aa r X i v : . [ m a t h . D S ] S e p Non-Archimedean dynamics of the complex shift
L. B. Tyapaev [email protected] Saratov State University, Saratov, Russia
An (asynchronous) automaton transformation of one-sided infinite words over p -letter alphabet F p = Z /p Z ,where p is a prime, is a continuous transformation (w.r.t. the p -adic metric) of the ring of p -adic integers Z p . Moreover, an automaton mapping generates a non-Archimedean dynamical system on Z p . Measure-preservation and ergodicity (w.r.t. the Haar measure) of such dynamical systems play an important rolein cryptography (e.g., in stream cyphers). The aim of this paper is to present a novel way of realizing acomplex shift in p -adics. In particular, we introduce conditions on the Mahler expansion of a transformationon the p -adics which are sufficient for it to be complex shift. Moreover, we have a sufficient condition ofergodicity of such mappings in terms of Mahler expansion. Keywords: p -adic numbers, automata, p -adic dynamical systems, measure-preservation, ergodicity. Introduction
Let X be a finite set; we call this set an alphabet . Given alphabet X , we denote by X ∗ a free monoidgenerated by the set X . The elements of the monoid X ∗ are expressed as words x x . . . x n − (includingthe empty word ∅ ). If u = x x . . . x n − ∈ X ∗ , then | u | = n is the length of the word u . The length of ∅ is equal to zero. Along with finite words from X ∗ we also consider infinite words of the form x x x . . . ,where x i ∈ X . The set of such infinite words is denote by X ∞ . For arbitrary u ∈ X ∗ and v ∈ X ∗ ∪ X ∞ ,we naturally defines the product (concatenation) uv ∈ X ∞ . A word u ∈ X ∗ is the beginning , or prefix ofa word w ∈ X ∗ ( ∈ X ∞ ) if w = uv for a certain v ∈ X ∗ ( ∈ X ∞ ). The set X ∞ is an infinite Cartesianproduct X N . We can introduce on the X N the topology of the direct Tikhonov product of finite discretetopological spaces X . In this topology X ∞ is homeomorphic to the Cantor set. Given finite word u ∈ X ∗ ,the set uX ∞ of all words beginning with u is closed and open simultaneously (i.e. is clopen ) in the giventopology; the family of all such sets { uX ∞ : u ∈ X ∗ } is the base of the topology.We put a metric d π on X ∞ by fixing a number π > d π ( u, v ) = π − ℓ , where ℓ is thelength of the longest common prefix of the words u and v . The distance between identical words is equalto zero.Thus the more that the initial terms of u and v agree, the closer they are to one another. It is easyto check that d π is a metric, and indeed a non-Archimedean metric , i.e. for any u, v, w ∈ X ∞ :0 d π ( u, v ) d π ( u, v ) = 0 ⇔ u = v ; d π ( u, w ) max { d π ( u, v ) , d π ( v, w ) } . The set uX ∞ of infinite words beginning with u is a ball B π −| u | ( w ) of radius π −| u | centered at arbitrary w ∈ uX ∞ .Speaking about an asynchronous automaton we always understand the “letter-to-word” transducer A = ( X , S , Y , h, g, s ), where1) X and Y are finite sets (the input and output alphabets , respectively);2) S is a set ( the set of internal states of automaton );3) h : X × S → S is a mapping ( transition function );4) g : X × S → Y ∗ is a mapping ( output function ), and5) s ∈ S is fixed ( initial state ).We assume that an asynchronous automaton A works in a framework of discrete time steps. Theautomaton reads one symbol at a time, changing its internal state and outputting a finite sequence ofsymbols at each step.The cardinality S of the set of states S of an automaton A is called the cardinality of the automaton.In particular, automaton A is finite if S < ∞ . If every value of the output function g ( · , · ) is a one-letter word, then automaton A is called a synchronous automaton. In the sequel denote a synchronousautomaton via B . The functions h and g can be continued to the set X ∗ × S ( X ∞ × S ).The state s ∈ S of the automaton A is called accessible if there exists a word w ∈ X ∗ such that h ( w, s ) = s . An automaton is called accessible if all its states are accessible. In the sequel, we consideronly accessible automata. n asynchronous automaton A is nondegenerate if and only if there do not exist any accessible state s ∈ S and an infinite word u ∈ X ∞ such that, for an arbitrary prefix w of the word u , the word g ( w, s )is empty.A mapping f : X ∞ → Y ∞ is said to be defined a nondegenerate automaton A if f ( u ) = g ( u, s ) ∈ Y ∞ for any u ∈ X ∞ . The mapping f : X ∞ → Y ∞ is continuous if and only if it is defined by a certainnondegenerate asynchronous automaton , see [2]. In the general case an asynchronous automaton defininga continuous mapping is infinite. p -adic numbers Let p be a fixed prime number. By the fundamental theorem of arithmetics, each non-zero integer n can be written uniquely as n = p ord p n ˆ n, where ˆ n is a non-zero integer, p ∤ ˆ n , and ord p n is a unique non-negative integer. The functionord p n : Z \{ } → N is called the p -adic valuation . If n, m ∈ Z , m = 0, then the p -adic valuation of x = n/m ∈ Q as ord p x = ord p n − ord p m. The p -adic valuation on Q is well defined; i.e., that ord p x of x does not depend on the fractional repre-sentation of x .By using the p -adic valuation we will define a new absolute value on the field Q of rational numbers.The p -adic absolute value of x ∈ Q \{ } is given by | x | p = p − ord p x and | | p = 0. The p -adic absolute value is non-Archimedean. It induces the p -adic metric d p ( x, y ) = | x − y | p which is non-Archimedean. The completion of Q w.r.t. p -adic metric is a field, the field of p -adic numbers, Q p . The p -adic absolute value is extended to Q p , and Q is dense in Q p . The space Q p is locally compactas topological space.As the absolute value | · | p may be only p k for some k ∈ Z , for p -adic balls (i.e., for balls in Q p ) wesee that B − p k ( x ) = { z ∈ Q p : d p ( z, x ) < p k } = B p k − ( x ) = { z ∈ Q p : d p ( z, x ) p k − } . Thus, we conclude that p -adic balls (of non-zero radii) are open and closed simultaneously; so B p k ( x ) isa clopen ball of radius p k centered at x ∈ Q p . Balls are compact; the set of all balls (of non-zero radii)form a topological base of a topology of a metric space. Thus, Q p is a totally disconnected topologicalspace.The ball B (0) = { x ∈ Q p : | x | p } is the ring of p -adic integers and denoted via Z p . The space Z p is a compact clopen totally disconnected metric subspace of Q p .The ball B − (0) = { x ∈ Z p : | x | p < } = B p − (0) = p Z p is a maximal ideal of the ring Z p . The factorring Z p /B − (0) = Z /p Z = F p is then a finite field F p of p elements; it is called the residue (class) field of Q p .Every p -adic number has a unique representation as a sum of a special convergent p -adic series whichis called a canonical representation , or a p -adic expansion. For x ∈ Z p there exist a unique sequence δ ( x ) , δ ( x ) , . . . ∈ { , , . . . , p − } such that x = P ∞ i =0 δ i ( x ) · p i = δ ( x ) + δ ( x ) · p + δ ( x ) · p + . . . . Thefunction δ i ( x ) is called the i -th coordinate function .Let x = δ ( x ) + δ ( x ) · p + δ ( x ) · p + . . . be a p -adic integer in its canonical representation. The mapmod p k : x = ∞ X i =0 δ i ( x ) · p i x mod p k = k − X i =0 δ i ( x ) · p i is a continuous ring epimorphism of the ring Z p onto the ring Z /p k Z of residues modulo p k ; it is called reduction map modulo p k . The kernel of the epimorphism mod p k is a ball p k Z p = B p − k (0) of radius p − k around 0. The rest p k − p − k are co-sets with respect to this epimorphism, e.g., B p − k (1) = 1 + p k Z p , is a co-set of 1, i.e. the sets of all p -adic integers that are congruent to 1 modulo p k .A p -adic integer x ∈ Z p is invertible in Z p , that is, has a multiplicative inverse x − ∈ Z p , x · x − = 1,if and only if δ ( x ) = 0; that is, if and only if x is invertible modulo p , meaning x mod p is invertible in F p . Invertible p -adic integer is also called a unit . The set Z ∗ p of all units of Z p is a group with respect o multiplication, called a group of units , or a multiplicative subgroup of Z p . The group of units Z ∗ p is a p -adic sphere S (0) of radius 1 around 0: Z ∗ p = { z ∈ Z p : | z | p = 1 } = Z p \ p Z p = B (0) \ B p − (0) = S (0) . Note that, the space Z p is a profinite algebra with the structure of an inverse limit , that is the ring Z p is an inverse limit of residue rings Z /p ℓ Z modulo p ℓ , for all ℓ = 1 , , . . . Z p ← . . . ← Z /p Z ← Z /p Z .
2. Automata functions
We identify n -letter words over F p = { , , . . . , p − } with non-negative integers in a natural way:Given an n -letter word u = x x . . . x n − , x i ∈ F p , we consider u as a base- p expansion of the number α ( u ) = x n − . . . x x = x + x · p + . . . + x n − · p n − . In turn, the latter number can be considered as anelement of the residue ring Z /p n Z = { , , . . . , p n − } modulo p n . Thus, every (synchronous) automaton B = ( F p , S , F p , h, g, s ) corresponds a map from Z /p n Z to Z /p n Z , for every n = 1 , , . . . .Moreover, given an infinite word u = x x x . . . over F p we consider u as the p -adic integer x = α ( u ) = . . . x x x whose canonical expansion is x = α ( u ) = x + x · p + x · p + . . . = P ∞ i =0 x i · p i . Thenevery (synchronous) automaton B = ( F p , S , F p , h, g, s ) defines a map f B from the ring of p -adic integers Z p to itself : For every x ∈ Z p we put δ i ( f B ( x )) = g ( δ i ( x ) , s i ), i = 0 , , , . . . where s i = h ( δ i − ( x ) , s i − ), i = 1 , , . . . . We say then that map f B is synchronous automaton function (or, automaton map ) of thesynchronous automaton B .Similarly way, a nondegenerate asynchronous automaton A = ( F p , S , F p , h, g, s ) naturally defines a continuous mapping (w.r.t. p -adic metric) f A from Z p to Z p .Note that, the automaton map f B : Z p → Z p of the (synchronous) automaton B = ( F p , S , F p , h, g, s ) satisfy the p -adic Lipschitz condition with constant | f B ( x ) − f B ( y ) | p | x − y | p for all x, y ∈ Z p . Conversely, for every -Lipschitz function f : Z p → Z p there exists an automaton B = ( F p , S , F p , h, g, s ) such that f = f B , see [1]. For example, every function defined by polynomialwith p -adic integers coefficients (in particular, with rational integers) is a 1-Lipschitz map, hence it is anautomaton map.Let n ∈ N be a natural number and let C ( n ) = ( X , S , Y , h, g, s ) be a nongenerate asynchronousautomaton, which is translated the infinite input word u = x x . . . x n − . . . into infinite output word w = ∅∅ . . . ∅ | {z } n times y n y n +1 . . . ; So, we have y i = g ( x i , s i ) = ∅ for i = 0 , , . . . , n − ,s i = h ( x i − , s i − ) for i = 1 , , . . . , n − , and y i = g ( x i , s i ) , s i +1 = h ( x i , s i )for all i = n, n + 1 , . . . .A unilateral shift is the transformation of the space of infinite words over alphabet X defined by therule x x x . . . x x x . . . . Note that, a unilateral shift is defined by an asynchronous automaton C (1) = ( X , S , X , h, g, s ), whoseoutput function g is expressed as follows g ( x , s ) = ∅ , and g ( x i , s i ) = x i for i = 1 , , . . .. The initial state of this automaton, irrespective of the incoming letter outputs an empty word, afterthat the automaton outputs the incoming words without changes. In this case, the automaton C (1) =( F p , S , F p , h, g, s ) naturally defines the p -adic shift (or, the one-sided Bernoulli shift , see, e.g., [3]); thatis, the p -adic shift σ : Z p → Z p is expressed as follows. If x = . . . x x x = x + x p + x p + . . . , where x i ∈ F p , we let σ ( x ) = x − x p = . . . x x x = x + x p + x p + · · · . In other words, the shift σ cuts offthe first digit term in the p -adic expansion of x ∈ Z p .We see that if σ n denotes the n -fold iterate of σ ,then we have σ n ( x ) = x − ( x + x p + ... + x n − p n − ) p n = x n + x n +1 p + . . . . Moreover, for x ∈ Z , it is the casethat σ n ( x ) = ⌊ xp n ⌋ were ⌊·⌋ is the greatest integer function.A function T : Z p → Q p is called a locally constant if for every x ∈ Z p there exist an open neighbour-hood U x (e.g., a ball of radius p − N for some N ∈ N centered at x , U x = { z ∈ Z p : | x − z | p < p − N } ) such hat T is a constant on U x . For example, for arbitrary i ∈ N a function δ i ( x ) is locally constant, because δ i remains unchanged if we replace x by any y , such that | x − y | p < p − i .Let D ⊂ Z p , not necessarily compact. A function T : Z p → Q p is called a step function on D if thereexists a positive integer ℓ such that T ( x ) = T ( y ) for all x, y ∈ D with | x − y | p p − ℓ . The smallest integer ℓ with this property is called the order of the step function T .It is clear from the definition that a step function is a locally constant on D . On Z p it also holds,that any locally constant function is a step function.Let f : Z p → Z p be a 1-Lipschitz function. Given natural n ∈ N , for all x ∈ Z p we can represent f as f ( x ) = ( f ( x mod p n )) mod p n + p n G z ( t ) , where t = p − n ( x − ( x mod p n )) ∈ Z p , z = x mod p n , and G z : Z p → Z p is a 1-Lipschitz function. It isclear that f is the automaton map of a synchronous automaton B = ( F p , S , F p , h, g, s ), and G z is anautomaton map of the automaton B z = ( F p , S , F p , h, g, s ( z )), where s ( z ) ∈ S is the accessible state ofthe automaton B , i.e. s ( z ) was reached after being feeded by the input word z = x mod p n .Similarly, for map f : Z p → Z p that defined by an asynchronous automaton C ( n ) , we can see that f ( x ) = G z ( t ) + T ( x ) , where for any z = x mod p n , the map G z : Z p → Z p is a 1-Lipschitz, T ( x ) is a step function of order notgreater than n , and t = p − n ( x − ( x mod p n )); and we say that f is a complex shift .For a complex shift f a following condition holds: There exist positive integer M ∈ N such that forevery i > M the i -th coordinate function δ i ( f ( x )) does not depend on δ i + k ( x ) for k = 1 , , . . . . Hence, acomplex shift is a locally 1-Lipschitz function . By the definition, a function F is a locally 1-Lipschitz iffor a given x ∈ Z p , there exist an open neighbourhood U x of x such that the inequality | F ( x ) − F ( y ) | p | x − y | p holds for all y ∈ U x . As Z p is compact, the function F : Z p → Z p is a locally 1-Lipschitz if and only ifthe latter inequality holds for all x, y ∈ Z p which are sufficiently close to one another.
3. Dynamical systems A dynamical system on a measurable space S is understood as a triple ( S , µ, f ), where S is a setendowed with a measure µ and f : S → S is a measurable function; that is, the f -preimage f − ( T ) of any µ -measurable subset T ⊂ S is a µ -measurable subset of S .An iteration of a function f A : Z p → Z p which is defined by (asynchronous) automaton A =( F p , S , F p , h, g, s ) generates a dynamical system ( Z p , µ p , f A ) on the space Z p . The space Z p is equippedwith a natural probability measure, namely, the Haar measure µ p normalized so that the measure of thewhole space is 1, µ p ( Z p ) = 1. Balls B p − k ( a ) of nonzero radii constitute the base of the corresponding σ -algebra of measurable subsets of Z p . That is, every element of the σ -algebra, the measurable subset of Z p , can be constructed from the elementary measurable subsets by taking complements and countableunions. We put µ p ( B p − k ( a )) = p − k .We remind that if a measure space S endowed with a probability measure µ is also a topologicalspace, the measure µ is called Borel if all Borel sets in S are µ -measurable. Recall that a Borel set isany element of σ -algebra generated by all open subsets of S ; that is, a Borel subset can be constructedfrom open subsets with the use of complements and countable unions. A probability measure µ is called regular if for all Borel sets X in S µ ( X ) = sup { µ ( A ) : A ⊆ X, A closed } = inf { µ ( B ) : X ⊆ B, B open } . The probability measure µ p is Borel and regular.A dynamical system ( Z p , µ p , f A ) is also topological since Z p are not only measurable space but alsometric space, and corresponding transformation f A are not only measurable but also continuous. More-over, this dynamical system is non-Archimedean, due to the fact that the space Z p is non-Archimedeanspace.A measurable mapping f : Z p → Z p is called measure-preserving if µ p ( f − ( S )) = µ p ( S ) for eachmeasurable subset S ⊂ Z p . A measure-preserving map f is said to be ergodic if for each measurablesubset S such that f − ( S ) = S holds either µ p ( S ) = 1 or µ p ( S ) = 0; so ergodicity of the map f justmeans that f has no proper invariant subsets; that is, invariant subsets whose measure is neither 0 nor 1.The following question arises. What continuous with respect to the metric µ p transformations aremeasure-preserving or ergodic with respect to the mentioned measure?For a given f : Z p → Z p and n ∈ N , let f k be a function defined on the ring Z /p n · k Z and valuated inthe ring Z /p n · ( k − Z , where k = 2 , , . . . . he following criterion of measure-preservation for a complex shift f is valid: A mapping f : Z p → Z p is measure-preserving if and only if the number f − k ( x ) of f k -preimages of the point x ∈ Z /p n · ( k − Z isequal to p n [4, 5].Given a map f : Z p → Z p , a point z ∈ Z p is said to be a periodic point if there exists r ∈ N such that f r ( z ) = z . The least r with this property is called the length of period of z . If z has period r , it iscalled an r -periodic point . The orbit of an r -periodic point z is { z , f ( z ) , . . . , f r − ( z ) } . This orbit iscalled an r -cycle .For a given n ∈ N , let f mod p k · n : Z /p k · n Z → Z /p k · n Z , for k = 1 , , , . . . ; and let γ k be an r k -cycle { z , z , . . . , z r k − } , where z j = ( f mod p k · n ) j ( z ), 0 j r k − k = 1 , , , . . . .The following condition of ergodicity holds: Let f : Z p → Z p be a complex shift and let f be a measure-preserving map. Then f is ergodic if for every k ∈ N γ k is a unique cycle [6].
4. Measure-preservation and ergodicity in terms of Mahler expansion
By Mahler’s Theorem, any continuous function F : Z p → Z p can be expressed in the form of auniformly convergent series, called its Mahler Expansion (or,
Mahler series ): F ( x ) = ∞ X m =0 a m (cid:18) xm (cid:19) , where a m = m X i =0 ( − m + i F ( i ) (cid:18) mi (cid:19) ∈ Z p and (cid:18) xm (cid:19) = x ( x − m ) · · · ( x − m + 1) m ! , m = 1 , , . . . , (cid:18) x (cid:19) = 1 . Mahler series converges uniformly on Z p if and only if p lim m →∞ a m = 0 . Hence uniformly convergent series defines a uniformly continuous function on Z p . The function f repre-sented by the Mahler series is uniformly differentiable everywhere on Z p if and only if p lim m →∞ a m + k m = 0for all k ∈ N .The function f is analytic on Z p if and only if p lim m →∞ a m m ! = 0 . Various properties of the function f can be expressed via properties of coefficients of its Mahlerexpansion.Let a ( n ) m be the n -th Mahler coefficient of the Bernoulli shift σ n . We have σ n ( x ) = ∞ X m =0 a ( n ) m (cid:18) xm (cid:19) . The coefficients a ( n ) m satisfy the following properties [3]: a ( n ) m = 0 for m < p n ; a ( n ) m = 1 for m = p n ;Suppose j > . Then, p j divides a ( n ) m for m > jp n − j + 1 (and so, | a ( n ) m | p /p j ). The following statement gives a description of 1-Lipschitz measure-preserving (respectively, of 1-Lipschitz ergodic) transformations on Z p [1]. The function f defines a -Lipschitz measure-preservingtransformation on Z p whenever the following conditions hold simultaneously: a p ) ; a m ≡ p ⌊ log p m ⌋ +1 ) , m = 2 , , . . . The function f defines a -Lipschitz ergodic transformation on Z p whenever the following conditions holdsimultaneously: a p ) ; ≡ p ) for p odd; a ≡ for p = 2 ; a m ≡ p ⌊ log p ( m +1) ⌋ +1 ) , m = 1 , , , . . . Moreover, in the case p = 2 these conditions are necessary. The following statement gives a description of complex shift in terms of Mahler expansion.
Theorem 1.
A function f : Z p → Z p is a complex shift if and only if | a m | p p −⌊ log pn m ⌋ +1 , where n ∈ N , m > . The following theorems give a description of measure-preservation and ergodicity for a complex shift.
Theorem 2.
A complex shift f : Z p → Z p is measure-preserving whenever the following conditions holdsimultaneously: a m p ) for m = p n ; a m ≡ p ⌊ log pn m ⌋ ) , m > p n ,where n ∈ N . Theorem 3.
A complex shift f : Z p → Z p is ergodic on Z p whenever the following conditions holdsimultaneously: a + a + . . . + a p n − ≡ p ) ; a m ≡ p ) for m = p n ; a m ≡ p ⌊ log pn m ⌋ ) , m > p n ,where n ∈ N . Let f : Z p → Z p be a complex shift, and let E k ( f ) be a set of all the following points e fk ( x ) of Euclideanunit square I = [0 , × [0 , ⊂ R for k = 1 , , , . . . [7]: e fk ( x ) = (cid:16) x mod p n + k p n + k , f ( x ) mod p k p k (cid:17) , where x ∈ Z p , n ∈ N . Note that x mod p n + k corresponds to the prefix of length n + k of the infinite word x ∈ Z p , i.e., to the input word of length n + k of the automaton C ( n ) ; while f ( x ) mod p k corresponds tothe respective output word of length k . Denote via E ( f ) the closure of the set E ( f ) = S ∞ k =1 E k ( f ) in thetopology of real plane R . As E ( f ) is closed, it is measurable with respect to the Lebesgue measure onreal plane R . Let λ ( f ) be the Lebesgue measure of E ( f ). Theorem 4.
For a given complex shift f : Z p → Z p the closure E ( f ) is nowhere dense in I , hence λ ( f ) = 0 . References [1] V. Anashin and A. Khrennikov,
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