Harmonic properties of some automatics flows Pierre Liardet (Joint - PowerPoint PPT Presentation

Journ´ ees Num´ eration Prague, Mai 26-30, 2008 Harmonic properties of some automatics flows Pierre Liardet (Joint work with Isabelle Abou) Universit´ e de Provence Laboratoire d’Analyse, Topologie et Probabilit´ e UMR-CNRS 6632 Marseille

Journ´ ees Num´ eration Prague, Mai 26-30, 2008 Contents Linear representation of a q -automatic sequence 1 Summation formula 2 3 q -stack-automata Chained sequences 4 Illustration 5 Generalisation 6 Laboratoire d’Analyse, Topologie et Probabilit´ e UMR-CNRS 6632 Marseille

Journ´ ees Num´ eration Prague, Mai 26-30, 2008 I. L INEAR REPRESENTATION OF A q - AUTOMATIC SEQUENCE I.1 q -automatic sequence Classical : A sequence u in a set X is said to be q -automatic if the set G ( u ) of subsequences n �→ u ( q k n + r ) , 0 ≤ t < q k ( k ∈ N is finite. Let G ( u ) := { g 0 , . . . , g m − 1 } with g 0 = u and define γ : n → X m by   g 0 ( n ) . . γ ( n ) =  .   .  g m − 1 ( n ) Laboratoire d’Analyse, Topologie et Probabilit´ e UMR-CNRS 6632 Marseille

Journ´ ees Num´ eration Prague, Mai 26-30, 2008 For any q -digits j = 0 , 1 , . . . , q − 1, there exists a matrix A j (called instruction map) with 0-1 entries defined by the relation γ ( qn + j ) = A j γ ( n ) ( n ∈ N ) Notice : • each row contains only one 1, • these 1 are symbolic (playing the rˆ ole of a selection operator). The sequence u is generated by the standard q -automaton defined by : space of states : γ ( N ) , initial state : γ ( 0 ) , instructions : A j ( 0 ≤ j < q ) , and the sequence u is obtained from the output map : γ ( n ) �→ g 0 ( n ) (first projection). Laboratoire d’Analyse, Topologie et Probabilit´ e UMR-CNRS 6632 Marseille

Journ´ ees Num´ eration Prague, Mai 26-30, 2008 I.2 Linear representation For a given automatic sequence u , there are many automata that generate u , but there always exists a linear model, i.e., the space of states is a subset of a linear space E ; each instruction can be extended to a linear endomorphism (say A j ) of E . For example, if u is real or complex valued, the above standard automaton furnished a standard linear model that generates u with a minimal number of states. → keep in mind that A j ( γ )( n ) = γ ( qn + j ) and A i A j ( γ )( n ) = γ ( q 2 n + qj + i ) Laboratoire d’Analyse, Topologie et Probabilit´ e UMR-CNRS 6632 Marseille

Journ´ ees Num´ eration Prague, Mai 26-30, 2008 A concrete example : Let u be the periodic sequence of period 3 defined by u ( 0 ) = 1 , u ( 1 ) = u ( 2 ) = − 1 . Classically, u is 2-automatic. If we introduce the translation map T : n �→ n + 1, one has in fact G ( u ) := { g 0 = u , g 1 = u ◦ T , g 1 = u ◦ T 2 } with instructions     1 0 0 0 0 1  , A 0 = 0 0 1 A 1 = 0 1 0    0 1 0 1 0 0    − 1   − 1  + 1 � �  ,  , and space of states γ ( N ) = − 1 − 1 + 1     − 1 + 1 − 1 Laboratoire d’Analyse, Topologie et Probabilit´ e UMR-CNRS 6632 Marseille

Journ´ ees Num´ eration Prague, Mai 26-30, 2008 II. S UMMATION FORMULA I.1 formal summation We use the standard model (to fix the ideas) r = 0 e r ( n ) q r (standard) : For n = � t γ ( n ) = A e 0 ( n ) . . . A e t ( n ) γ ( 0 ) . And we are interested in the formal sum n < N γ ( n ) . z n Γ( N ; z ) := � which is a comfortable manner :=)) to write the object ( γ ( 0 ) , γ ( 1 ) , . . . , γ ( N − 1 ) , ∅ , ∅ , ∅ , . . . ) Laboratoire d’Analyse, Topologie et Probabilit´ e UMR-CNRS 6632 Marseille

Journ´ ees Num´ eration Prague, Mai 26-30, 2008 Now we introduce the formal sum A [ z ] := A 0 + zA 1 + · · · + z q − 1 A q − 1 (= ( A 0 , . . . , A q − 1 , ∧ , ∧ , ∧ . . . ) ( ∧ for cancel operator ) which acts on Γ( N ; z ) by distributivity of the local actions z k A k ( z m γ ( m )) = z k + m γ ( qm + k ) so that Γ( qN , z ) = A ( z )Γ( N , z q ) and (”block” summation) : Γ( q m , z ) = A ( z ) A ( z q ) . . . A ( z q m − 1 )Γ( 1 ; z q m ) , Γ( 1 ; z q m ) = γ ( 0 ) Laboratoire d’Analyse, Topologie et Probabilit´ e UMR-CNRS 6632 Marseille

Journ´ ees Num´ eration Prague, Mai 26-30, 2008 Formal sum (continued) Set Π m ( z ) = A ( z ) A ( z q ) . . . A ( z q m − 1 ) 0 ≤ r ≤ K e r ( N ) q r For N = � ( e k ( N ) � = 0) and for 0 ≤ m ≤ K , define the m -tail � e r ( N ) q r t m = and t K + 1 = 0 . m ≤ r ≤ K From above we derive the formal summation �� �� m ≥ 0 z t m + 1 Π m ( z ) e m ( 1 , N , z q m ) γ N Γ( N , z ) = � q m + 1 with e m ( A , N , · ) = ∧ if e m ( N ) = 0 and e m ( A , N , z q m ) = z jq m A j � otherwise. j < e m ( N ) Laboratoire d’Analyse, Topologie et Probabilit´ e UMR-CNRS 6632 Marseille

Journ´ ees Num´ eration Prague, Mai 26-30, 2008 Interest ? (1) If X is a compact space, in order to study u from a statistical and harmonical point of view, it is classical to replace u ( n ) by f ( u ( n )) where f is a continuous map. (2) If X is a compact metrizable group, it is useful to introduce irreducible representations ρ of X and then to replace u ( n ) by orthogonal matrices ρ ( u ( n )) . In both case z figure a complex number of modulus 1. With a linear representation of the automaton, the instructions turn to be matrices, the operator A ( z ) can be viewed as a matrix and then, formal sums become summation in a suitable linear space. The usual goal is to estimate these sums (used in the ergodic machinery). A way to attack this problem is to compute the quadratic operator norm of the matrix A ( z ) : � max { eigenvalues of A ( ζ ) ∗ A ( ζ ) } . || A ( ζ ) || 2 = Laboratoire d’Analyse, Topologie et Probabilit´ e UMR-CNRS 6632 Marseille

Journ´ ees Num´ eration Prague, Mai 26-30, 2008 III. q - STACK - AUTOMATA III.1 Some generalisation Going back to the output formula γ ( n ) = A e 0 ( n ) . . . A e t ( n ) γ ( 0 ) one can decide to change the automaton on line, at each step, taking care that the corresponding output e 0 ( n ) . . . A ( t ) γ ( n ) = A 0 e t ( n ) γ ( 0 ) is meaning full. Summation formula remains unchanged and estimation by quadratic norm can be a fruitful tool. Laboratoire d’Analyse, Topologie et Probabilit´ e UMR-CNRS 6632 Marseille

Journ´ ees Num´ eration Prague, Mai 26-30, 2008 A nice example (for me, but for you ?) Choose the following linear realizations of Thue-Morse sequence � � � � � � � � � � + 1 − 1 1 0 0 1 S ( 0 ) = , A ( 0 ) , A ( 0 ) = = ; , − 1 + 1 0 1 0 1 1 0 and Rudin-Shapiro sequence � � � � � � � � � � � � + 1 + 1 − 1 − 1 1 0 0 0 � � , A ( 1 ) , A ( 1 ) S ( 1 ) = 0 = 1 = , , , . + 1 − 1 + 1 − 1 1 0 1 − 1 Fine ! the matrices A ( 0 ) and A ( 0 ) also act on S ( 1 ) , leading to the 0 1 following 2 -stack-automatic sequences : Laboratoire d’Analyse, Topologie et Probabilit´ e UMR-CNRS 6632 Marseille

Journ´ ees Num´ eration Prague, Mai 26-30, 2008 Construction Choose ( ε n ) n ∈ { 0 , 1 } N and define γ ( ε ) : N → {− 1 , + 1 } 2 by � + 1 � γ ( ε ) ( n ) = A ε 0 e 0 ( n ) . . . A ( ε t ) e t ( n ) − 1 We have for | z | = 1 (easy) √ || A ( 0 ) ( z ) || 2 ≤ 2 and || A ( 1 ) ( z ) || ≤ 2 . Therefore 1 || Γ( q m , z ) || 2 ≤ c . 2 2 ( ε 0 + ··· + ε m − 1 ) and 2 ( ε 0 + ··· + ε t − 1 ) ; 1 || Γ( N , z ) || 2 ≤ C . � ∞ r = 0 e r ( N ) 2 finally we are able to infer many interesting properties of the dynamical system built from the sequences γ ( ǫ ) . Laboratoire d’Analyse, Topologie et Probabilit´ e UMR-CNRS 6632 Marseille

Journ´ ees Num´ eration Prague, Mai 26-30, 2008 IV. C HAINED SEQUENCES IV.1 Main definitions X is a compact metrizable group denoted by G . Definition. A map f : { 0 , 1 , . . . , q − 1 } ∗ → G is (q)-chained if f ( empty word ) = 1 G and for all digits a , b and all digital words w one has f ( abw ) = f ( ab ) f ( b ) − 1 f ( bw ) . Consequently f ( a 1 a 2 · · · a s w ) = f ( a 1 a 2 ) f ( a 2 ) − 1 . . . f ( a s − 1 a s ) f ( a s ) − 1 f ( a s w ) . The chained map is said to be left regular if f ( 0 w ) = f ( w ) for any digital word w . Laboratoire d’Analyse, Topologie et Probabilit´ e UMR-CNRS 6632 Marseille

Journ´ ees Num´ eration Prague, Mai 26-30, 2008 Definition. a sequence u : N → G is chained if there exists a left regular chained map f such that t � e r ( n ) q r ) . u n = f ( e t ( n ) · · · e 0 ( n )) ( n = r = 0 Typical examples : – Completely q -multiplicative sequences are chained, in particular the sum-of-digits function ; – Rudin-Shapiro sequence is chained (it is a 2-bloc map). Notice that the underlying chained maps f are right regular i.e., f ( w 0 ) = f ( w ) for all digital words. Laboratoire d’Analyse, Topologie et Probabilit´ e UMR-CNRS 6632 Marseille

Journ´ ees Num´ eration Prague, Mai 26-30, 2008 Transition matrix T a T b = f ( ab ) f ( b ) − 1 . For any irreducible representation ρ of G the matrices ρ T = ( ρ ( a T b )) a , b verifies : √ q ≤ || ρ T || 2 ≤ q . The chaines sequence is called : • contractive if || ρ T || 2 < q , • Hadamard if || ρ T || 2 = √ q , for all irreducible non trivial representations of G . The case || ρ T || = 2 is typically represented by   1 0 · · · 0 0   i ρ T j =  .  .  i B j  .   0 Laboratoire d’Analyse, Topologie et Probabilit´ e UMR-CNRS 6632 Marseille