ xy andφ xy(t) =0forsomex, y X and some t>0; DC2 Φ xy andφ xy (t) < forsomex, y X and some t>0. While DC3 means that there are points x, y X such that Φ xy Φ xy. In general, these notions are mutually not equivalent but DC DC2 DC3, and DC2 implies that there exists a Li-Yorke pair of f. f is distributionally chaotic if and only if, for some n, f n is. Theorem 2. [26] f is chaotic in the sense of Takens-Ruelle if Tran(f) PQW(f) or Tran(f) PW(f). Theorem 2.2 [8] There exists a Li-Yorke pair (x, y) off if PQW(f). Theorem 2.3 [8] There exists a Li-Yorke pair (x, y) off if PW(f). Proof By the similar idea to the proof of Theorem 2.2, we can easily prove Theorem 2.3. In fact, such a map f satisfying PQW(f) or PW(f) has countable Li-Yorke pairs. Before proving the result, we give a lemma as follows. Lemma 2. Let x PQW(f). Then f n (x) PQW(f) for any n>0. Proof For any n 0, by [32] we know f n (x) QW(f), thus it suffices to prove that f n (x) / W (f). Suppose on the contrary that there exists n>0 such that f n (x) W(f). Then by the definition of weakly almost periodic point, for any neighborhood U of f n (x), we have k #({l : f l (f n (x)) U, 0 l<k}) > 0. (2.) Noting that x PQW(f), that is x QW(f) but x/ W(f). Thus there exists a neighborhood V of x such that lim sup k #({l : f l (x) V,0 l<k}) > 0, (2.2) but k #({l : f l (x) V,0 l<k}) =0. (2.3)

3fi ΠΞ,»ß : Some Results and Problems on Quasi Weakly Almost Periodic Points 325 From (2.2), we get that there exists m>nsuch that f m (x) V,namely,f m n (f n (x)) V which implies f n (x) f n m (V ). Hence, f n m (V )isanopenneighborhoodoff n (x). Following (2.), we obtain that k #({l : f l (f n (x)) f n m (V ), 0 l<k}) > 0. (2.4) Notice that for any k>0, s {l : f l (f n (x)) f n m (V ), 0 l<k} if and only if s {l + m : f l+m (x) V,0 l<k}. Accordingto(2.3),wehave k #({l + m : f l+m (x) V,0 l<k}) =0. (2.5) Therefore there exists a contradiction between (2.4) and (2.5). This contradiction yields the result of Lemma 2.. Lemma 2.2 Let x PW(f). Then f n (x) PW(f) for any n>0. Proof By a proof similar to Lemma 2., we can easily get this lemmas, so we omit it. Theorem 2.4 f is countable Li-Yorke chaotic if PQW(f) or PW(f). Proof We only prove the result in the case of PQW(f). According to Theorem 2.2, there exist x PQW(f) andy A(f) such that (x, y) is a Li-Yorke pair of f. Thus it suffices to prove that (f n (x),f n (y)) is also a Li-Yorke pair of f for any n>0. Let x n = f n (x), y n = f n (y), n 0. Since (x, y) is a Li-Yorke pair of f, d(x n,y n )=0, lim sup d(x n,y n ) > 0. Hence, there exist two sequences of positive integers {n i } and {m j } such that lim i d(x n i,y ni )=0, lim d(x m j,y mj ) > 0. (2.6) j Since X is a compact metric space, there exists z X such that lim i x ni = z (otherwise, we take a subsequence). By the same argument, we get that there exists z 2 X such that lim i y ni = z 2. From the first formula of (2.6), we easily obtain z = z 2. By the continuity of f n,wegetthat lim d(f ni (x n ),f ni (y n )) = lim d(f ni+n (x),f ni+n (y)) i i = lim d(f n (x ni ),f n (y ni )) = 0. i Therefore d(f k (x n ),f k (y n )) = 0. (2.7) From Lemma 2., for any n 0, we have f n (x) PQW(f). And obviously, f n (y) A(f). Thus from the definitions of quasi weakly almost periodic point and minimal point, we easily get that lim sup d(f k (x n ),f k (y n )) > 0. (2.8) Formula (2.8) together with (2.7) shows that (x n,y n ) is also a Li-Yorke pair of f.

326 fl ν 43Ψ We obtain some chaotic results of f if PQW(f) or PW(f), but all the given results do not imply that f is chaotic in the sense of Li-Yorke. Thus we give the following problem. Problem 2. Is f chaotic in the sense of Li-Yorke if PQW(f) or PW(f)? In [8], the authors got the following result. Theorem 2.5 [8] There exists a skew-product map F of the space C I, wherec is the Cantor set and I =[0, ] the unit interval such that () PQW(F ) ; (2) F is DC2 but not DC. This theorem reveals that a dynamical system (X, f) with PQW(f) may not be DC, namely, the existence of a proper quasi weakly almost periodic point does not imply DC, and of course, does not imply Schweizer-Smital chaos. Hence, the following problem is natural. Problem 2.2 Is f DC2 if PQW(f) or PW(f)? For interval self-maps and sub-shifts of finite type, the above problem has a positive answer since the existence of a proper quasi weakly almost periodic point or a proper weakly almost periodic point is equivalent to such maps being chaotic in the sense of Li-Yorke and Schweizer- Smital chaos. 3 Quasi Weakly Almost Periodic Point and Invariant Measure Zhou et al. proved that quasi weakly almost periodic points have different ergodic properties from those of weakly almost periodic points as follows: Theorem 3. [32] Let (X, f) be a dynamical system, x R(f). The following conditions are equivalent: () x W(f); (2) P x (V (x, ε)) > 0, ε >0; (3) x C x = S m, m M x ; (4) S m = ω(x, f), m M x. Theorem 3.2 [32] Let (X, f) be a dynamical system, x R(f). The following statements are equivalent: () x QW(f); (2) P x (V (x, ε)) > 0, ε >0; (3) x C x ; (4) ω(x, f) = m M x S m ; (5) C x = ω(x, f). Remark 3. For all concepts used here, we refer to [30, 32]. One can easily obtain from Theorems 3. and 3.2 that () x QW(f) if and only if x C x, S m C x for some m M x,andm x is not a singleton. (2) x W(f) if and only if x S m and S m = C x = ω(x, f) for all m M x.inthiscasem x may or may not be a singleton. According to Theorem 3.2, the following corollary is clear.

3fi ΠΞ,»ß : Some Results and Problems on Quasi Weakly Almost Periodic Points 327 Corollary 3. Let (X, f) be a dynamical system, x R(f) \ QW(f), C be the closure of the union of S µ for all μ M x.thencis properly contained in ω(x, f). Thus given x PQW(f), there is a natural problem (see [32, Open problem 4]) that whether there is an m M x such that S m = C x? In 203, He et al. [2] gave examples in a symbol system showing that there exists a dynamical system (X, f) with x PQW(f) such that S m C x for any m M x and also there exists a dynamical system (Y,g) with z PQW(g) andμ M z such that S µ = C z. Thus these examples answer the above problem sufficiently. In 2006, Zhou posed an open problem as follows. Problem 3. [33] Dose f have positive positive topological entropy if f is topologically transitive and the measure center M(f) off is not minimal? Now, we answer this problem negatively. Theorem 3.3 There exists a dynamical system (X, f) such that f is topologically transitive and the measure center M(f) off is not minimal, but f has zero topological entropy. Proof Let f be the skew-product map given in Theorem.. From Theorem., we know that PQW(f) and f has zero topological entropy. Hence, the measure center M(f) off is not minimal. Moreover, from the proof of Theorem. (see [8, Theorem ]), it is not difficult to get that f is topologically transitive. So Theorem 3.3 is proved. References [] Abraham, C., Biau, G. and Cadre, B., Chaotic properties of mappings on a probability space, J. Math. Anal. Appl., 2002, 266(2): 420-43. [2] Akin, E., The General Topology of Dynamical Systems, Graduate Studies in Mathematics, Vol., Providence: AMS, 993. [3] Akin, E., Auslander, J. and Berg, K., When is a transitive map chaotic? Convergence in ergodic theory and probability (Columbus, OH, 993), De Gruyter, Berlin: Ohio University Math. Res. Inst. Pub., 996, 5: 25-40. [4] Balibrea, F., Smítal, J. and Šefánková, M., The three versions of distributional chaos, Chaos, Solitons & Fractals, 2005, 23(5): 58-583. [5] Banks, J., Brooks, J., Cairns, G., Davis, G. and Stacey, P., On Devaney s definition of chaos, Amer. Math. Monthly, 992, 99(4): 332-334. [6] Bowen, R., ω-limit sets for axiom A diffeomorphisms, J. Diff. Eq., 975, 8(2): 333-339. [7] Cheng, C. and Zhang, Z., Chaos Caused by a Topologically Mixing Map, Singapore: In a Dynamical System and Related Topic (Nagoya, 992), World Scientific, 99, 550-572. [8] Devaney, R.L., A First Course in Chaotic Dynamical systems: Theory and Experiment, New York: Westview Press, 992. [9] Fu, X., Chen, F. and Zhao, X., Dynamical properties of 2-torus parabolic maps, Nonlinear Dyn., 2007, 50(3): 539-549. [0] Furstenberg, H., Recurrence in Ergodic Theory and Combinatorial number theory, Princeton: Princeton University Press, 98. [] Glasner, E. and Weiss, B., Sensitive dependence on initial conditions, Nonlinearity, 993, 6(6): 067-075. [2] He, W., Yin, J. and Zhou, Z., On quasi-weakly almost periodic points, Science China Math., 203, 56(3): 597-606. [3] Huang, W. and Ye, X.D., An explicit scattering non-weak mixing example and weak disjointness, Nonlinearity, 2002, 5: -4.

328 fl ν 43Ψ [4] Huang, W. and Ye, X.D., Devaney s chaos or 2-scattering implies Li-Yorke s Chaos, Topology Appl., 2002, 7(3): 259-272. [5] Huang, W. and Ye, X., Topological complexity, return times and weak disjointness, Ergod. Th. Dynam. Sys., 2004, 24(3): 825-846. [6] Li, T. and Yorke, J., Period three implies chaos, Amer. Math. Monthly, 975, 82(0): 985-992. [7] Martelli, M., Dang, M. and Seph, T., Defining chao, Math. Mag., 998, 7(2): 2-22. [8] Obadalová, L. and Smítal, J., Distributional chaos and irregular recurrence, Nonlinear Anal., 200, 72(5): 290-294. [9] Petersen, K., Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc., 970, 24(2): 278-280. [20] Ruelle, D. and Takens, F., On the nature of turbulence, Comm. Math. Phys., 97, 20(3): 67-92. [2] Schweizer, B. and Smital, J., Measures of chaos and a spectral decomposition of dynamical systems of the interval, Trans. Amer. Math. Soc., 994, 344(2): 737-754. [22] Smítal, J. andsĕfánková, M., Distributional chaos for triangular maps, Chaos, Solitons & Fractals, 2004, 2(5): 25-28. [23] Walter, P., An introduction to ergodic theory, New York: Springer-Verlag, 982. [24] Xiong, J., Lü, J. and Tan, F., Furstenberg family and chaos, Sci. China Ser. A, 2007, 50(9): 325-333. [25] Ye, X., Huang, W. and Shao, S., An Introduction of Topological Dynamical Systems, Beijing: Scientific and Technological Press, 2008 (in Chinese). [26] Yin, J. and Zhou, Z., The equivalent definitions of quasi-weakly almost periodic points and chaos, J. Sys. Sci. Math., 200, 30(8): 56-62 (in Chinese). [27] Yang, R., Topologically ergodic maps, Acta Math. Sinica, 200, 44(6): 063-068 (in Chinese). [28] Yang, R., Topological ergodicity and topological double ergodicity, Acta Math. Sin., Chin. Ser., 2003, 46(3): 555-560 (in Chinese). [29] Yang, R. and Shen, L., Pseudo-orbit-tracing and completely positive entropy, Acta Math. Sin., Chin. Ser., 999, 42(): 99-04 (in Chinese). [30] Zhou, Z., Weakly almost periodic point and ergodic measure, Chin. Ann. Math. Ser. B, 992, 3(2): 37-42. [3] Zhou, Z., Symbolic Dynamics, Shanghai: Shanghai Scientific and Technological Education Press, 997 (in Chinese). [32] Zhou, Z. and He, W., Level of the orbit s topological structure and semi-conjugacy, Sci. China Ser. A, 995, 25(5): 897-907 (in Chinese). [33] Zhou, Z. and Le, J., Some problems on fractal geometry and dynamical systems, J. Engineering Math., 2006, 23(5): 76-766 (in Chinese). [34] Zhou, Z., Liao, G. and Wang, L., The positive entropy not equivalent to chaos-a class of subshifts, Sci. China Ser. A, 994, 24(3): 653-660 (in Chinese). [35] Zhou, Z., Yin, J. and Xu, S., Topological Dynamics Systems-from Topological Methods to Ergodic Methods, Beijing: Science Press, 20 (in Chinese). χ/')"!(fiff.,$&ψfl%+* 632, 574 (Ω fl, Ω, ±ffl, 33003) 0-8T_X GO=HVR], >FIZ@[KNCA^M<;YWDJ9P:IY W?EUQ. ffi#ρ KNCA^M<; B`; SLa