In [Proc. ECIT-89, World Scientific, (1991), 177–183], A. N. Sharkovski˘ı and S.F. Kolyada stated the problem of characterization skew-product maps having zero topological entropy. It is known that, even under some additional assumptions, this aim has not been reached. In [J. Math. Anal. Appl., 287, (2003), 516–521], J. L. G. Guirao and J. Chudziak partially solved the problem in the class of skew-product maps with base map having closed set of periodic points. The present paper has two aims for this class of maps, on one hand to improve that solution showing the equivalence between the property “to have zero topological entropy” and the fact “not to be Li-Yorke chaotic in the union of the ω-limit sets of recurrent points”. On other hand, we show that the properties “to have closed set of periodic points” and “all
nonwandering points are periodic” are not mutually equivalent properties, for doing this we disprove a result from Efremova of 1990

García Guirao, Juan Luis

López Guerrero, Miguel Ángel

idUS. Depósito de Investigación Universidad de Sevilla

XX Congreso de Ecuaciones Diferenciales y AplicacionesX Congreso de Matema´tica AplicadaSevilla, 24-28 septiembre 2007(pp. 1–6)Skew-product maps with base having closed set of periodicpoints.Juan Luis Garc´ıa Guirao1, Miguel A´ngel Lo´pezGuerrero21 Dpto. de Matema´tica Aplicada y Estad´ıstica Campus de la Muralla, Universidad de Polite´cnica deCartagena, E-30203 Cartagena. E-mails: juan.garcia@upct.es,http://www.dmae.upct.es/ jlguirao.2 Dpto. de Matema´ticas Universidad de Castilla-La Mancha, E.U. Polite´cnica de Cuenca, E-16071Cuenca. E-mail: mangel.lopez@uclm.es.Key words: discrete dynamical system; skew-product map; topological entropy; Li-Yorke chaos;non-wandering pointsAbstractIn [Proc. ECIT-89, World Scientific, (1991), 177–183], A. N. Sharkovski˘ı and S.F. Kolyada stated the problem of characterization skew-product maps having zerotopological entropy. It is known that, even under some additional assumptions, thisaim has not been reached. In [J. Math. Anal. Appl., 287, (2003), 516–521], J. L.G. Guirao and J. Chudziak partially solved the problem in the class of skew-productmaps with base map having closed set of periodic points. The present paper hastwo aims for this class of maps, on one hand to improve that solution showing theequivalence between the property “to have zero topological entropy” and the fact “notto be Li-Yorke chaotic in the union of the ω-limit sets of recurrent points”. On otherhand, we show that the properties “to have closed set of periodic points” and “allnonwandering points are periodic” are not mutually equivalent properties, for doingthis we disprove a result from Efremova of 1990.1 Introduction, Notation and Statement of the main resultsOur frame of working will be discrete dynamical systems induced by skew-product mapsdefined on the unit square I2 = [0, 1]× [0, 1], i.e., continuous transformations from I2 intoitself of the form F : (x, y) → (f(x), g(x, y)). The maps f and g are respectively calledthe base and the fiber map of F . Obviously, for every x ∈ I, the maps gx defined by1Juan L.G. Guirao, M.A. Lo´pezgx(y) := g(x, y) form a system of one-dimensional maps depending continuously on x. Forhaving more information on this type of systems, see for instance [1], [3], [4], [14] or [15].By C∆(I2) we denote the class of skew-product maps on the unit square. Let F bean element of C∆(I2): for every x ∈ I2 and every integer n ≥ 1, we define Fn(x) =F (Fn−1(x)) and F 0 as the identity map on I2. A point x ∈ I2 is said to be periodic byF if there exists a positive integer n such that Fn(x) = x. The smallest of the valuesn satisfying the previous condition is called the period of x. By P(F ) we denote the setof all periodic points by F and by Per(F ) we denote the set of all periods of the pointsof P(F ). For an x ∈ I2, we define the ω-limit set ωF (x) of x by F as the set of ally ∈ I2 such that there exists a sequence of positive integers {nk}∞k=0 holding Fnk(x)→ y,where k → ∞. Let Rec(F ) be the set of recurrent points of F , i.e., the set of all x ∈ I2such that x is an accumulation point of (Fn(x))∞n=0. A closed invariant set M ⊆ I2 (i.e.,F (M) ⊆ M) is called minimal by F if it is non-empty and it does not contain properclosed invarinat subsets. By UR(F ) we denote the set of uniformly recurrent points of F ,i.e., all recurrent points with minimal ω-limit sets. A pair of points {x, y} ⊂ I2 is said to bea Li-Yorke pair of a map F , if simultaneously holds lim infn→∞ d(Fn(x), Fn(y)) = 0 andlim supn→∞ d(Fn(x), Fn(y)) > 0. A point x ∈ I2 is a nonwandering point of F providedthat for any neighborhood U of x there exists a positive integerm such that Fm(U)∩U 6= ∅.The set of nonwandering points of F is denoted by Ω(F ). Given a subset A ⊆ I2, we saythat F |A is chaotic (in the sense of Li-Yorke, see [17]) if A contains a Li-Yorke pair of F .To detect the presence of simple dynamic in a given discrete system is an importantproblem in mathematics. The Bowen’s definition (see [5]) of the notion of topologicalentropy is a good tool for reaching this aim. In this setting, if the system has zerotopological entropy the dynamical behaviour can be understood as simple. On the contraryif the entropy is positive a complex dynamics appear. Therefore, a natural problem arrives:to find topological characterizations of the notion of zero topological entropy. For intervalsystems, i.e., discrete systems of the form (I, f), where f is a continuous self-map of I,there exists a long list of equivalent properties to (P1): f has zero entropy (h(f)=0) (see[19]). Some of the most representative of such properties are the following:(P2): the topological entropy of f|Rec(f) is 0 (h(f|Rec(f))= 0),(P3): f|Rec(f) is non-chaotic,(P4): every recurrent point of f is uniformly recurrent (Rec(f)=UR(f)),(P5): the period of every periodic point is power of two.The equivalence between (P1) - (P5) for the interval case establishes an useful proce-dure for discover dynamical simplicity.In 1989, A.N. Sharkovski˘ı and S.F. Kolyada (see [18]) formulated the problem of study-ing the relations between the properties (P1)− (P5) on the setting of skew-product mapsof the unit square. It is well known that they are not mutually equivalent (see [2], [8], [12],[14], [16]). Moreover, even under some additional assumptions on the skew-product mapF , the equivalence is not reached. In [13], Z. Kocˇan proved that in the case of skew-productmaps non-decreasing on the fibres (i.e., on sets of the form Ix = {x}× I, x ∈ I) conditions(P1), (P2) and (P5) are equivalent, (P3) implies (P4) and (P4) implies (P1). However(see [13, cf. Lemma 4.2]) there exists an example of a skew-product map non-decreasingon the fibres holding (P2) but nor (P3) neither (P4) (this example is based on the ideasfrom [9]). The implication from (P4) to (P3) has been recently disproved by J. Chudziaket al. in [7] by taking an appropriate Floyd-Auslander minimal system and then taking2Skew-product maps with base having closed set of periodic pointsits appropriate continuous extension to a skew-product map of the square non-decreasingon the fibres.In [11], J.L.G. Guirao and J. Chudziak considered the problem of the equivalence of(P1)− (P5) in the class of skew-product maps with base map having closed set of periodicpoints. Under such assumption was proved that conditions (P1) − (P5) are mutuallyequivalent. In that setting the relation between (P1) and (P3) states that the property ofhaving zero topological entropy is equivalent to the presence of some “order” (in the sensethat there is no chaos) in the set of recurrent points for maps in C∆(I2) having bases withclosed sets of periodic points. The first objective of the present paper is to remark thatthe previous result can be easily improve by proving the equivalence between (P1) anda stronger property than (P3), in the sense that we will have no chaos in a potentiallybigger set than Rec(F ), we called this new property (SP3) (strong (P3)).On other hand, the class of skew-product maps having base map with a closed set ofperiodic point, which works for solving the problem stated by Sharkovski˘ı and Kolyada,was studied by L. Efremova [6] in the nineties of the last century. One of the main resultsin the dynamics of these maps is the following theorem.Theorem 1 (Efremova) Let F ∈ C∆(I2) with base map f such that P(f) is closed. ThenΩ(F ) =⋃x∈P(f){x} × Ω(F px |Ix), (1)where px is the period of x.This result has some important implications, one of them is the equivalence betweenthe following two properties:• P(F ) is closed,• P(F ) = Ω(F ).The second a central aim of this paper is to show that the previous equivalence doesnot hold by disproving Theorem 1.The statement of our main results on the two problems is:Theorem A. Let F ∈ C∆(I2) with base map f such that P(f) is closed. Then the followingproperties are equivalent:(P1) F has zero topological entropy,(SP3) F restricted to⋃(x,y)∈Rec(F )ωF (x, y) is non-chaotic.Theorem B. There exists G ∈ C∆(I2) with base map g such that P(g) is closed, holdingΩ(G) 6=⋃x∈P(g){x} × Ω(Gpx |Ix). (2)3Juan L.G. Guirao, M.A. Lo´pez2 Proof of the main resultsProof of Theorem A.On one hand, by definition (SP3) implies (P3) and by [11], (P3) implies (P1).Let h(F ) = 0. By [10], Rec(F ) = UR(F ), and thus⋃x∈Rec(F )ωF (x) =⋃x∈UR(F )ωF (x) = UR(F ).Indeed, by definition every uniformly recurrent point belongs to its own ω-limit set andtherefore to⋃x∈UR(F ) ωF (x). On the other hand, let y ∈⋃x∈UR(F ) ωF (x), so there existsx0 ∈ UR(F ) that y ∈ ωF (x0).By definition follows that ωF (y) ⊂ ωF (x0) and since x is uniformly recurrent ωF (x0)is a minimal set and therefore ωF (y) = ωF (x0) from which we have that y is a uniformlyrecurrent point.Thus, F |⋃x∈Rec(F ) ωF (x) = F |Rec(F ) and it is non-chaotic, by [10].Proof of Theorem B. As a base map let g be a continuous interval map having P(g) closedwith an attracting fixed point y∗ isolated in the set P(g).For constructing the fiber map, let h be a continuous interval map with a no nonwan-dering point z∗ but, such that a small modification on the definition of h(z) becames z∗on a nonwandering point of the modified map. Indeed, for example we consider as a maph the piecewise linear map such that 0→ 78 , 14 → 38 , 12 → 12 , 34 → 78 and 1→ 1. Obviously,z∗ = 14 does not belong to Ω(h). On other hand, if we consider hδ (with δ <116) thepiecewise linear map such that hδ(z∗−δ) = z∗−δ and hδ(z) = h(z) for any z ∈ I\(z∗−δ),the point z∗ ∈ Ω(hδ).Since the fixed point y∗ is attracting, there exists an open neighborhood V of y∗ suchthat diam(V) < 116 . We define a skew-product map G : V × I → V × I of the formG(y, z) = (g(y), hδ(z)) where δ = |y− y∗|. By [10] this map can be extended continuouslyto the whole square I2 remaining the skew-product morphology.Let m∗ = (y∗, z∗). We shall show that m∗ ∈ Ω(G)\⋃x∈P(g){x} × Ω(Gpx |Ix). Indeed,y∗ is isolated in P(g) and hy∗ = h, therefore Ω(hy∗) = Ω(h). Since z∗ /∈ Ω(h) and the setof nonwandering points is closed, m∗ /∈ ⋃x∈P(g){x} × Ω(Gpx |Ix).On the converse, we shall verify that m∗ ∈ Ω(G). By the construction of the map h,for any left-side open neighborhoodW of the fixed point c = 12 , there exists a non-negativeinteger m such thatz∗ ∈ hm−1(W). (3)Let M = V ′ × L ⊂ I2 be an open neighborhoof of m∗ where we assume that V ′,L ⊂I and V ′ ⊂ V. We have to show that there exists a non-negative integer l such thatGl(M) ∩M 6= ∅.By definition, the point y∗ is an attracting fixed point for the base map g, thereforethere exists y ∈ V ′\{y∗} such that gn(y) ∈ V ′ for each non-negative integer n. Thus,for ending the proof is enough to show the existence a non-negative integer l such thatz∗ ∈ Kl where by Kl we denote the projection on the second coordinate of Gl(M∩ Iy).By definition of the skew-product map G, Gl(M∩ Iy) = g(G(M∩ Iy)) for every non-negative integer l. Now, since G(M∩ Iy) contains certain left-side open neighborhood Wof the fixed point c, (3) ends the proof showing m∗ ∈ Ω(G).4Skew-product maps with base having closed set of periodic pointsReferences1. C. Arteaga, Smooth triangilar Maps of the Square with Closed Set of Periodic Points.Journal of Mathematical Analysis and Applications. 196 (1995), 987–997.2. F. Balibrea, F. Esquembre and A. Linero, Smooth triangular Maps of type 2∞ withPositive Topological Entropy. Inter. Journal of Bifurcation and Chaos 5 (1995), 1319–1324.3. —–, J.L.G. Guirao and J.I. Mun˜oz, Description of ω-limit sets of a triangular mapon I2. Far East Journal of Dynamical Systems. 3(1) (2001), 87–101.4. —–, —– and —–, A Triangular map on I2 whose ω-limit Sets are all Compact Inter-vals of {0} × I. Discrete and Continuous Dynamical Systems 8(4) (2002), 983–994.5. R. Bowen, Entropy for Group Endomorphism and Homogeneous Spaces, Trans. Amer.Math. Soc. 153 (1971), 401–414.6. L. S. Efremova, On the nonwandering set and the center of triangular maps withclosed set of periodic points in the base, Dynamical Systems and Nonlinear Phenom-ena, Inst. Math. NAS Ukraine, Kiev, 1990, 15 – 25 (in Russian).7. J. Chudziak, L’. Snoha and Sˇpitalsky´, From a Floyd-Auslander minimal system toan odd triangular map, J. Math. Anal. Appl. 296 (2004), 316-318.8. G. L. Forti, L. Paganoni and J. Smı´tal, Strange Triangular Maps of the Square, Bull.Austral. Math. Soc. 51 (1995), 395–415.9. —–, —– and —–, Dynamics of Homeomorphisms on Minimal sets Generated byTriangular mappings, Bull. Austral. Math. Soc. 59 (1999), 1–20.10. M. Grinc and L. Snoha, Jungck theorem for triangular maps and related results ,Applied General Topology. 1 (2000), 83-92.11. J.L.G. Guirao and J. Chudziak, A characterization of zero topological entropy for aclass of triangular mappings, J. Math. Anal.Appl. 287 (2003), 516–521.12. Z. Kocˇan, The Problem of Classification of Triangular maps with Zero TopologicalEntropy, Annales Mathematicae Silesianae, 13 (1999), 181–192.13. —– , Triangular maps Non-Decreasing on the Fibres, Preprint Series of Silesian Uni-versity at Opava. Mathematical Institute at Opava. Preprint MA 31/2001.14. S.F. Kolyada. On Dynamics of Triangular Maps of the Square, Ergodic Theory andDynamical Systems. 12 (1992) 749–768.15. —– and L’. Snoha, On ω-limit Sets of Triangular Maps, Real Analysis Exchange.18(1) (1992/93), 115–130.16. —– , M. Misiurewicz and Lˇ. Snoha, Topological Entropy of Nonautonomous PiecewiseMonotone Dynamical Systems on the Interval, Fundam. Math. 160, No.2 (1999),161–181.5Juan L.G. Guirao, M.A. Lo´pez17. T. Y. Li and J. A. Yorke, Period Three implies Chaos, Amer. Math. Monthly. 82(1975), 985–992.18. A.N. Sharkovski˘ı and S.F. Kolyada, On Topological Dynamics of Triangular Maps ofthe Plane, Proceedings of ECIT-89, Batschums, Austria, World Scientific, Singapore(1991), 177–183.19. —– , —– , A.G. Sivak and V.V. Fedorenko, Dynamics of One Dimensional Maps,(1997) Kluwer Academic Publishers.6

Skew-product maps with base having closed set of periodic points

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Skew-product maps with base having closed set of periodic points

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