Dendric words and dendric subshifts Paulina CECCHI B . (Joint work with Val´ e ) erie Berth´ Institute de Recherche en Informatique Fondamentale Universit´ e Paris Diderot - Paris 7 Departamento de Matem´ atica y Ciencia de la Computaci´ on Facultad de Ciencia. Universidad de Santiago de Chile S´ eminaire CALIN, Paris, July 2018 Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 1 / 35
Let A be a finite alphabet and consider x ∈ A N , x = x 0 x 1 x 2 x 3 · · · Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 2 / 35
Let A be a finite alphabet and consider x ∈ A N , x = x 0 x 1 x 2 x 3 · · · The language L ( x ) of x is it set of finite subwords or factors. Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 2 / 35
Let A be a finite alphabet and consider x ∈ A N , x = x 0 x 1 x 2 x 3 · · · The language L ( x ) of x is it set of finite subwords or factors. The factor complexity of a x is the map p x : N → N defined by p x ( n ) = |L ( x ) ∩ A n | . Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 2 / 35
Let A be a finite alphabet and consider x ∈ A N , x = x 0 x 1 x 2 x 3 · · · The language L ( x ) of x is it set of finite subwords or factors. The factor complexity of a x is the map p x : N → N defined by p x ( n ) = |L ( x ) ∩ A n | . Given a factor w of x , a right extension of w is a letter a ∈ A such that aw ∈ L ( x ). We define analogously a left extension ( wb ∈ L ( x )) and a biextension ( awb ∈ L ( x )) of w . Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 2 / 35
Let A be a finite alphabet and consider x ∈ A N , x = x 0 x 1 x 2 x 3 · · · The language L ( x ) of x is it set of finite subwords or factors. The factor complexity of a x is the map p x : N → N defined by p x ( n ) = |L ( x ) ∩ A n | . Given a factor w of x , a right extension of w is a letter a ∈ A such that aw ∈ L ( x ). We define analogously a left extension ( wb ∈ L ( x )) and a biextension ( awb ∈ L ( x )) of w . We are interested in a family of words with linear complexity and some restrictions on the possible extensions of their factors. Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 2 / 35
Outline Dendric words. Dendric subshifts (symbolic dynamical systems). Exploting properties of extension graphs. ◮ Balance in dendric words. ◮ Invariant measures and orbit equivalence. Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 3 / 35
Dendric words Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 4 / 35
Dendric words Let A be a finite non-empty alphabet and x ∈ A N or A Z . For w ∈ L ( x ), the extensions of w are the following sets, L ( w ) = { a ∈ A | aw ∈ L ( x ) } R ( w ) = { a ∈ A | wa ∈ L ( x ) } B ( w ) = { ( a , b ) ∈ A × A | awb ∈ L ( x ) } . Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 5 / 35
Dendric words Let A be a finite non-empty alphabet and x ∈ A N or A Z . For w ∈ L ( x ), the extensions of w are the following sets, L ( w ) = { a ∈ A | aw ∈ L ( x ) } R ( w ) = { a ∈ A | wa ∈ L ( x ) } B ( w ) = { ( a , b ) ∈ A × A | awb ∈ L ( x ) } . The extension graph E x ( w ) of w is the undirected bipartite graph whose set of vertices is the disjoint union of L ( w ) and R ( w ) and whose edges are the pairs ( a , b ) ∈ B ( w ). Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 5 / 35
Dendric words Example: Consider the Fibonacci word in { a , b } x = abaababaabaababaababa · · · produced by the substitution ϕ : a �→ ab , b �→ a ( x = ϕ ω ( a )). Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 6 / 35
Dendric words Example: Consider the Fibonacci word in { a , b } x = abaababaabaababaababa · · · produced by the substitution ϕ : a �→ ab , b �→ a ( x = ϕ ω ( a )). The extension graphs of a and b are E x ( a ) E x ( b ) a a a a b b L 3 ( x ) = { aba , baa , aab , bab } . Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 6 / 35
Dendric words Example: Consider the Fibonacci word in { a , b } x = abaababaabaababaababa · · · produced by the substitution ϕ : a �→ ab , b �→ a . The extension graphs of aa , ab , ba are E x ( aa ) E x ( ab ) E x ( ba ) a a a a b b b b L 4 ( x ) = { abaa , baab , aaba , abab , baba } . Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 7 / 35
Dendric words If for all w ∈ L ( x ) the graph E x ( w ) is a tree, x is said to be dendric. Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 8 / 35
Dendric words If for all w ∈ L ( x ) the graph E x ( w ) is a tree, x is said to be dendric. The Fibonacci word is a dendric word. Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 8 / 35
Dendric words If for all w ∈ L ( x ) the graph E x ( w ) is a tree, x is said to be dendric. The Fibonacci word is a dendric word. Consider the Thue-Morse word in { a , b } given by y = abbabaabbaababba · · · produced by the Thue-Morse substitution σ : a �→ ab , b �→ ba . This word is not dendric. The extension graph of ǫ is E y ( ǫ ) a a b b L 2 ( y ) = { aa , ab , ba , bb } . Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 8 / 35
Dendric words: examples Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 9 / 35
Dendric words: examples Sturmian words: aperiodic (bi)infinite words with factor complexity p x ( n ) = n + 1 . Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 9 / 35
Dendric words: examples Sturmian words: aperiodic (bi)infinite words with factor complexity p x ( n ) = n + 1 . Arnoux-Rauzy words. Consider the alphabet A = { 1 , 2 , · · · , d } , x ∈ A N or A Z is an Arnoux-Rauzy word if every factor appears infinitely often; for all n , p n ( x ) = ( d − 1) n + 1 , Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 9 / 35
Dendric words: examples Sturmian words: aperiodic (bi)infinite words with factor complexity p x ( n ) = n + 1 . Arnoux-Rauzy words. Consider the alphabet A = { 1 , 2 , · · · , d } , x ∈ A N or A Z is an Arnoux-Rauzy word if every factor appears infinitely often; for all n , p n ( x ) = ( d − 1) n + 1 , and there exists exactly one left special factor ( L ( w ) ≥ 2) and one rigth special factor ( R ( v ) ≥ 2) of each given length. Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 9 / 35
Dendric words: examples Sturmian words: aperiodic (bi)infinite words with factor complexity p x ( n ) = n + 1 . Arnoux-Rauzy words. Consider the alphabet A = { 1 , 2 , · · · , d } , x ∈ A N or A Z is an Arnoux-Rauzy word if every factor appears infinitely often; for all n , p n ( x ) = ( d − 1) n + 1 , and there exists exactly one left special factor ( L ( w ) ≥ 2) and one rigth special factor ( R ( v ) ≥ 2) of each given length. Sturmian words are Arnoux-Rauzy words for d = 2. Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 9 / 35
Dendric words: examples Codings of regular interval exchanges. a b c p I I ( b ) I ( c ) I ( a ) Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 10 / 35
Dendric words: examples Codings of regular interval exchanges. a b c p I I ( b ) I ( c ) I ( a ) � Regular means that the orbits of nonzero separation points are infinite and disjoint. Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 10 / 35
Dendric words: examples Codings of regular interval exchanges. a b c p I I ( b ) I ( c ) I ( a ) � Regular means that the orbits of nonzero separation points are infinite and disjoint. x p = · · · abbacb · bacbbacbba · · · Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 10 / 35
Dendric subshifts Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 11 / 35
Dendric subshifts The set A Z equipped with the product topology of the discrete topology is a Cantor space. Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 12 / 35
Dendric subshifts The set A Z equipped with the product topology of the discrete topology is a Cantor space. The shift map T acts on A Z as T (( x n ) n ∈ Z ) = ( x n +1 ) n ∈ Z . Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 12 / 35
Dendric subshifts The set A Z equipped with the product topology of the discrete topology is a Cantor space. The shift map T acts on A Z as T (( x n ) n ∈ Z ) = ( x n +1 ) n ∈ Z . A subshift is a dynamical system ( X , T ) where X is a closed shift-invariant subset of A Z . The language L X of the subshift is defined as the union of the languaje of its elements. Paulina CECCHI B. (IRIF/USACh) Dendric words and dendric subshifts LIPN, July 2018 12 / 35
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