Toposym 2016 Dynamical systems S. Garcia-Ferreira Ellis Semigroup - PowerPoint PPT Presentation

Countable spaces Dynamical systems S. Garcia-Ferreira Ellis Semigroup Theorem, 2015 Countable spaces Let ( X , f ) be a dynamical system such that X is a compact metric Ultrafilters countable space and every accumulation point of X is periodic. Then p-limit points either all function of E ( X , f ) ∗ are continuous or all of them are p-iterate discontinuous. Cardinality Countable case Questions Theorem, 2015 Let ( X , f ) be a dynamical system such that X is a compact metric countable space. If X has finitely many accumulation points, then either all function of E ( X , f ) ∗ are continuous or all of them are discontinuous.

Example Dynamical systems S. Garcia-Ferreira Ellis Semigroup Example, 2015 Countable spaces Ultrafilters There is a dynamical system ( X , f ) where X is a compact metric p-limit points countable space such that the orbit of each accumulation point is p-iterate finite and that there are f 0 , f 1 ∈ E ( X , f ) ∗ so that f 0 is continuous on Cardinality X and f 1 is discontinuous on X . Countable case Questions The space X is the countable ordinal space ω 2 + 1 which is identified with a suitable subspace of R .

Example Dynamical systems S. Garcia-Ferreira Ellis Semigroup Example, 2015 Countable spaces Ultrafilters There is a dynamical system ( X , f ) where X is a compact metric p-limit points countable space such that the orbit of each accumulation point is p-iterate finite and that there are f 0 , f 1 ∈ E ( X , f ) ∗ so that f 0 is continuous on Cardinality X and f 1 is discontinuous on X . Countable case Questions The space X is the countable ordinal space ω 2 + 1 which is identified with a suitable subspace of R .

Example Dynamical systems S. Garcia-Ferreira Ellis Semigroup Example, 2015 Countable spaces Ultrafilters There is a dynamical system ( X , f ) where X is a compact metric p-limit points countable space such that the orbit of each accumulation point is p-iterate finite and that there are f 0 , f 1 ∈ E ( X , f ) ∗ so that f 0 is continuous on Cardinality X and f 1 is discontinuous on X . Countable case Questions The space X is the countable ordinal space ω 2 + 1 which is identified with a suitable subspace of R .

Example Dynamical systems S. Garcia-Ferreira Ellis Semigroup Example, 2015 Countable spaces Ultrafilters There is a dynamical system ( X , f ) where X is a compact metric p-limit points countable space such that the orbit of each accumulation point is p-iterate finite and that there are f 0 , f 1 ∈ E ( X , f ) ∗ so that f 0 is continuous on Cardinality X and f 1 is discontinuous on X . Countable case Questions The space X is the countable ordinal space ω 2 + 1 which is identified with a suitable subspace of R .

Example Dynamical systems S. Garcia-Ferreira Ellis Semigroup Example, 2015 Countable spaces Ultrafilters There is a dynamical system ( X , f ) where X is a compact metric p-limit points countable space such that the orbit of each accumulation point is p-iterate finite and that there are f 0 , f 1 ∈ E ( X , f ) ∗ so that f 0 is continuous on Cardinality X and f 1 is discontinuous on X . Countable case Questions The space X is the countable ordinal space ω 2 + 1 which is identified with a suitable subspace of R .

Example Dynamical systems S. Garcia-Ferreira Ellis Semigroup Example, 2015 Countable spaces Ultrafilters There is a dynamical system ( X , f ) where X is a compact metric p-limit points countable space such that the orbit of each accumulation point is p-iterate finite and that there are f 0 , f 1 ∈ E ( X , f ) ∗ so that f 0 is continuous on Cardinality X and f 1 is discontinuous on X . Countable case Questions The space X is the countable ordinal space ω 2 + 1 which is identified with a suitable subspace of R .

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup In this talk we were interested on the cardinality of the Ellis Countable spaces semigroup E ( X , f ). The work of A. K¨ ohler (1995) and M. E. Glasner Ultrafilters and Megrehisvili (2006) contain very interesting results about the p-limit points cardinality of E ( X , f ). Indeed, M. E. Glasner and Megrehisvili p-iterate stablished the Bourgain-Fremlin-Talagrand dichotomy for dynamical Cardinality systems : Either | E ( X , f ) | ≤ c or E ( X , f ) contains a copy of β N . Countable case Questions We willl be mostly concerned with countable compact metrizable spaces. In this case, it is evident that | E ( X , f ) | ≤ c . Moreover, since E ( X , f ) is a separable metric space, then E ( X , f ) is either countable or has cardinality c .

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup In this talk we were interested on the cardinality of the Ellis Countable spaces semigroup E ( X , f ). The work of A. K¨ ohler (1995) and M. E. Glasner Ultrafilters and Megrehisvili (2006) contain very interesting results about the p-limit points cardinality of E ( X , f ). Indeed, M. E. Glasner and Megrehisvili p-iterate stablished the Bourgain-Fremlin-Talagrand dichotomy for dynamical Cardinality systems : Either | E ( X , f ) | ≤ c or E ( X , f ) contains a copy of β N . Countable case Questions We willl be mostly concerned with countable compact metrizable spaces. In this case, it is evident that | E ( X , f ) | ≤ c . Moreover, since E ( X , f ) is a separable metric space, then E ( X , f ) is either countable or has cardinality c .

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup In this talk we were interested on the cardinality of the Ellis Countable spaces semigroup E ( X , f ). The work of A. K¨ ohler (1995) and M. E. Glasner Ultrafilters and Megrehisvili (2006) contain very interesting results about the p-limit points cardinality of E ( X , f ). Indeed, M. E. Glasner and Megrehisvili p-iterate stablished the Bourgain-Fremlin-Talagrand dichotomy for dynamical Cardinality systems : Either | E ( X , f ) | ≤ c or E ( X , f ) contains a copy of β N . Countable case Questions We willl be mostly concerned with countable compact metrizable spaces. In this case, it is evident that | E ( X , f ) | ≤ c . Moreover, since E ( X , f ) is a separable metric space, then E ( X , f ) is either countable or has cardinality c .

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup In this talk we were interested on the cardinality of the Ellis Countable spaces semigroup E ( X , f ). The work of A. K¨ ohler (1995) and M. E. Glasner Ultrafilters and Megrehisvili (2006) contain very interesting results about the p-limit points cardinality of E ( X , f ). Indeed, M. E. Glasner and Megrehisvili p-iterate stablished the Bourgain-Fremlin-Talagrand dichotomy for dynamical Cardinality systems : Either | E ( X , f ) | ≤ c or E ( X , f ) contains a copy of β N . Countable case Questions We willl be mostly concerned with countable compact metrizable spaces. In this case, it is evident that | E ( X , f ) | ≤ c . Moreover, since E ( X , f ) is a separable metric space, then E ( X , f ) is either countable or has cardinality c .

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup In this talk we were interested on the cardinality of the Ellis Countable spaces semigroup E ( X , f ). The work of A. K¨ ohler (1995) and M. E. Glasner Ultrafilters and Megrehisvili (2006) contain very interesting results about the p-limit points cardinality of E ( X , f ). Indeed, M. E. Glasner and Megrehisvili p-iterate stablished the Bourgain-Fremlin-Talagrand dichotomy for dynamical Cardinality systems : Either | E ( X , f ) | ≤ c or E ( X , f ) contains a copy of β N . Countable case Questions We willl be mostly concerned with countable compact metrizable spaces. In this case, it is evident that | E ( X , f ) | ≤ c . Moreover, since E ( X , f ) is a separable metric space, then E ( X , f ) is either countable or has cardinality c .

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup In this talk we were interested on the cardinality of the Ellis Countable spaces semigroup E ( X , f ). The work of A. K¨ ohler (1995) and M. E. Glasner Ultrafilters and Megrehisvili (2006) contain very interesting results about the p-limit points cardinality of E ( X , f ). Indeed, M. E. Glasner and Megrehisvili p-iterate stablished the Bourgain-Fremlin-Talagrand dichotomy for dynamical Cardinality systems : Either | E ( X , f ) | ≤ c or E ( X , f ) contains a copy of β N . Countable case Questions We willl be mostly concerned with countable compact metrizable spaces. In this case, it is evident that | E ( X , f ) | ≤ c . Moreover, since E ( X , f ) is a separable metric space, then E ( X , f ) is either countable or has cardinality c .

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup In this talk we were interested on the cardinality of the Ellis Countable spaces semigroup E ( X , f ). The work of A. K¨ ohler (1995) and M. E. Glasner Ultrafilters and Megrehisvili (2006) contain very interesting results about the p-limit points cardinality of E ( X , f ). Indeed, M. E. Glasner and Megrehisvili p-iterate stablished the Bourgain-Fremlin-Talagrand dichotomy for dynamical Cardinality systems : Either | E ( X , f ) | ≤ c or E ( X , f ) contains a copy of β N . Countable case Questions We willl be mostly concerned with countable compact metrizable spaces. In this case, it is evident that | E ( X , f ) | ≤ c . Moreover, since E ( X , f ) is a separable metric space, then E ( X , f ) is either countable or has cardinality c .

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup In this talk we were interested on the cardinality of the Ellis Countable spaces semigroup E ( X , f ). The work of A. K¨ ohler (1995) and M. E. Glasner Ultrafilters and Megrehisvili (2006) contain very interesting results about the p-limit points cardinality of E ( X , f ). Indeed, M. E. Glasner and Megrehisvili p-iterate stablished the Bourgain-Fremlin-Talagrand dichotomy for dynamical Cardinality systems : Either | E ( X , f ) | ≤ c or E ( X , f ) contains a copy of β N . Countable case Questions We willl be mostly concerned with countable compact metrizable spaces. In this case, it is evident that | E ( X , f ) | ≤ c . Moreover, since E ( X , f ) is a separable metric space, then E ( X , f ) is either countable or has cardinality c .

Content Dynamical systems S. Garcia-Ferreira 1 Ellis semigroup Ellis Semigroup 2 Countable spaces Countable spaces Ultrafilters 3 Ultrafilters p-limit points p-iterate Cardinality 4 The p-limit points Countable case Questions 5 p-iterate 6 Cardinality 7 Countable case 8 Questions

Ultrafilters Dynamical systems S. Garcia-Ferreira Ellis Semigroup β ( N ) will denote the set of all ultrafilters on N y N ∗ = β ( N ) \ N will Countable spaces Ultrafilters be the set of all free ultrafilters on N . p-limit points p-iterate Cardinality β ( N ) is the Stone-ˇ Cech compactification of the natural numbers N Countable case with the discrete topology. Questions If A ⊆ N , then ˆ A = { p ∈ β ( N ) : A ∈ p } is a basic open subset of β ( N ) and A ∗ = ˆ A \ N is a basic open subset of N ∗ .

Ultrafilters Dynamical systems S. Garcia-Ferreira Ellis Semigroup β ( N ) will denote the set of all ultrafilters on N y N ∗ = β ( N ) \ N will Countable spaces Ultrafilters be the set of all free ultrafilters on N . p-limit points p-iterate Cardinality β ( N ) is the Stone-ˇ Cech compactification of the natural numbers N Countable case with the discrete topology. Questions If A ⊆ N , then ˆ A = { p ∈ β ( N ) : A ∈ p } is a basic open subset of β ( N ) and A ∗ = ˆ A \ N is a basic open subset of N ∗ .

Ultrafilters Dynamical systems S. Garcia-Ferreira Ellis Semigroup β ( N ) will denote the set of all ultrafilters on N y N ∗ = β ( N ) \ N will Countable spaces Ultrafilters be the set of all free ultrafilters on N . p-limit points p-iterate Cardinality β ( N ) is the Stone-ˇ Cech compactification of the natural numbers N Countable case with the discrete topology. Questions If A ⊆ N , then ˆ A = { p ∈ β ( N ) : A ∈ p } is a basic open subset of β ( N ) and A ∗ = ˆ A \ N is a basic open subset of N ∗ .

Ultrafilters Dynamical systems S. Garcia-Ferreira Ellis Semigroup β ( N ) will denote the set of all ultrafilters on N y N ∗ = β ( N ) \ N will Countable spaces Ultrafilters be the set of all free ultrafilters on N . p-limit points p-iterate Cardinality β ( N ) is the Stone-ˇ Cech compactification of the natural numbers N Countable case with the discrete topology. Questions If A ⊆ N , then ˆ A = { p ∈ β ( N ) : A ∈ p } is a basic open subset of β ( N ) and A ∗ = ˆ A \ N is a basic open subset of N ∗ .

Ultrafilters Dynamical systems S. Garcia-Ferreira Ellis Semigroup β ( N ) will denote the set of all ultrafilters on N y N ∗ = β ( N ) \ N will Countable spaces Ultrafilters be the set of all free ultrafilters on N . p-limit points p-iterate Cardinality β ( N ) is the Stone-ˇ Cech compactification of the natural numbers N Countable case with the discrete topology. Questions If A ⊆ N , then ˆ A = { p ∈ β ( N ) : A ∈ p } is a basic open subset of β ( N ) and A ∗ = ˆ A \ N is a basic open subset of N ∗ .

Content Dynamical systems S. Garcia-Ferreira 1 Ellis semigroup Ellis Semigroup 2 Countable spaces Countable spaces Ultrafilters 3 Ultrafilters p-limit points p-iterate Cardinality 4 The p-limit points Countable case Questions 5 p-iterate 6 Cardinality 7 Countable case 8 Questions

p -limit points Dynamical systems Definition [Several mathematicians] S. Garcia-Ferreira Let p ∈ N ∗ . Let X a space and ( x n ) n ∈ N a sequence in X . We say that Ellis Semigroup x ∈ X is a p - limit of ( x n ) n ∈ N if for every neighborhood V of x we Countable spaces have that { n ∈ N : x n ∈ V } ∈ p . Ultrafilters p-limit points p-iterate Cardinality We write x = p − lim n →∞ x n . Countable case Questions x ∈ X is an accumulation point of { x n : n ∈ N } iff there is p ∈ N ∗ such that x = p − lim n →∞ x n . In a compact space, every sequence has a p -limit point for every p ∈ N ∗ .

p -limit points Dynamical systems Definition [Several mathematicians] S. Garcia-Ferreira Let p ∈ N ∗ . Let X a space and ( x n ) n ∈ N a sequence in X . We say that Ellis Semigroup x ∈ X is a p - limit of ( x n ) n ∈ N if for every neighborhood V of x we Countable spaces have that { n ∈ N : x n ∈ V } ∈ p . Ultrafilters p-limit points p-iterate Cardinality We write x = p − lim n →∞ x n . Countable case Questions x ∈ X is an accumulation point of { x n : n ∈ N } iff there is p ∈ N ∗ such that x = p − lim n →∞ x n . In a compact space, every sequence has a p -limit point for every p ∈ N ∗ .

p -limit points Dynamical systems Definition [Several mathematicians] S. Garcia-Ferreira Let p ∈ N ∗ . Let X a space and ( x n ) n ∈ N a sequence in X . We say that Ellis Semigroup x ∈ X is a p - limit of ( x n ) n ∈ N if for every neighborhood V of x we Countable spaces have that { n ∈ N : x n ∈ V } ∈ p . Ultrafilters p-limit points p-iterate Cardinality We write x = p − lim n →∞ x n . Countable case Questions x ∈ X is an accumulation point of { x n : n ∈ N } iff there is p ∈ N ∗ such that x = p − lim n →∞ x n . In a compact space, every sequence has a p -limit point for every p ∈ N ∗ .

p -limit points Dynamical systems Definition [Several mathematicians] S. Garcia-Ferreira Let p ∈ N ∗ . Let X a space and ( x n ) n ∈ N a sequence in X . We say that Ellis Semigroup x ∈ X is a p - limit of ( x n ) n ∈ N if for every neighborhood V of x we Countable spaces have that { n ∈ N : x n ∈ V } ∈ p . Ultrafilters p-limit points p-iterate Cardinality We write x = p − lim n →∞ x n . Countable case Questions x ∈ X is an accumulation point of { x n : n ∈ N } iff there is p ∈ N ∗ such that x = p − lim n →∞ x n . In a compact space, every sequence has a p -limit point for every p ∈ N ∗ .

p -limit points Dynamical systems Definition [Several mathematicians] S. Garcia-Ferreira Let p ∈ N ∗ . Let X a space and ( x n ) n ∈ N a sequence in X . We say that Ellis Semigroup x ∈ X is a p - limit of ( x n ) n ∈ N if for every neighborhood V of x we Countable spaces have that { n ∈ N : x n ∈ V } ∈ p . Ultrafilters p-limit points p-iterate Cardinality We write x = p − lim n →∞ x n . Countable case Questions x ∈ X is an accumulation point of { x n : n ∈ N } iff there is p ∈ N ∗ such that x = p − lim n →∞ x n . In a compact space, every sequence has a p -limit point for every p ∈ N ∗ .

Content Dynamical systems S. Garcia-Ferreira 1 Ellis semigroup Ellis Semigroup 2 Countable spaces Countable spaces Ultrafilters 3 Ultrafilters p-limit points p-iterate Cardinality 4 The p-limit points Countable case Questions 5 p-iterate 6 Cardinality 7 Countable case 8 Questions

p-iterate Dynamical systems S. Garcia-Ferreira Definition Ellis Semigroup Countable spaces Let ( X , f ) a dynamical system. For each p ∈ N ∗ , we define Ultrafilters f p : X → X as f p ( x ) = p − lim n →∞ f n ( x ) for all x ∈ X . p-limit points p-iterate Cardinality f p is called the p - iterate of f , for p ∈ N ∗ . Countable case Questions Unfortunately, f p is not in general continuous.

p-iterate Dynamical systems S. Garcia-Ferreira Definition Ellis Semigroup Countable spaces Let ( X , f ) a dynamical system. For each p ∈ N ∗ , we define Ultrafilters f p : X → X as f p ( x ) = p − lim n →∞ f n ( x ) for all x ∈ X . p-limit points p-iterate Cardinality f p is called the p - iterate of f , for p ∈ N ∗ . Countable case Questions Unfortunately, f p is not in general continuous.

p-iterate Dynamical systems S. Garcia-Ferreira Definition Ellis Semigroup Countable spaces Let ( X , f ) a dynamical system. For each p ∈ N ∗ , we define Ultrafilters f p : X → X as f p ( x ) = p − lim n →∞ f n ( x ) for all x ∈ X . p-limit points p-iterate Cardinality f p is called the p - iterate of f , for p ∈ N ∗ . Countable case Questions Unfortunately, f p is not in general continuous.

p-iterate Dynamical systems S. Garcia-Ferreira Definition Ellis Semigroup Countable spaces Let ( X , f ) a dynamical system. For each p ∈ N ∗ , we define Ultrafilters f p : X → X as f p ( x ) = p − lim n →∞ f n ( x ) for all x ∈ X . p-limit points p-iterate Cardinality f p is called the p - iterate of f , for p ∈ N ∗ . Countable case Questions Unfortunately, f p is not in general continuous.

Example Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces Ultrafilters Example p-limit points p-iterate Let X = [0 , 1] and let f : [0 , 1] → [0 , 1] any continuous function such Cardinality that f (0) = 0, f (1) = 1 and f ( t ) < 1 for all t ∈ (0 , 1). Then, f is a Countable case continuous function such that f p [[0 , 1)] = 0 and f p (1) = 1, for each Questions p ∈ N ∗ . Therefore, f p is not continuous at 1, for any p ∈ N ∗ .

Example Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces Ultrafilters Example p-limit points p-iterate Let X = [0 , 1] and let f : [0 , 1] → [0 , 1] any continuous function such Cardinality that f (0) = 0, f (1) = 1 and f ( t ) < 1 for all t ∈ (0 , 1). Then, f is a Countable case continuous function such that f p [[0 , 1)] = 0 and f p (1) = 1, for each Questions p ∈ N ∗ . Therefore, f p is not continuous at 1, for any p ∈ N ∗ .

Example Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces Ultrafilters Example p-limit points p-iterate Let X = [0 , 1] and let f : [0 , 1] → [0 , 1] any continuous function such Cardinality that f (0) = 0, f (1) = 1 and f ( t ) < 1 for all t ∈ (0 , 1). Then, f is a Countable case continuous function such that f p [[0 , 1)] = 0 and f p (1) = 1, for each Questions p ∈ N ∗ . Therefore, f p is not continuous at 1, for any p ∈ N ∗ .

Example Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces Ultrafilters Example p-limit points p-iterate Let X = [0 , 1] and let f : [0 , 1] → [0 , 1] any continuous function such Cardinality that f (0) = 0, f (1) = 1 and f ( t ) < 1 for all t ∈ (0 , 1). Then, f is a Countable case continuous function such that f p [[0 , 1)] = 0 and f p (1) = 1, for each Questions p ∈ N ∗ . Therefore, f p is not continuous at 1, for any p ∈ N ∗ .

Example Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces Ultrafilters Example p-limit points p-iterate Let X = [0 , 1] and let f : [0 , 1] → [0 , 1] any continuous function such Cardinality that f (0) = 0, f (1) = 1 and f ( t ) < 1 for all t ∈ (0 , 1). Then, f is a Countable case continuous function such that f p [[0 , 1)] = 0 and f p (1) = 1, for each Questions p ∈ N ∗ . Therefore, f p is not continuous at 1, for any p ∈ N ∗ .

Ellis semigroup Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces By using the p -iteration, for p ∈ β ( N ), we can see that Ultrafilters p-limit points E ( X , f ) = { f p : p ∈ β ( N ) } p-iterate Cardinality and Countable case E ( X , f ) ∗ ⊆ { f p : p ∈ N ∗ } Questions for any dynamical system ( X , f ).

Ellis semigroup Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces By using the p -iteration, for p ∈ β ( N ), we can see that Ultrafilters p-limit points E ( X , f ) = { f p : p ∈ β ( N ) } p-iterate Cardinality and Countable case E ( X , f ) ∗ ⊆ { f p : p ∈ N ∗ } Questions for any dynamical system ( X , f ).

Ellis semigroup Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces By using the p -iteration, for p ∈ β ( N ), we can see that Ultrafilters p-limit points E ( X , f ) = { f p : p ∈ β ( N ) } p-iterate Cardinality and Countable case E ( X , f ) ∗ ⊆ { f p : p ∈ N ∗ } Questions for any dynamical system ( X , f ).

Ellis semigroup Dynamical systems Definition S. Garcia-Ferreira For p ∈ β ( N ) and n ∈ N , we define Ellis Semigroup Countable spaces p + n = p − l´ m →∞ ( m + n ) ım Ultrafilters p-limit points p-iterate Cardinality Folklore Countable case Now, if p , q ∈ β ( N ), then we define Questions p + q = q − l´ n →∞ p + n . ım We know that β ( N ) and N ∗ with this operation + are a semigroups.

Ellis semigroup Dynamical systems Definition S. Garcia-Ferreira For p ∈ β ( N ) and n ∈ N , we define Ellis Semigroup Countable spaces p + n = p − l´ m →∞ ( m + n ) ım Ultrafilters p-limit points p-iterate Cardinality Folklore Countable case Now, if p , q ∈ β ( N ), then we define Questions p + q = q − l´ n →∞ p + n . ım We know that β ( N ) and N ∗ with this operation + are a semigroups.

Ellis semigroup Dynamical systems Definition S. Garcia-Ferreira For p ∈ β ( N ) and n ∈ N , we define Ellis Semigroup Countable spaces p + n = p − l´ m →∞ ( m + n ) ım Ultrafilters p-limit points p-iterate Cardinality Folklore Countable case Now, if p , q ∈ β ( N ), then we define Questions p + q = q − l´ n →∞ p + n . ım We know that β ( N ) and N ∗ with this operation + are a semigroups.

Ellis semigroup Dynamical systems Definition S. Garcia-Ferreira For p ∈ β ( N ) and n ∈ N , we define Ellis Semigroup Countable spaces p + n = p − l´ m →∞ ( m + n ) ım Ultrafilters p-limit points p-iterate Cardinality Folklore Countable case Now, if p , q ∈ β ( N ), then we define Questions p + q = q − l´ n →∞ p + n . ım We know that β ( N ) and N ∗ with this operation + are a semigroups.

Ellis semigroup Dynamical systems S. Garcia-Ferreira Ellis Semigroup Theorem, Folklore Countable spaces Ultrafilters If ( X , f ) is a dynamical system, then, p-limit points f p ◦ f q = f q + p , p-iterate Cardinality Countable case for every p , q ∈ β ( N ). Questions Notice that if f p is continuous for some p ∈ N ∗ , then f p + n is also continuous for all n ∈ N .

Ellis semigroup Dynamical systems S. Garcia-Ferreira Ellis Semigroup Theorem, Folklore Countable spaces Ultrafilters If ( X , f ) is a dynamical system, then, p-limit points f p ◦ f q = f q + p , p-iterate Cardinality Countable case for every p , q ∈ β ( N ). Questions Notice that if f p is continuous for some p ∈ N ∗ , then f p + n is also continuous for all n ∈ N .

Ellis semigroup Dynamical systems S. Garcia-Ferreira Ellis Semigroup Theorem, Folklore Countable spaces Ultrafilters If ( X , f ) is a dynamical system, then, p-limit points f p ◦ f q = f q + p , p-iterate Cardinality Countable case for every p , q ∈ β ( N ). Questions Notice that if f p is continuous for some p ∈ N ∗ , then f p + n is also continuous for all n ∈ N .

Content Dynamical systems S. Garcia-Ferreira 1 Ellis semigroup Ellis Semigroup 2 Countable spaces Countable spaces Ultrafilters 3 Ultrafilters p-limit points p-iterate Cardinality 4 The p-limit points Countable case Questions 5 p-iterate 6 Cardinality 7 Countable case 8 Questions

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Theorem Countable spaces Ultrafilters Let ( X , f ) be a dynamical system. Then E ( X , f ) is finite iff there p-limit points exist M > 0 such that |O f ( x ) | < M for each x ∈ X . p-iterate Cardinality Countable case It is noteworthy that E ( X , f ) ∗ could be finite and E ( X , f ) could be Questions infinite. For instance, if X is a convergent sequence with its limit point and f is the shift function, then E ( X , f ) is infinite and E ( X , f ) ∗ has only one point.

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Theorem Countable spaces Ultrafilters Let ( X , f ) be a dynamical system. Then E ( X , f ) is finite iff there p-limit points exist M > 0 such that |O f ( x ) | < M for each x ∈ X . p-iterate Cardinality Countable case It is noteworthy that E ( X , f ) ∗ could be finite and E ( X , f ) could be Questions infinite. For instance, if X is a convergent sequence with its limit point and f is the shift function, then E ( X , f ) is infinite and E ( X , f ) ∗ has only one point.

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Theorem Countable spaces Ultrafilters Let ( X , f ) be a dynamical system. Then E ( X , f ) is finite iff there p-limit points exist M > 0 such that |O f ( x ) | < M for each x ∈ X . p-iterate Cardinality Countable case It is noteworthy that E ( X , f ) ∗ could be finite and E ( X , f ) could be Questions infinite. For instance, if X is a convergent sequence with its limit point and f is the shift function, then E ( X , f ) is infinite and E ( X , f ) ∗ has only one point.

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Theorem Countable spaces Ultrafilters Let ( X , f ) be a dynamical system. Then E ( X , f ) is finite iff there p-limit points exist M > 0 such that |O f ( x ) | < M for each x ∈ X . p-iterate Cardinality Countable case It is noteworthy that E ( X , f ) ∗ could be finite and E ( X , f ) could be Questions infinite. For instance, if X is a convergent sequence with its limit point and f is the shift function, then E ( X , f ) is infinite and E ( X , f ) ∗ has only one point.

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Theorem Countable spaces Ultrafilters Let ( X , f ) be a dynamical system. Then E ( X , f ) is finite iff there p-limit points exist M > 0 such that |O f ( x ) | < M for each x ∈ X . p-iterate Cardinality Countable case It is noteworthy that E ( X , f ) ∗ could be finite and E ( X , f ) could be Questions infinite. For instance, if X is a convergent sequence with its limit point and f is the shift function, then E ( X , f ) is infinite and E ( X , f ) ∗ has only one point.

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Theorem Countable spaces Ultrafilters Let ( X , f ) be a dynamical system. Then E ( X , f ) is finite iff there p-limit points exist M > 0 such that |O f ( x ) | < M for each x ∈ X . p-iterate Cardinality Countable case It is noteworthy that E ( X , f ) ∗ could be finite and E ( X , f ) could be Questions infinite. For instance, if X is a convergent sequence with its limit point and f is the shift function, then E ( X , f ) is infinite and E ( X , f ) ∗ has only one point.

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces For a dynamical system ( X , f ), the ω − limit set of x ∈ X , denoted by Ultrafilters ω f ( x ), is the set of points y ∈ X for which there exists an increasing p-limit points sequence ( n k ) k ∈ N such that f n k ( x ) → y . p-iterate Cardinality Countable case Theorem Questions Let ( X , f ) be a dynamical system. E ( X , f ) ∗ is finite iff there is M ∈ N such that | ω f ( x ) | ≤ M for each x ∈ X .

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces For a dynamical system ( X , f ), the ω − limit set of x ∈ X , denoted by Ultrafilters ω f ( x ), is the set of points y ∈ X for which there exists an increasing p-limit points sequence ( n k ) k ∈ N such that f n k ( x ) → y . p-iterate Cardinality Countable case Theorem Questions Let ( X , f ) be a dynamical system. E ( X , f ) ∗ is finite iff there is M ∈ N such that | ω f ( x ) | ≤ M for each x ∈ X .

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces For a dynamical system ( X , f ), the ω − limit set of x ∈ X , denoted by Ultrafilters ω f ( x ), is the set of points y ∈ X for which there exists an increasing p-limit points sequence ( n k ) k ∈ N such that f n k ( x ) → y . p-iterate Cardinality Countable case Theorem Questions Let ( X , f ) be a dynamical system. E ( X , f ) ∗ is finite iff there is M ∈ N such that | ω f ( x ) | ≤ M for each x ∈ X .

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces For a dynamical system ( X , f ), the ω − limit set of x ∈ X , denoted by Ultrafilters ω f ( x ), is the set of points y ∈ X for which there exists an increasing p-limit points sequence ( n k ) k ∈ N such that f n k ( x ) → y . p-iterate Cardinality Countable case Theorem Questions Let ( X , f ) be a dynamical system. E ( X , f ) ∗ is finite iff there is M ∈ N such that | ω f ( x ) | ≤ M for each x ∈ X .

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces Ultrafilters p-limit points For a dynamical system ( X , f ), let P f denote the set of all periods of p-iterate the periodic points of ( X , f ) which are accumulation points. Cardinality Countable case Theorem Questions Let ( X , f ) be a dynamical system. If E ( X , f ) ∗ is finite, then P f is finite.

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces Ultrafilters p-limit points For a dynamical system ( X , f ), let P f denote the set of all periods of p-iterate the periodic points of ( X , f ) which are accumulation points. Cardinality Countable case Theorem Questions Let ( X , f ) be a dynamical system. If E ( X , f ) ∗ is finite, then P f is finite.

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces Ultrafilters p-limit points For a dynamical system ( X , f ), let P f denote the set of all periods of p-iterate the periodic points of ( X , f ) which are accumulation points. Cardinality Countable case Theorem Questions Let ( X , f ) be a dynamical system. If E ( X , f ) ∗ is finite, then P f is finite.

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces Ultrafilters p-limit points For a dynamical system ( X , f ), let P f denote the set of all periods of p-iterate the periodic points of ( X , f ) which are accumulation points. Cardinality Countable case Theorem Questions Let ( X , f ) be a dynamical system. If E ( X , f ) ∗ is finite, then P f is finite.

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces Theorem Ultrafilters Let ( X , f ) be a dynamical system. If P f is infinite, then E ( X , f ) has p-limit points at least size c . p-iterate Cardinality Countable case Theorem Questions Let ( X , f ) be a dynamical system and assume that X has a point with dense orbit. If f p is continuous for every p ∈ N ∗ , then | E ( X , f ) ∗ | ≤ | X | .

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces Theorem Ultrafilters Let ( X , f ) be a dynamical system. If P f is infinite, then E ( X , f ) has p-limit points at least size c . p-iterate Cardinality Countable case Theorem Questions Let ( X , f ) be a dynamical system and assume that X has a point with dense orbit. If f p is continuous for every p ∈ N ∗ , then | E ( X , f ) ∗ | ≤ | X | .

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces Theorem Ultrafilters Let ( X , f ) be a dynamical system. If P f is infinite, then E ( X , f ) has p-limit points at least size c . p-iterate Cardinality Countable case Theorem Questions Let ( X , f ) be a dynamical system and assume that X has a point with dense orbit. If f p is continuous for every p ∈ N ∗ , then | E ( X , f ) ∗ | ≤ | X | .

Cardinality Dynamical systems S. Garcia-Ferreira Ellis Semigroup Countable spaces Theorem Ultrafilters Let ( X , f ) be a dynamical system. If P f is infinite, then E ( X , f ) has p-limit points at least size c . p-iterate Cardinality Countable case Theorem Questions Let ( X , f ) be a dynamical system and assume that X has a point with dense orbit. If f p is continuous for every p ∈ N ∗ , then | E ( X , f ) ∗ | ≤ | X | .

Content Dynamical systems S. Garcia-Ferreira 1 Ellis semigroup Ellis Semigroup 2 Countable spaces Countable spaces Ultrafilters 3 Ultrafilters p-limit points p-iterate Cardinality 4 The p-limit points Countable case Questions 5 p-iterate 6 Cardinality 7 Countable case 8 Questions

Countable case Dynamical systems S. Garcia-Ferreira Theorem, S. Mazurkiewicz and W. Sierpinski, 1920 Ellis Semigroup Countable spaces Every compact metric countable space is homeomorphic to a Ultrafilters countable ordinal with the order topology. p-limit points p-iterate Cardinality In what follows, our phase space will be the compact metric space Countable case ω α + 1 where α is a countable ordinal. Questions For our convenience, X ′ will denote the set of limit points of X , d will stand for the unique point of ω α + 1 of CB -rank α and { d n : n ∈ N } will be the collection of all its points with CB -rank equal to α − 1.

Countable case Dynamical systems S. Garcia-Ferreira Theorem, S. Mazurkiewicz and W. Sierpinski, 1920 Ellis Semigroup Countable spaces Every compact metric countable space is homeomorphic to a Ultrafilters countable ordinal with the order topology. p-limit points p-iterate Cardinality In what follows, our phase space will be the compact metric space Countable case ω α + 1 where α is a countable ordinal. Questions For our convenience, X ′ will denote the set of limit points of X , d will stand for the unique point of ω α + 1 of CB -rank α and { d n : n ∈ N } will be the collection of all its points with CB -rank equal to α − 1.

Countable case Dynamical systems S. Garcia-Ferreira Theorem, S. Mazurkiewicz and W. Sierpinski, 1920 Ellis Semigroup Countable spaces Every compact metric countable space is homeomorphic to a Ultrafilters countable ordinal with the order topology. p-limit points p-iterate Cardinality In what follows, our phase space will be the compact metric space Countable case ω α + 1 where α is a countable ordinal. Questions For our convenience, X ′ will denote the set of limit points of X , d will stand for the unique point of ω α + 1 of CB -rank α and { d n : n ∈ N } will be the collection of all its points with CB -rank equal to α − 1.

Countable case Dynamical systems S. Garcia-Ferreira Theorem, S. Mazurkiewicz and W. Sierpinski, 1920 Ellis Semigroup Countable spaces Every compact metric countable space is homeomorphic to a Ultrafilters countable ordinal with the order topology. p-limit points p-iterate Cardinality In what follows, our phase space will be the compact metric space Countable case ω α + 1 where α is a countable ordinal. Questions For our convenience, X ′ will denote the set of limit points of X , d will stand for the unique point of ω α + 1 of CB -rank α and { d n : n ∈ N } will be the collection of all its points with CB -rank equal to α − 1.

Countable case Dynamical systems S. Garcia-Ferreira Theorem, S. Mazurkiewicz and W. Sierpinski, 1920 Ellis Semigroup Countable spaces Every compact metric countable space is homeomorphic to a Ultrafilters countable ordinal with the order topology. p-limit points p-iterate Cardinality In what follows, our phase space will be the compact metric space Countable case ω α + 1 where α is a countable ordinal. Questions For our convenience, X ′ will denote the set of limit points of X , d will stand for the unique point of ω α + 1 of CB -rank α and { d n : n ∈ N } will be the collection of all its points with CB -rank equal to α − 1.

Countable case Dynamical systems S. Garcia-Ferreira Theorem, S. Mazurkiewicz and W. Sierpinski, 1920 Ellis Semigroup Countable spaces Every compact metric countable space is homeomorphic to a Ultrafilters countable ordinal with the order topology. p-limit points p-iterate Cardinality In what follows, our phase space will be the compact metric space Countable case ω α + 1 where α is a countable ordinal. Questions For our convenience, X ′ will denote the set of limit points of X , d will stand for the unique point of ω α + 1 of CB -rank α and { d n : n ∈ N } will be the collection of all its points with CB -rank equal to α − 1.

Countable case Dynamical systems S. Garcia-Ferreira Lema Ellis Semigroup Let ( ω α + 1 , f ) be a dynamical system with α ≥ 1 a countable Countable spaces Ultrafilters successor ordinal, such that there exists w ∈ ω α + 1 with a dense p-limit points orbit. Then the following conditions hold: p-iterate ( i ) f ( y ) is a limit point for every y ∈ ( ω α + 1) ′ . Cardinality ( ii ) The range of f is ω α + 1 \ { w } . Countable case Questions ( iii ) If x ∈ ( ω α + 1) ′ , then ∅ � = f − 1 ( x ) ⊆ ( ω α + 1) ′ . ( iv ) 1 ≤ CB ( f ( d )).

Countable case Dynamical systems S. Garcia-Ferreira Lema Ellis Semigroup Let ( ω α + 1 , f ) be a dynamical system with α ≥ 1 a countable Countable spaces Ultrafilters successor ordinal, such that there exists w ∈ ω α + 1 with a dense p-limit points orbit. Then the following conditions hold: p-iterate ( i ) f ( y ) is a limit point for every y ∈ ( ω α + 1) ′ . Cardinality ( ii ) The range of f is ω α + 1 \ { w } . Countable case Questions ( iii ) If x ∈ ( ω α + 1) ′ , then ∅ � = f − 1 ( x ) ⊆ ( ω α + 1) ′ . ( iv ) 1 ≤ CB ( f ( d )).

Countable case Dynamical systems S. Garcia-Ferreira Lema Ellis Semigroup Let ( ω α + 1 , f ) be a dynamical system with α ≥ 1 a countable Countable spaces Ultrafilters successor ordinal, such that there exists w ∈ ω α + 1 with a dense p-limit points orbit. Then the following conditions hold: p-iterate ( i ) f ( y ) is a limit point for every y ∈ ( ω α + 1) ′ . Cardinality ( ii ) The range of f is ω α + 1 \ { w } . Countable case Questions ( iii ) If x ∈ ( ω α + 1) ′ , then ∅ � = f − 1 ( x ) ⊆ ( ω α + 1) ′ . ( iv ) 1 ≤ CB ( f ( d )).

Countable case Dynamical systems S. Garcia-Ferreira Lema Ellis Semigroup Let ( ω α + 1 , f ) be a dynamical system with α ≥ 1 a countable Countable spaces Ultrafilters successor ordinal, such that there exists w ∈ ω α + 1 with a dense p-limit points orbit. Then the following conditions hold: p-iterate ( i ) f ( y ) is a limit point for every y ∈ ( ω α + 1) ′ . Cardinality ( ii ) The range of f is ω α + 1 \ { w } . Countable case Questions ( iii ) If x ∈ ( ω α + 1) ′ , then ∅ � = f − 1 ( x ) ⊆ ( ω α + 1) ′ . ( iv ) 1 ≤ CB ( f ( d )).

Countable case Dynamical systems S. Garcia-Ferreira Lema Ellis Semigroup Let ( ω α + 1 , f ) be a dynamical system with α ≥ 1 a countable Countable spaces Ultrafilters successor ordinal, such that there exists w ∈ ω α + 1 with a dense p-limit points orbit. Then the following conditions hold: p-iterate ( i ) f ( y ) is a limit point for every y ∈ ( ω α + 1) ′ . Cardinality ( ii ) The range of f is ω α + 1 \ { w } . Countable case Questions ( iii ) If x ∈ ( ω α + 1) ′ , then ∅ � = f − 1 ( x ) ⊆ ( ω α + 1) ′ . ( iv ) 1 ≤ CB ( f ( d )).

Countable case Dynamical systems S. Garcia-Ferreira Lema Ellis Semigroup Let ( ω α + 1 , f ) be a dynamical system with α ≥ 1 a countable Countable spaces Ultrafilters successor ordinal, such that there exists w ∈ ω α + 1 with a dense p-limit points orbit. Then the following conditions hold: p-iterate ( i ) f ( y ) is a limit point for every y ∈ ( ω α + 1) ′ . Cardinality ( ii ) The range of f is ω α + 1 \ { w } . Countable case Questions ( iii ) If x ∈ ( ω α + 1) ′ , then ∅ � = f − 1 ( x ) ⊆ ( ω α + 1) ′ . ( iv ) 1 ≤ CB ( f ( d )).

Countable case Dynamical systems S. Garcia-Ferreira Lema Ellis Semigroup Let ( ω α + 1 , f ) be a dynamical system with α ≥ 1 a countable Countable spaces Ultrafilters successor ordinal, such that there exists w ∈ ω α + 1 with a dense p-limit points orbit. Then the following conditions hold: p-iterate ( i ) f ( y ) is a limit point for every y ∈ ( ω α + 1) ′ . Cardinality ( ii ) The range of f is ω α + 1 \ { w } . Countable case Questions ( iii ) If x ∈ ( ω α + 1) ′ , then ∅ � = f − 1 ( x ) ⊆ ( ω α + 1) ′ . ( iv ) 1 ≤ CB ( f ( d )).

Countable case Dynamical systems S. Garcia-Ferreira Lemma Ellis Semigroup Let ( ω α + 1 , f ) be a dynamical system with α ≥ 1 a countable Countable spaces successor ordinal, such that there exists w ∈ ω α + 1 with a dense Ultrafilters p-limit points orbit. Then the following conditions hold: p-iterate ( i ) Let x ∈ ( ω α + 1) ′ so that CB ( x ) = γ < α and CB ( y ) < γ for Cardinality every y ∈ f − 1 ( x ). If ( x n ) n ∈ N is a sequence such that x n → x and Countable case CB ( x n ) = γ − 1, for each n ∈ N , then there is N ∈ N such that Questions if n ≥ N , then CB ( z ) < CB ( x n ) for all z ∈ f − 1 ( x n ). ( ii ) f ( d ) = d .

Countable case Dynamical systems S. Garcia-Ferreira Lemma Ellis Semigroup Let ( ω α + 1 , f ) be a dynamical system with α ≥ 1 a countable Countable spaces successor ordinal, such that there exists w ∈ ω α + 1 with a dense Ultrafilters p-limit points orbit. Then the following conditions hold: p-iterate ( i ) Let x ∈ ( ω α + 1) ′ so that CB ( x ) = γ < α and CB ( y ) < γ for Cardinality every y ∈ f − 1 ( x ). If ( x n ) n ∈ N is a sequence such that x n → x and Countable case CB ( x n ) = γ − 1, for each n ∈ N , then there is N ∈ N such that Questions if n ≥ N , then CB ( z ) < CB ( x n ) for all z ∈ f − 1 ( x n ). ( ii ) f ( d ) = d .

Countable case Dynamical systems S. Garcia-Ferreira Lemma Ellis Semigroup Let ( ω α + 1 , f ) be a dynamical system with α ≥ 1 a countable Countable spaces successor ordinal, such that there exists w ∈ ω α + 1 with a dense Ultrafilters p-limit points orbit. Then the following conditions hold: p-iterate ( i ) Let x ∈ ( ω α + 1) ′ so that CB ( x ) = γ < α and CB ( y ) < γ for Cardinality every y ∈ f − 1 ( x ). If ( x n ) n ∈ N is a sequence such that x n → x and Countable case CB ( x n ) = γ − 1, for each n ∈ N , then there is N ∈ N such that Questions if n ≥ N , then CB ( z ) < CB ( x n ) for all z ∈ f − 1 ( x n ). ( ii ) f ( d ) = d .