Chaos in hyperspaces of nonautonomous discrete systems Hugo Villanueva Méndez Joint work with Iván Sánchez and Manuel Sanchis Facultad de Ciencias en Física y Matemáticas Universidad Autónoma de Chiapas México Twelfth Symposium on General Topology and its Relations to Modern Analysis and Algebra 2016 Hugo Villanueva Méndez Joint work with Iván Sánchez and Manuel Sanchis Chaos in hyperspaces of nonautonomous discrete systems
Hyperspaces of continua Definition Given a topological space X , let f n : X → X be a continuous function for each positive integer n . Denote by f ∞ the sequence ( f 1 , f 2 , . . . ). We say that the pair ( X , f ∞ ) is the nonautonomous discrete dynamical system (NDS, for short) in which the orbit of a point x ∈ X under f ∞ is defined as the set orb ( x , f ∞ ) = { x , f 1 ( x ) , f 2 1 ( x ) , . . . , f n 1 ( x ) , . . . } , where f n 1 := f n ◦ f n − 1 ◦ · · · ◦ f 2 ◦ f 1 , for each positive integer n . In particular, when f ∞ is the constant sequence ( f , f , . . . , f , . . . ), the pair ( X , f ∞ ) is the usual (autonomous) discrete dynamical system given by the continuous function f on X and it will denoted by ( X , f ). Hugo Villanueva Méndez Joint work with Iván Sánchez and Manuel Sanchis Chaos in hyperspaces of nonautonomous discrete systems
Hyperspaces of continua Definition Given a topological space X , let f n : X → X be a continuous function for each positive integer n . Denote by f ∞ the sequence ( f 1 , f 2 , . . . ). We say that the pair ( X , f ∞ ) is the nonautonomous discrete dynamical system (NDS, for short) in which the orbit of a point x ∈ X under f ∞ is defined as the set orb ( x , f ∞ ) = { x , f 1 ( x ) , f 2 1 ( x ) , . . . , f n 1 ( x ) , . . . } , where f n 1 := f n ◦ f n − 1 ◦ · · · ◦ f 2 ◦ f 1 , for each positive integer n . In particular, when f ∞ is the constant sequence ( f , f , . . . , f , . . . ), the pair ( X , f ∞ ) is the usual (autonomous) discrete dynamical system given by the continuous function f on X and it will denoted by ( X , f ). Hugo Villanueva Méndez Joint work with Iván Sánchez and Manuel Sanchis Chaos in hyperspaces of nonautonomous discrete systems
Hyperspaces of continua Definition Given a topological space X , let f n : X → X be a continuous function for each positive integer n . Denote by f ∞ the sequence ( f 1 , f 2 , . . . ). We say that the pair ( X , f ∞ ) is the nonautonomous discrete dynamical system (NDS, for short) in which the orbit of a point x ∈ X under f ∞ is defined as the set orb ( x , f ∞ ) = { x , f 1 ( x ) , f 2 1 ( x ) , . . . , f n 1 ( x ) , . . . } , where f n 1 := f n ◦ f n − 1 ◦ · · · ◦ f 2 ◦ f 1 , for each positive integer n . In particular, when f ∞ is the constant sequence ( f , f , . . . , f , . . . ), the pair ( X , f ∞ ) is the usual (autonomous) discrete dynamical system given by the continuous function f on X and it will denoted by ( X , f ). Hugo Villanueva Méndez Joint work with Iván Sánchez and Manuel Sanchis Chaos in hyperspaces of nonautonomous discrete systems
Hyperspaces of continua Definition Given a topological space X , let f n : X → X be a continuous function for each positive integer n . Denote by f ∞ the sequence ( f 1 , f 2 , . . . ). We say that the pair ( X , f ∞ ) is the nonautonomous discrete dynamical system (NDS, for short) in which the orbit of a point x ∈ X under f ∞ is defined as the set orb ( x , f ∞ ) = { x , f 1 ( x ) , f 2 1 ( x ) , . . . , f n 1 ( x ) , . . . } , where f n 1 := f n ◦ f n − 1 ◦ · · · ◦ f 2 ◦ f 1 , for each positive integer n . In particular, when f ∞ is the constant sequence ( f , f , . . . , f , . . . ), the pair ( X , f ∞ ) is the usual (autonomous) discrete dynamical system given by the continuous function f on X and it will denoted by ( X , f ). Hugo Villanueva Méndez Joint work with Iván Sánchez and Manuel Sanchis Chaos in hyperspaces of nonautonomous discrete systems
Definitions Definition A NDS ( X , f ∞ ) is topologically transitive if for any two non-empty open sets U and V in X , there exists a positive integer k such that f k 1 ( U ) ∩ V � = ∅ ; said to satisfy Banks’ condition if for any three non-empty open sets U , V , W in X , there exists a positive integer k such that f k 1 ( U ) ∩ V � = ∅ and f k 1 ( U ) ∩ W � = ∅ ; weakly mixing if for any four non-empty open sets U 1 , U 2 , V 1 , V 2 in X , there exists a positive integer k such that f k 1 ( U i ) ∩ V i � = ∅ , for each i ∈ { 1 , 2 } . If ( X , f ∞ ) is weakly mixing, then it has Banks’ condition and, this implies that it is transitive. Hugo Villanueva Méndez Joint work with Iván Sánchez and Manuel Sanchis Chaos in hyperspaces of nonautonomous discrete systems
Definitions Definition A NDS ( X , f ∞ ) is topologically transitive if for any two non-empty open sets U and V in X , there exists a positive integer k such that f k 1 ( U ) ∩ V � = ∅ ; said to satisfy Banks’ condition if for any three non-empty open sets U , V , W in X , there exists a positive integer k such that f k 1 ( U ) ∩ V � = ∅ and f k 1 ( U ) ∩ W � = ∅ ; weakly mixing if for any four non-empty open sets U 1 , U 2 , V 1 , V 2 in X , there exists a positive integer k such that f k 1 ( U i ) ∩ V i � = ∅ , for each i ∈ { 1 , 2 } . If ( X , f ∞ ) is weakly mixing, then it has Banks’ condition and, this implies that it is transitive. Hugo Villanueva Méndez Joint work with Iván Sánchez and Manuel Sanchis Chaos in hyperspaces of nonautonomous discrete systems
Definitions Definition A NDS ( X , f ∞ ) is topologically transitive if for any two non-empty open sets U and V in X , there exists a positive integer k such that f k 1 ( U ) ∩ V � = ∅ ; said to satisfy Banks’ condition if for any three non-empty open sets U , V , W in X , there exists a positive integer k such that f k 1 ( U ) ∩ V � = ∅ and f k 1 ( U ) ∩ W � = ∅ ; weakly mixing if for any four non-empty open sets U 1 , U 2 , V 1 , V 2 in X , there exists a positive integer k such that f k 1 ( U i ) ∩ V i � = ∅ , for each i ∈ { 1 , 2 } . If ( X , f ∞ ) is weakly mixing, then it has Banks’ condition and, this implies that it is transitive. Hugo Villanueva Méndez Joint work with Iván Sánchez and Manuel Sanchis Chaos in hyperspaces of nonautonomous discrete systems
Definitions Definition A NDS ( X , f ∞ ) is topologically transitive if for any two non-empty open sets U and V in X , there exists a positive integer k such that f k 1 ( U ) ∩ V � = ∅ ; said to satisfy Banks’ condition if for any three non-empty open sets U , V , W in X , there exists a positive integer k such that f k 1 ( U ) ∩ V � = ∅ and f k 1 ( U ) ∩ W � = ∅ ; weakly mixing if for any four non-empty open sets U 1 , U 2 , V 1 , V 2 in X , there exists a positive integer k such that f k 1 ( U i ) ∩ V i � = ∅ , for each i ∈ { 1 , 2 } . If ( X , f ∞ ) is weakly mixing, then it has Banks’ condition and, this implies that it is transitive. Hugo Villanueva Méndez Joint work with Iván Sánchez and Manuel Sanchis Chaos in hyperspaces of nonautonomous discrete systems
Definitions Definition A NDS ( X , f ∞ ) is topologically transitive if for any two non-empty open sets U and V in X , there exists a positive integer k such that f k 1 ( U ) ∩ V � = ∅ ; said to satisfy Banks’ condition if for any three non-empty open sets U , V , W in X , there exists a positive integer k such that f k 1 ( U ) ∩ V � = ∅ and f k 1 ( U ) ∩ W � = ∅ ; weakly mixing if for any four non-empty open sets U 1 , U 2 , V 1 , V 2 in X , there exists a positive integer k such that f k 1 ( U i ) ∩ V i � = ∅ , for each i ∈ { 1 , 2 } . If ( X , f ∞ ) is weakly mixing, then it has Banks’ condition and, this implies that it is transitive. Hugo Villanueva Méndez Joint work with Iván Sánchez and Manuel Sanchis Chaos in hyperspaces of nonautonomous discrete systems
Definitions Let X be a topological space. The symbol K ( X ) will denote the hyperspace of all non-empty compact subsets of X endowed with the Vietoris topology. Induced NDS Given a continuous function f : X → X , it induces a continuous function on K ( X ), f : K ( X ) → K ( X ) defined by f ( K ) = f ( K ) for every K ∈ K ( X ). Let ( X , f ∞ ) be a NDS and f n the induced continuous function of f n on K ( X ), for each positive integer n . Then, the sequence f ∞ = ( f 1 , f 2 , . . . , f n , . . . ) induces a nonautonomous discrete dynamical n system ( K ( X ) , f ∞ ). In this case, f 1 = f n ◦ · · · ◦ f 2 ◦ f 1 . Hugo Villanueva Méndez Joint work with Iván Sánchez and Manuel Sanchis Chaos in hyperspaces of nonautonomous discrete systems
Definitions Let X be a topological space. The symbol K ( X ) will denote the hyperspace of all non-empty compact subsets of X endowed with the Vietoris topology. Induced NDS Given a continuous function f : X → X , it induces a continuous function on K ( X ), f : K ( X ) → K ( X ) defined by f ( K ) = f ( K ) for every K ∈ K ( X ). Let ( X , f ∞ ) be a NDS and f n the induced continuous function of f n on K ( X ), for each positive integer n . Then, the sequence f ∞ = ( f 1 , f 2 , . . . , f n , . . . ) induces a nonautonomous discrete dynamical n system ( K ( X ) , f ∞ ). In this case, f 1 = f n ◦ · · · ◦ f 2 ◦ f 1 . Hugo Villanueva Méndez Joint work with Iván Sánchez and Manuel Sanchis Chaos in hyperspaces of nonautonomous discrete systems
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