Complexity, periodicity, and expansiveness Van Cyr Northwestern - PowerPoint PPT Presentation

Complexity, periodicity, and expansiveness Van Cyr Northwestern University June 4, 2013 University of Crete Van Cyr Complexity, periodicity, and expansiveness

Introduction A =finite set, X = A Z d . For α, β ∈ X , d ( α, β ) = 2 − min {� � x � : α ( � x ) � = β ( � x ) } . For S ⊂ Z d and α ∈ A Z d , α | S =restriction of α to S . T � v : X → X is the translation by � v : ( T � v α )( � x ) := α ( � x + � v ). Van Cyr Complexity, periodicity, and expansiveness

Introduction A =finite set, X = A Z d . For α, β ∈ X , d ( α, β ) = 2 − min {� � x � : α ( � x ) � = β ( � x ) } . For S ⊂ Z d and α ∈ A Z d , α | S =restriction of α to S . T � v : X → X is the translation by � v : ( T � v α )( � x ) := α ( � x + � v ). If η ∈ X , the block complexity function P α : N d → N is ˛ ˛ v ∈ Z d } P η ( n 1 , . . . , n d ) := ˛ { ( T � v α ) | [1 , n 1 ] ×···× [1 , n d ] : � ˛ . ˛ ˛ X η := O ( η ). Van Cyr Complexity, periodicity, and expansiveness

Introduction A =finite set, X = A Z d . For α, β ∈ X , d ( α, β ) = 2 − min {� � x � : α ( � x ) � = β ( � x ) } . For S ⊂ Z d and α ∈ A Z d , α | S =restriction of α to S . T � v : X → X is the translation by � v : ( T � v α )( � x ) := α ( � x + � v ). If η ∈ X , the block complexity function P α : N d → N is ˛ ˛ v ∈ Z d } P η ( n 1 , . . . , n d ) := ˛ { ( T � v α ) | [1 , n 1 ] ×···× [1 , n d ] : � ˛ . ˛ ˛ X η := O ( η ). Theorem (Morse-Hedlund, 1940): α ∈ A Z is periodic iff there exists n ∈ N such that P α ( n ) ≤ n . Van Cyr Complexity, periodicity, and expansiveness

Introduction A =finite set, X = A Z d . For α, β ∈ X , d ( α, β ) = 2 − min {� � x � : α ( � x ) � = β ( � x ) } . For S ⊂ Z d and α ∈ A Z d , α | S =restriction of α to S . T � v : X → X is the translation by � v : ( T � v α )( � x ) := α ( � x + � v ). If η ∈ X , the block complexity function P α : N d → N is ˛ ˛ v ∈ Z d } P η ( n 1 , . . . , n d ) := ˛ { ( T � v α ) | [1 , n 1 ] ×···× [1 , n d ] : � ˛ . ˛ ˛ X η := O ( η ). Theorem (Morse-Hedlund, 1940): α ∈ A Z is periodic iff there exists n ∈ N such that P α ( n ) ≤ n . Conjecture (M. Nivat, 1997): If α ∈ A Z 2 and there exist n , k ∈ N such that P η ( n , k ) ≤ nk , then α is periodic. Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results Cassaigne (‘99): Classified all η whose complexity function is given by P η ( n , k ) = nk + 1. Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results Cassaigne (‘99): Classified all η whose complexity function is given by P η ( n , k ) = nk + 1. (No minimal examples exist) Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results Cassaigne (‘99): Classified all η whose complexity function is given by P η ( n , k ) = nk + 1. (No minimal examples exist) Berthe-Vuillon (‘00): Can code two circle rotations and get complexity P η ( n , k ) = nk + n + k . Example of a uniformly recurrent coloring with complexity P η ( n , k ) = nk + min( n , k ). Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results Cassaigne (‘99): Classified all η whose complexity function is given by P η ( n , k ) = nk + 1. (No minimal examples exist) Berthe-Vuillon (‘00): Can code two circle rotations and get complexity P η ( n , k ) = nk + n + k . Example of a uniformly recurrent coloring with complexity P η ( n , k ) = nk + min( n , k ). Sander-Tijdeman (‘00): For d > 2 and all n , there exists aperiodic η ∈ A Z d such that P η ( n , . . . , n ) = 2 n d − 1 + 1. Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results Cassaigne (‘99): Classified all η whose complexity function is given by P η ( n , k ) = nk + 1. (No minimal examples exist) Berthe-Vuillon (‘00): Can code two circle rotations and get complexity P η ( n , k ) = nk + n + k . Example of a uniformly recurrent coloring with complexity P η ( n , k ) = nk + min( n , k ). Sander-Tijdeman (‘00): For d > 2 and all n , there exists aperiodic η ∈ A Z d such that P η ( n , . . . , n ) = 2 n d − 1 + 1. Sander-Tijdeman (‘02): If there exists n such that P η ( n , 2) ≤ 2 n then η is periodic. Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results Cassaigne (‘99): Classified all η whose complexity function is given by P η ( n , k ) = nk + 1. (No minimal examples exist) Berthe-Vuillon (‘00): Can code two circle rotations and get complexity P η ( n , k ) = nk + n + k . Example of a uniformly recurrent coloring with complexity P η ( n , k ) = nk + min( n , k ). Sander-Tijdeman (‘00): For d > 2 and all n , there exists aperiodic η ∈ A Z d such that P η ( n , . . . , n ) = 2 n d − 1 + 1. Sander-Tijdeman (‘02): If there exists n such that P η ( n , 2) ≤ 2 n then η is periodic. nk Epifanio-Koskas-Mignosi (‘03): If there exist n , k ∈ N such that P η ( n , k ) ≤ 144 , then nk η is periodic. (also claimed 100 works) Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results Cassaigne (‘99): Classified all η whose complexity function is given by P η ( n , k ) = nk + 1. (No minimal examples exist) Berthe-Vuillon (‘00): Can code two circle rotations and get complexity P η ( n , k ) = nk + n + k . Example of a uniformly recurrent coloring with complexity P η ( n , k ) = nk + min( n , k ). Sander-Tijdeman (‘00): For d > 2 and all n , there exists aperiodic η ∈ A Z d such that P η ( n , . . . , n ) = 2 n d − 1 + 1. Sander-Tijdeman (‘02): If there exists n such that P η ( n , 2) ≤ 2 n then η is periodic. nk Epifanio-Koskas-Mignosi (‘03): If there exist n , k ∈ N such that P η ( n , k ) ≤ 144 , then nk η is periodic. (also claimed 100 works) Quas-Zamboni (‘04): If there exist n , k ∈ N such that P η ( n , k ) ≤ nk 16 , then η is periodic. Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results Cassaigne (‘99): Classified all η whose complexity function is given by P η ( n , k ) = nk + 1. (No minimal examples exist) Berthe-Vuillon (‘00): Can code two circle rotations and get complexity P η ( n , k ) = nk + n + k . Example of a uniformly recurrent coloring with complexity P η ( n , k ) = nk + min( n , k ). Sander-Tijdeman (‘00): For d > 2 and all n , there exists aperiodic η ∈ A Z d such that P η ( n , . . . , n ) = 2 n d − 1 + 1. Sander-Tijdeman (‘02): If there exists n such that P η ( n , 2) ≤ 2 n then η is periodic. nk Epifanio-Koskas-Mignosi (‘03): If there exist n , k ∈ N such that P η ( n , k ) ≤ 144 , then nk η is periodic. (also claimed 100 works) Quas-Zamboni (‘04): If there exist n , k ∈ N such that P η ( n , k ) ≤ nk 16 , then η is periodic. Durand-Rigo (’11): Nivat-like bound on the recurrent complexity function is equivalent to definability in Presburger arithmetic. Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results Cassaigne (‘99): Classified all η whose complexity function is given by P η ( n , k ) = nk + 1. (No minimal examples exist) Berthe-Vuillon (‘00): Can code two circle rotations and get complexity P η ( n , k ) = nk + n + k . Example of a uniformly recurrent coloring with complexity P η ( n , k ) = nk + min( n , k ). Sander-Tijdeman (‘00): For d > 2 and all n , there exists aperiodic η ∈ A Z d such that P η ( n , . . . , n ) = 2 n d − 1 + 1. Sander-Tijdeman (‘02): If there exists n such that P η ( n , 2) ≤ 2 n then η is periodic. nk Epifanio-Koskas-Mignosi (‘03): If there exist n , k ∈ N such that P η ( n , k ) ≤ 144 , then nk η is periodic. (also claimed 100 works) Quas-Zamboni (‘04): If there exist n , k ∈ N such that P η ( n , k ) ≤ nk 16 , then η is periodic. Durand-Rigo (’11): Nivat-like bound on the recurrent complexity function is equivalent to definability in Presburger arithmetic. (works in dimension d ) Van Cyr Complexity, periodicity, and expansiveness

Background and Main Results Cassaigne (‘99): Classified all η whose complexity function is given by P η ( n , k ) = nk + 1. (No minimal examples exist) Berthe-Vuillon (‘00): Can code two circle rotations and get complexity P η ( n , k ) = nk + n + k . Example of a uniformly recurrent coloring with complexity P η ( n , k ) = nk + min( n , k ). Sander-Tijdeman (‘00): For d > 2 and all n , there exists aperiodic η ∈ A Z d such that P η ( n , . . . , n ) = 2 n d − 1 + 1. Sander-Tijdeman (‘02): If there exists n such that P η ( n , 2) ≤ 2 n then η is periodic. nk Epifanio-Koskas-Mignosi (‘03): If there exist n , k ∈ N such that P η ( n , k ) ≤ 144 , then nk η is periodic. (also claimed 100 works) Quas-Zamboni (‘04): If there exist n , k ∈ N such that P η ( n , k ) ≤ nk 16 , then η is periodic. Durand-Rigo (’11): Nivat-like bound on the recurrent complexity function is equivalent to definability in Presburger arithmetic. (works in dimension d ) C.-Kra (‘12): If there exist n , k ∈ N such that P η ( n , k ) ≤ nk 2 , then η is periodic. Van Cyr Complexity, periodicity, and expansiveness