"A classification of triangular maps of the square" . . . "SK - Slovensk\u00E1 republika" . "[7C02E498183F]" . "0862-9544" . "Acta Mathematica Universitatis Comenianae" . "A classification of triangular maps of the square"@en . . "2" . "463427" . "P(GA201/03/1153), Z(MSM4781305904)" . . . "triangular map; topological entropy; w-limit set"@en . "12"^^ . . "It is well-known that, for a continuous map $\\varphi$ of the interval, the condition {P1} $\\varphi$ has zero topological entropy, is equivalent, e.g., to any of the following: { P2} any $\\omega $-limit set contains a unique minimal set; { P3} the period of any cycle of $\\varphi$ is a power of two; { P4} any $\\omega$-limit set either is a cycle or contains no cycle; {P5} if $\\omega _\\varphi(\\xi)=\\omega_{\\varphi^2}(\\xi)$, then $\\omega_\\varphi (\\xi)$ is a fixed point; {P6} $\\varphi $ has no homoclinic trajectory; {P7} there is no countably infinite $\\omega$-limit set; {P8} trajectories of any two points are correlated; {P9} there is no closed invariant subset $A$ such that $\\varphi ^m|A$ is topologically almost conjugate to the shift, for some $m\\ge 1$. In the paper we exhibit the relations between these properties in the class $(x,y)\\mapsto (f(x),g_x(y))$ of triangular maps of the square. This contributes to the solution of a longstanding open problem of Sharkovsky."@en . "RIV/47813059:19610/06:#0000097!RIV07-GA0-19610___" . "Je zn\u00E1mo, \u017Ee pro spojit\u00E1 zobrazen\u00ED $\\varphi$ na intervalu je podm\u00EDnka P1 $\\varphi$ m\u00E1 nulovou topologickou entropii, ekvivalentn\u00ED s ka\u017Edou z n\u00E1sleduj\u00EDc\u00EDch: P2 ka\u017Ed\u00E1 $\\omega$-limitn\u00ED mno\u017Eina obsahuje jedinou minim\u00E1ln\u00ED mno\u017Einu; P3 perioda ka\u017Ed\u00E9ho cyklu je mocnina dvou; P4 ka\u017Ed\u00E1 $\\omega$-limitn\u00ED mno\u017Eina je cyklus nebo \u017E\u00E1dn\u00FD cyklus neobsahuje; P5 jestli\u017Ee $\\omega_\\varphi(\\xi)=\\omega_{\\varphi^2}(\\xi)$, pak $\\omega_\\varphi (\\xi)$ je pevn\u00FD bod; P6 $\\varphi$ nem\u00E1 \u017E\u00E1dn\u00E9 homoklinick\u00E9 trajektorie; P7 neexistuj\u00ED \u017E\u00E1dn\u00E9 nekone\u010Dn\u00E9 spo\u010Detn\u00E9 $omega$-limitn\u00ED mno\u017Einy; P8 trajektorie ka\u017Ed\u00FDch dvou bod\u016F jsou korelovan\u00E9; P9 neexistuje \u017E\u00E1dn\u00E1 uzav\u0159en\u00E1 invariantn\u00ED podmno\u017Eina $A$ takov\u00E1, aby pro n\u011Bjak\u00E9 p\u0159irozen\u00E9 \u010D\u00EDslo $m$ bylo zobrazen\u00ED $\\varphi ^m|A$ topologicky skoro konjugovan\u00E9 s shiftem. V tomto \u010Dl\u00E1nku ukazujeme vztahy mezi t\u011Bmito vlastnostmi pro t\u0159\u00EDdu $(x,y)\\mapsto (f(x),g_x(y))$ troj\u00FAheln\u00EDkov\u00FDch zobrazen\u00ED na \u010Dtverci. Tento v\u00FDsledek p\u0159isp\u00EDv\u00E1 k vy\u0159e\u0161en\u00ED dlouhotrvaj\u00EDc\u00EDho otev\u0159en\u00E9ho Sharkovsk\u00E9ho probl\u00E9mu."@cs . . "241-252" . . . . . "Klasifikace troj\u00FAheln\u00EDkov\u00FDch zobrazen\u00ED na \u010Dtverci"@cs . "A classification of triangular maps of the square" . "It is well-known that, for a continuous map $\\varphi$ of the interval, the condition {P1} $\\varphi$ has zero topological entropy, is equivalent, e.g., to any of the following: { P2} any $\\omega $-limit set contains a unique minimal set; { P3} the period of any cycle of $\\varphi$ is a power of two; { P4} any $\\omega$-limit set either is a cycle or contains no cycle; {P5} if $\\omega _\\varphi(\\xi)=\\omega_{\\varphi^2}(\\xi)$, then $\\omega_\\varphi (\\xi)$ is a fixed point; {P6} $\\varphi $ has no homoclinic trajectory; {P7} there is no countably infinite $\\omega$-limit set; {P8} trajectories of any two points are correlated; {P9} there is no closed invariant subset $A$ such that $\\varphi ^m|A$ is topologically almost conjugate to the shift, for some $m\\ge 1$. In the paper we exhibit the relations between these properties in the class $(x,y)\\mapsto (f(x),g_x(y))$ of triangular maps of the square. This contributes to the solution of a longstanding open problem of Sharkovsky." . . "1"^^ . . . "Klasifikace troj\u00FAheln\u00EDkov\u00FDch zobrazen\u00ED na \u010Dtverci"@cs . . "1"^^ . "Korneck\u00E1, Veronika" . . . "RIV/47813059:19610/06:#0000097" . "19610" . "75" . "A classification of triangular maps of the square"@en .