Let M denote the maximal ideal of the ring of integers of a non-Archimedean ﬁeld K with residue class ﬁeld k whose invertible elements, we denote k , and a uniformizer we denote π. In this paper, we consider the map T : M → M deﬁned by v(x ) T (x ) = − b(x ), v(x ) where b(x ) denotes the equivalence class to which belongs in k . We show that T preserves Haar measure μ on the compact abelian topological group M.Let B denote the Haar σ -algebra on M. We show the natural extension of the dynamical system #(k) (M, B,μ, T ) is Bernoulli and has entropy log(#(k)). The ﬁrst of these two v × #(k ) properties is used to study the average behaviour of the convergents arising from T . Here for a ﬁnite set A its cardinality has been denoted by #(A). In the case K = Q , i.e. the ﬁeld of p-adic numbers, the map T reduces to the well-studied continued fraction map due to Schneider. Keywords Schneider’s continued fraction map · Non-Archimedean ﬁelds · Bernoulli processes · Entropy Mathematics Subject Classiﬁcation Primary 11K50 · Secondary 11A55 · 11J70 1 Introduction The purpose of this paper is to calculate the entropy of T. Schneider’s continued fraction map, and to show the map has a natural extension which is Bernoulli. This is B R. Nair a.haddley@liv.ac.uk ; nair@liv.ac.uk Mathematical Sciences, The University of Liverpool, Peach Street, Liverpool L69 7ZL, UK 123 A. Haddley, R. Nair then used to study the behaviour of averages of convergents arising from Schneider’s map. Schneider’s map is usually deﬁned on the p-adic ﬁeld for the rational prime p, see [26]. In fact we work in a more general setting which we now describe. Let K denote a topological ﬁeld. By this, we mean that the ﬁeld K is a locally compact group under the addition, with respect to a topology. This ensures that there is a translation invariant Haar measure μ on K , which is unique up to scalar multiplication. In the non-Archimedean examples that concern us in this paper, this topology will always be discrete. For an element a ∈ K , we are now able to deﬁne its absolute value, as μ(aF ) |a|= , μ(F ) for every μ measurable F ⊆ K of ﬁnite positive μ measure. Let R denote the set ≥0 of all non-negative real numbers. An absolute value is a function |.|: K → R such ≥0 that (i) |a|= 0 if and only if a = 0; (ii) |ab|=|a||b| for all a, b ∈ K and (iii) |a + b|≤|a|+|b| for all pairs a, b ∈ K . The absolute value just deﬁned gives rise to a metric deﬁned by d(a, b) =|a − b| with a, b ∈ K , whose topology coincides with original topology on the ﬁeld K . Topological ﬁelds come in two types. The ﬁrst where (iii) can be replaced by the stronger condition (iii)* |a + b|≤ max(|a|, |b|) a, b ∈ K , called non-Archimedean ﬁelds and ﬁelds where (iii)* is not true called Archimedean spaces. In this paper, we shall concern ourselves solely with non-Archimedean ﬁelds. Another approach to deﬁning a non-Archimedean ﬁeld is via discrete valuations. Denote the real numbers ∗ ∗ ∗ by R.Let K = K \{0}.Amap v : K → R is a valuation if (a) v(K ) ={0}; (b) v(xy) = v(x ) + v(y) for x , y ∈ K and (c) v(x + y) ≥ min{v(x ), v(y)}.Two valuations v and cv,for c > 0 a real constant, are called equivalent. We extend v to K formally by letting v(0) =∞. The image v(K ) is an additive subgroup of R called the value group of v. If the value group is isomorphic to Z,wesay v is a discrete valuation. Here Z denotes the set of integers. If v(K ) = Z, we call v a normalised discrete valuation. To our initial absolute value we associate the valuation described v(a) as follows. Pick 0 <α < 1 and write |a|= α , i.e., let v(a) = log |a|. Then v(a) is a valuation, an additive version of |a|. Let v : K → R be a valuation corresponding to the absolute value |.|: K → R . ≥0 Then O = O := {x ∈ K : v(x ) ≥ 0}= O := {x ∈ K :|x|≤ 1} v K is a ring, called the valuation ring of v and K is its ﬁeld of fractions. The set of units in O is O ={x ∈ K : v(x ) = 0}={x ∈ K :|x|= 1} and M ={x ∈ K : v(x)> 0}={x ∈ K :|x | < 1} is an ideal in O.Note O = O ∪ M. Because M is a maximal ideal, we know k = O/M is a ﬁeld, called the residue ﬁeld of v or of K . In the sequel throughout this paper, we assume that k is a ﬁnite ﬁeld. Suppose the valuation v : K → Z is normalised and discrete. Take π ∈ M such that v(π ) = 1. We call π a uniformizer. Then every x ∈ K can be written uniquely as x = uπ with × n u ∈ O and n ∈ Z. In particular every x ∈ M can be written uniquely as x = uπ for a unit u ∈ O and n ≥ 1. 123 On Schneider’s Continued Fraction Map... We now consider two examples. a) p-adic numbers : Let Q denote the rational numbers. For r = p in Q with −v u and v coprime to p and each other, let |r | = p . Then d (r , r ) =|r − r | for p p p r ∈ Q deﬁnes a metric on Q. The completion of Q with respect to the metric d is a ﬁeld denoted Q referred to as the p-adic numbers. We also use Z to denote {x ∈ Q : p p p |x | ≤ 1}—the ring of p-adic integers. | It is worth keeping in mind that the metric d p p has the ultrametric property, namely that d (r , r ) ≤ max(d (r , r ), d (r , r )) for p p p all r , r and r ∈ Q . The main characteristics of the ﬁeld Q that distinguish it from p p the ﬁeld R stem from the ultrametric property. It turns out that Q is a locally compact abelian ﬁeld and hence comes endowed with a translation invariant Haar measure. In this instance, K = Q , O = Z , M = pZ, π = p and k = Z/pZ. See [16]for a p p clear and succinct introduction to p-adic numbers. b) The ﬁeld of formal Laurent series in ﬁnite characteristic :Let q be a power of a prime p and let F be the ﬁnite ﬁeld with q elements. Denote by F [X ] and q q F (X ) the ring of polynomials with coefﬁcients in F and the quotient ﬁeld of F [X ], q q q deg(P)−deg(Q) respectively. For each P, Q ∈ F [X ] set |P/Q|:= q , where for an −1 element g ∈ F [X ] we have denoted its degree by deg(g).Let F ((X )) denote the p q ﬁeld of formal Laurent series, i.e. −1 n −1 F ((X )) = a X + ··· + a + a X + ··· : n ∈ Z, a ∈ F . q n 0 −1 i q Also, d (x , y) =|x − y| for x , y ∈ F (X ) deﬁnes a metric on F (X ). The metric q q q −1 extends to F ((X )) by completion and by implication to its subset L ={x ∈ −1 F ((X )) :|x|≤ 1}. Note that this metric is non-Archimedean since |x + y|≤ −1 max(|x |, |y|). In this example, K = F ((X )), O = L, M = X L, π = X and k = L/X L = F . See [30] for more information on the ﬁeld of Laurent expansions over a ﬁnite ﬁeld. The only two types of non-Archimedean local ﬁelds there are ﬁnite extensions of the ﬁeld of p-adic numbers for some rational prime p and the ﬁeld of formal Laurent series over a ﬁnite ﬁeld. For more details and background to this discussion of non- Archimedean ﬁelds see Chapter 2 of [10], and Chapter 4 of [23]. Our primary object of study in this paper is the map T : M → M deﬁned by v(x ) T (x ) = − b(x ), v(x ) where b(x ) denotes the residue class to which belongs in k. This gives rise to the continued fraction expansion of x ∈ M in the form x = , (1) b + b + b + 123 A. Haddley, R. Nair where b ∈ k , a ∈ N for n = 1, 2,....Here N denotes the set of natural numbers. n n The rational approximants to x ∈ M arise in a manner similar to that in the case of the real numbers as follows. We suppose A = b , B = 1, A = b b +π , B = b . 0 0 0 1 0 1 1 1 Then set a a n n A = π A + b A and B = π B + b B (2) n n−2 n n−1 n n−2 n n−1 for n ≥ 2. A simple inductive argument, for n = 1, 2,... gives n a +···+a 1 n A B − A B = (−1) π . (3) n−1 n n n−1 The map T : M → M preserves Haar measure on M. By this we mean, for −1 each Haar measurable set A contained in M we have μ(T (A)) = μ(A).Here −1 T (A) := {x ∈ M : T (x ) ∈ A}. To prove that T preserves Haar measure on M v v we only need to check it for special sets of the form πa + π O, where a ∈ O.This is because sets of this form generate the Haar σ -algebra on M. Suppose c ∈ k\{0} and let m, n be positive integers. Then T = πa + π O. c + πa + π O It follows −1 n T (πa + π O) = . c + πa + π O m=1 c ∈k\{0} Since m m π π n+m = + π O, c + πa + π O c + πa 0 0 1−m−n which has measure #(k) . Recall here #(A) denotes the cardinality of the ﬁnite set A. It follows −1 n n+m μ(T (πa + π O)) = μ + π O , c + πa c ∈k\{0} m=1 n+m = μ π O , c ∈k\{0} m=1 1−n−m = #(k) , c ∈k\{0} m=1 1−n n = #(k) = μ(π + π O), as required. So, T preserves Haar measure on M. 123 On Schneider’s Continued Fraction Map... In the case where K = Q the map T reduces to the original Schneider’s continued p v fraction map T , which motivates this whole investigation and is deﬁned as follows. For x ∈ pZ deﬁne the map T : pZ → pZ by p p p p v(x ) v(x ) a(x ) p p p T (x ) = − mod p = − b(x ), (4) x x x where v(x ) is the p-adic valuation of x, a(x ) ∈ N and b(x ) ∈{1, 2,..., p − 1}. Then using the continued fraction algorithm for x we get the expansion, x = , (5) b + b + b + where b ∈{1, 2,..., p − 1}, a ∈ N for n = 1, 2,.... n n We now deﬁne the measure-theoretic entropy. Let (X , A, m) be a probability space where X is a set, A is a σ -algebra of its subsets and m is a probability measure. A partition of (X , A, m) is deﬁned as a denumerable collection of elements α = {A , A ,...} of A such that A ∩ A =∅ for all i , j ∈ with i = j and A = 1 2 i j i i ∈ X . Here is a denumerable index set. For a measure-preserving transformation T , −1 −1 we have T α ={T A |A ∈ α} which is also a denumerable partition. Given i i partitions α ={A , A ,...} and β ={B , B ,...}, we deﬁne the join of α and 1 2 1 2 β to be the partition α ∨ β ={A ∩ B |A ∈ α, B ∈ β}. For a ﬁnite partition i j i j α ={A ,..., A }, we deﬁne its entropy H (α) := − m(A ) log m(A ). Let 1 n i i i =1 A ⊂ A be a sub-σ -algebra. Then we deﬁne the conditional entropy of α given A to be H (α|A ) := − m(A |A ) log m(A |A ).Here m(A|A ) denotes the m- i i i =1 conditional probability of A with respect to the σ -algebra A . See [22] for more details about conditional probability. The entropy of a measure-preserving transformation T relative to a partition α is deﬁned to be n−1 −i h (T ,α) = lim H T α , n→∞ i =0 where the limit always exists. The alternative formula for h (T ,α) which is used for calculating entropy is n ∞ −i −i h (T ,α) = lim H α| T α = H α| T α . (6) n→∞ i =1 i =1 We deﬁne the measure-theoretic entropy of T with respect to the measure m to be h (T ) = sup h (T ,α). Here the supremum is taken over all ﬁnite or countable m m partitions α from A with H (α) < ∞. 123 A. Haddley, R. Nair Two measure-preserving transformations (X ,β , m , T ) and (X ,β , m , T ) are 1 1 1 1 2 2 2 2 said to be isomorphic if there exist sets M ⊆ X and M ⊆ X with m (M ) = 1 1 1 2 2 1 1 and m (M ) = 1 such that T (M ) ⊆ M and T (M ) ⊆ M and such that there 2 2 1 1 1 2 2 2 exists a map φ : M → M satisfying φT (x ) = T φ(x ) for all x ∈ M and 1 2 1 2 1 −1 m (φ (A)) = m (A) for all A ∈ β . The importance of measure theoretic entropy is 1 2 2 that two dynamical systems with different entropies can not be isomorphic. For more on measure-theoretic entropy and isomorphism, see [31]. The following is our ﬁrst result. Theorem 1.1 Let B denote the Haar σ -algebra restricted to M and let μ denote the Haar measure on M. Then the measure-preserving transformation (M, B,μ, T ) has #(k) log(#(k)). measure-theoretic entropy #(k ) The measure-preserving transformation (pZ , B,μ, T ) is known to be ergodic p p [14]. Moreover, in [12] it was proved that (pZ , B,μ, T ) is exact. We forgo the p p deﬁnition of exactness here, however, as we do not use the concept in this paper. The exactness of (pZ , B,μ, T ) implies other weaker properties including mixing, p p which implies weak-mixing implying ergodicity, all implications being strict. Suppose (Y ,α, l) is a probability space, and let Y = (Y ,α, l) for each n ∈ Z. Set (X,β, m) = Y , i.e. the bi-inﬁnite product probability space. For the shift map τ({x }) = n n n∈Z ({x }), the measure-preserving transformation (X,β, m,τ) is called the Bernoulli n+1 process with state space (Y ,α, l).Here {x } is a bi-inﬁnite sequence of elements of the set Y . Any measure-preserving transformation isomorphic to a Bernoulli process will be referred to as Bernoulli. The fundamental fact about Bernoulli processes, famously proved by D. Ornstein, is that Bernoulli processes with the same entropy are isomorphic [27]. To any measure-preserving transformation, (X,β, m, T ) we can associate another called its natural extension. Originally introduced by V. A. Rokhlin [24], the natural extension is deﬁned as follows. Set X ={(x , x , x ,...) : x = T (x ), x ∈ X , n = 0, 1, 2,...}, T 0 1 2 n 0 n+1 n and let T : X → X be deﬁned by T T 0 0 T ((x , x ,...,)) = (T (x ), x , x ,...,). 0 1 0 0 0 1 The map T is 1 −1on X .If T preserves a measure m, then we can deﬁne a measure T 0 m on X , by deﬁning m on the cylinder sets C (A , A ,..., A ) ={{x }: x ∈ A , x ∈ A ,..., x ∈ A } 0 1 k n 0 0 1 1 k k by −k −k+1 m(C (A , A ,..., A )) = m(T (A ) ∩ T (A ) ∩ ... ∩ A ), 0 1 k 0 1 k 0 0 for k ≥ 1. One can check that the transformation (X , β, m, T ) is measure- T 0 preserving as a consequence of the measure preservation of the transformation (X,β, m, T ). Our second theorem is the following. 123 On Schneider’s Continued Fraction Map... Theorem 1.2 Suppose (M, B,μ, T ) is as in our ﬁrst theorem. Then the dynamical system (M, B,μ, T ) has a natural extension that is Bernoulli. In last two sections of this paper, this theorem is combined with subsequence pointwise ergodic theorems and the moving average ergodic theorem [4], respectively, to study the average behaviour of the convergents arising from the map T . These results in the special case K = Q already appear in [12]. Our two theorems above together tell us that as a dynamical system, the isomorphism class to which T belongs is determined solely by its residue class ﬁeld. This is irrespective of the characteristic of the underlying global ﬁeld. For instance, for each rational prime p the corresponding Schneider map has entropy log(p), so we know these maps are mutually non- p−1 isomorphic. Each of them is, however, isomorphic to the analogue of the Schneider map on the ﬁeld of formal power series with coefﬁcient ﬁeld the ﬁnite ﬁeld of p elements. Henceforth, for a real number y let {y} denote the fractional part of y. The study of the properties of (M, B,μ, T ) parallels that of the Gauss map deﬁned on [0, 1] by if x = 0; Tx = 0if x = 0. This map preserves the measure deﬁned for Lebesgue measurable A ⊆[0, 1] by 1 dx η(A) = . log 2 x + 1 Like (M, B,μ, T ),if L denotes the Lebesgue σ -algebra on [0, 1], the transformation ([0, 1], L,η, T ) also has a Bernoulli natural extension and in this case has entropy . Evidently the Gauss map cannot be isomorphic to (M, B,μ, T ), for any v. 6log(2) Analogously to the Gauss map [13], the map which governs the regular continued fraction on the real numbers, the measure-preserving transformation (pZ , B,μ, T ) p p via (4) gives rise to an integer recurrence relationship. This is as follows. We Suppose A = b , B = 1, A = b b + p , B = b . Then set 0 0 0 1 0 1 1 1 a a n n A = p A + b A and B = p B + b B (7) n n−2 n n−1 n n−2 n n−1 for n ≥ 2. A simple inductive argument gives for n = 1, 2,.... n a +...+a 1 n A B − A B = (−1) p . (8) n−1 n n n−1 Because p does not divide B we deduce that the integers A and B are coprime. n n n n ∞ The sequence of rationals ( ) are the convergents to x in pZ arising from B n=1 (5). Naturally one of the ﬁrst things one might try to do is explore the extent to which, theorems true for continued fractions on the real numbers extend to the p-adic numbers. For the most part, one can extend the regular continued fraction expansion and its properties to the ﬁeld of formal Laurent series over a ﬁnite ﬁeld, in a relatively trouble-free manner. This is primarily because the ﬁeld of formal Laurent series over 123 A. Haddley, R. Nair a ﬁnite ﬁeld is a Euclidean domain. In the context of the p-adic numbers the direct analogue of the regular continued fraction is the Ruban continued fraction [25]. Here, however, there are problems. The p-adic numbers are not a Euclidean domain. It is possible to deﬁne a sequence of rationals analogous to the convergents of the regular continued fractions. Their convergence to the number they are supposed to represent is not assured, however. This problem can be got round using a system of weights. This is what leads to Schneider’s continued fraction expansion. This is at a cost, however. Some partial success at recovering analogues of standard properties of the regular continued fraction for the real numbers is possible on the p-adic numbers. See, for instance, [1–3,8,11], where the issues of when a p-adic continued fraction is either ﬁnite or periodic is explored. One cannot, however, hope to have a theory as satisfactory or as useful as that offered by the regular continued fraction expansion. The situation is just more complex. For instance, unlike the sequence of convergents n ∞ of the regular continued fraction expansion, the sequence ( ) does not always B n=1 provide a sequence of best approximants to the p-adic number they approximate. Other solutions to this particular problem are available, though not using Schneider’s map however [20], [21]. All this said, as observed in [7], while not as versatile as the regular continued fraction, Schneider continued fraction can be a powerful tool in a number of situations. It is sometimes very useful in delicate constructions on the p-adic numbers. In [7] for instance it is used to construct numbers that distinguish between the Mahler and Koksma schemes of approximation to a speciﬁed degree. Speciﬁcally for a p-adic number η let w(n) denote its Mahler function on N deﬁned to be the supremum of all real numbers w such that the inequality −w−1 0 < |P(η)| ≤ H (P) is satisﬁed by inﬁnitely many polynomials P over Z of degree at most n.Here H denotes the height of the polynomial P, deﬁned to be the maximum of the absolute values of the coefﬁcients of P. Analogously, to η we can also associate the Koksma function w (n) which is deﬁned to be the supremum over all real numbers w such that the inequality −w−1 0 < |η − ξ | ≤ H (ξ ) is satisﬁed by inﬁnitely many algebraic numbers ξ of degree at most n. In this instance H (ξ ) denotes the height of the minimal polynomial deﬁning ξ. The relationship between the numbers w(n) and w (n) is a complex and unresolved issue. Restricting to the case n = 2 some progress has been made, though even here this is not an easy ∗ ∗ ∗ matter. It is known w(2) ∈[w (2), w (2) + 1]. We also know that w(2) = w (2) for almost all η in Q . Methods of diophantine approximation have been used to show there are p-adic numbers η such that w(2) = w (2) + δ for each δ ∈[0, 1). Con- structing ξ such that w(2) = w (2) + 1 has so for only proved possible using the Schneider continued fraction. The method has the additional advantage over diophan- tine approximation methods of being constructive. The details of this are to be found in [7]. See also [5,6] for related applications. 123 On Schneider’s Continued Fraction Map... Another interesting application of Schneider’s continued fraction is to deciding the algebraic independence of a set of p-adic numbers. See [9,19] for details. For background on the theory of regular continued fractions and its ergodic theory see for instance [13,15]. As is well known, if you restrict the Gauss map to the rational numbers you get the Euclidean algorithm. If you set p = 2 and restrict the Schneider map to the rational numbers what you get is the Binary Euclidean algorithm. This is another way of calculating the highest common factor of two integers, particularly well adapted to efﬁcient implementation on binary machines. The algorithm was ﬁrst published by Josef Stein [29] but is also attributed to Roland Silver and John Terzian in unpublished form [17]. The algorithm may, however, be much older. Knuth [17] cites a verbal description of the algorithm in the ﬁrst-century A.D. Chinese text “Chiu Chang Suan Shu”. 2 The Entropy of Schneider’s Continued Fraction Map In this section, we will prove the ﬁrst result of the paper. One can see that it can be complicated to compute entropy from its deﬁnition, so there is the following theorem due to Ya. G. Sinai which is the main tool. The proof of the theorem and more information about entropy can be found in Chapter 4 of [31]. n−1 −i Theorem 2.1 If α is a strong generator, i. e. T α → A as n →∞, and if i =0 H (α) < ∞ then h (T ) = h (T ,α). m m Proof of Theorem 1.1 Let B = k ×N and let j = ( j , j ,...) be a countable sequence 1 2 of elements of B. For a particular element j = (b, a) ∈ B, deﬁne the cylinder-set ( j ) by v(x ) ( j ) = x ∈ M : v(x ) = a and mod π = b . (1) (0) Now let = M and let = ( j ), where j is the ﬁrst element of the sequence 1 1 j. Next deﬁne (2) ={x ∈ M : x ∈ ( j ) and T (x ) ∈ ( j )}. 1 v 2 Proceeding inductively, we get (n) n−1 ={x ∈ M : x ∈ ( j ), T (x ) ∈ ( j ),..., T (x ) ∈ ( j )}. 1 v 2 n j v (n) So, is the set of all x ∈ M with continued fraction expansion starting with (n) j , j ,..., j . This means that depends only on the ﬁrst n terms of j.If J = 1 2 n n 123 A. Haddley, R. Nair ( j , j ,..., j ) ∈ B ,wehave 1 2 n (n) M = for all n ≥ 1 J ∈B such that (n) (n−1) j j j ∈B (n) (n−1) T ( ) = j j and (1) T ( ) = M. Now take j = (b , a ), n = 1, 2,... with j = j if r = s and let α = n n n r s { ( j ), ( j ), ( j ),...} be the partition. Notice that 1 2 3 (n) −1 −2 −(n−1) = ( j ) ∩ T ( ( j )) ∩ T ( ( j )) ∩ ··· ∩ T ( ( j )) 1 2 3 n j v v v (1) (2) (3) = ∩ ∩ ∩ ··· ∩ j j j J ∈B 1 J ∈B (n) n−1 J ∈B n−1 To compute entropy, we ﬁrst need to ﬁnd the conditional information function n−1 −i I (α| T α) which is deﬁned as i =1 v −(n−1) −(n−1) −1 −1 I (α|T α ∨ ··· ∨ T α) =− χ (x ) log μ( ( j )|T α ∨ ··· ∨ T α). v ( j ) v v v ( j )∈α Here, for a partition φ the symbol μ(A|φ) denotes the μ-conditional probability of A (n) with respect to the σ -algebra generated by the partition φ.If x ∈ , then χ (x ) = ( j ) j 1 1 and χ (x ) = 0 for all i ≥ 2. So we get ( j ) −1 −(n−1) −1 −(n−1) I (α|T α ∨ ··· ∨ T α) =− log μ( ( j )|T α ∨ ··· ∨ T α). v v v v The conditional probability is μ( ( j ) ∩ C ) −1 −(n−1) μ( ( j )|T α ∨ ··· ∨ T α) = χ (x ) . 1 C v v μ(C ) −(n−1) −1 C ∈T α∨···∨T α v v 123 On Schneider’s Continued Fraction Map... (n) −(n−1) −1 −2 If x ∈ ,weset C = T ( ( j )) ∩ T ( ( j )) ∩ ··· ∩ T ( ( j )). Then 1 2 3 v n v v we can see that χ (x ) = 1 and for other −1 −(n−1) C = T ( ( j )) ∩ ··· ∩ T ( ( j )), i 2 n v v where i ≥ 2we have χ (x ) = 0. Thus, we obtain −1 −(n−1) μ( ( j )|T α ∨ ··· ∨ T α) v v −(n−1) −1 μ( ( j ) ∩ T ( ( j )) ∩ ··· ∩ T ( ( j ))) 1 2 v n −(n−1) −1 μ(T ( ( j )) ∩ ··· ∩ T ( ( j ))) v 2 v n (n) μ( ) = . (n−1) μ( ) (n) A simple computation shows that μ( ) = , where N = a + a + ··· + a . 1 2 n #(k) Thus, we have 1 1 −1 −(n−1) −a μ( ( j )|T α ∨ ··· ∨ T α) = = #(k) v v N N −a #(k) #(k) and the conditional information function is −1 −(n−1) −a I (α|T α ∨ ··· ∨ T α) =− log(#(k) ) = a log(#(k)). v v By (6), we see that the entropy of T relative to the partition α is n−1 −i h (T ,α) = lim H α| T α , μ v n→∞ i =1 where n−1 n−1 −i −i H α| T α = I α| T α dμ. v v i =1 i =1 So, we get h (T ,α) = lim a log(#(k)) dμ. μ v 1 n→∞ Notice that a (x ) = v(x ) and we have #(k) h (T ,α) = lim v(x ) log(#(k)) dμ = log(#(k)). μ v n→∞ #(k ) 123 A. Haddley, R. Nair We claim that α is a strong generator for T . This is because for almost every x , y ∈ M if x = y, the points x and y have distinct Schneider continued fraction expansions. This implies the partition α separates almost every pair of points. Hence, by Sinai’s Theorem 2.1, the entropy of T with respect to μ is #(k) h (T ) = h (T ,α) = log(#(k)). μ v μ v #(k ) 3 Proof of the Bernoulli Property Let P = (p , p ,...) and Q = (q , q ,...) denote two μ-measurable denumerable 1 2 1 2 partitions of the same set X. Then P and Q are said to be ε-independent and we write P⊥ Q if |μ(p ∩ q ) − μ(p )μ(q )| <ε. i j i j i j A denumerable partition P is called weak Bernoulli with respect to an invertible, measure-preserving transformation T if for each ε> 0 there exists a positive constant K = K (ε) such that for every n ≥ 0we have 0 K +n i ε i T P ⊥ T P. i =−n i =K Note this is not the only way to formulate this property. As observed in [28]for a non-invertible transformation, its natural extension is weakly Bernoulli, if there is a denumerable partition such that for each ε> 0 there exists K = K (ε) and every n ≥ 0we have n K +2n −i ε −i T P ⊥ T P. i =0 i =K +n The isomorphism to a Bernoulli shift is then ensured by the following theorem which was proved by Friedmann and Ornstein, see [27]. Theorem 3.1 A weak Bernoulli (invertible) transformation is isomorphic to a Bernoulli shift with the same entropy. We now complete the proof of our second theorem. Proof of Theorem 1.2 Set (n) −n−l A = T (A) ∩ v j 123 On Schneider’s Continued Fraction Map... then we get −n dμ(T (x )) v −n μ(A ) = dμ(x ) = dμ(T (x )). −l −l dμ(x ) T A T A v v (n) −n 1 Lemma 3.1 dμ(T (x )) = dμ(x ) where N = a + ··· + a (on ). 1 n v N #(k) (n) th A Proof For x ∈ , suppose its n convergent is deﬁned by the recurrence relation (4) . Using (4) and (5) one checks that n N xA + B A (−1) x π n−1 n n −n T (x ) = = + . xB + B B (xB + B )B n−1 n n n−1 n n × N −N As B is in O and multiplication by π scales Haar measure by |π | , this lemma is proved if we show that the map t : M → M deﬁned by t (x ) = preserves xB +B n−1 n Haar measure. Fix L ∈ N and y ∈ k[π ] (the ring of polynomials in π over the residue class ring). One checks readily that t maps the coset π y + π O bijectively to the coset t (π y) + π M. Cosets of this type form a basis for the open sets of M and have the same measure, so their measure is preserved. Hence, our lemma is proved. We, therefore, have 1 1 1 −l μ(A ) = dμ(x ) = μ(T A) = μ(A). N N N −l #(k) #(k) #(k) T A (n) Recall that = μ( ). So we get #(k) (n) (n) −n−l μ(T (A) ∩ ) = μ( )μ(A). j j (n) −i Suppose both ( j ) and A belong to T α, where i =0 v α ={ ( j ), ( j ), ( j ),...} 1 2 3 k+2n −l−n −i is a generator for T . Then = T A ∈ T α and we get v i =k+n v (n) (n) μ( ∩ ) − μ( )μ( ) = 0 j j which implies (n) (n) μ( ∩ ) − μ( )μ( ) = 0 <ε. j j (n) k+2n −i n −i ∈ T α ∈ T α v v i =0 i =k+n Thus, the generator α for T is weak Bernoulli which by the above theorem means that the natural extension of T is isomorphic to a Bernoulli shift with the entropy #(k) log(#(k)). #(k ) 123 A. Haddley, R. Nair 4 Application of the Pointwise Subsequence Ergodic Theorems ∞ x ∞ 1+x n 2 n Recall the elementary identities nx = and n x = for 2 3 n=1 n=1 (1−x ) (1−x ) |x | < 1. Also as is easily veriﬁed #(k ) μ({x : v(x ) = n}) = (n = 1, 2,...). #(k) From this, we get #(k) v(x )dμ(x ) = nμ({x : v(x ) = n}) = #(k ) n=1 and #(k) (#(k) + 1) 2 2 |v(x )| dμ(x ) = n μ({x : v(x ) = n}) = . × 2 #(k ) n=1 We now describe the elements of subsequence ergodic theory, which we use to study convergents. ∞ p A sequence of integers (a ) is called L -good universal if for each dynamical n=1 system (X , B,μ, T ) and f ∈ L (X , B,μ),wehave f (x ) = lim f (T x ) existing μ almost everywhere. N →∞ n=1 Recall that we say a sequence of real numbers (x ) is uniformly distributed modulo n=1 one if for each interval I ⊆[0, 1), denoting its length by |I |,wehave lim #{n ≤ N :{x }∈ I}=|I |. N →∞ N See [18] for further background. The reference [12] contains an extensive list of p 1 sequences of natural numbers, that are L -good universal for all p > 1. Some are L - good universal as well. All the examples mentioned in the reference have the additional property that ({k ψ }) is uniformly distributed for each irrational number ψ.We n n≥1 will call a sequence of natural numbers (k ) that is both L -good universal and such n n≥1 that ({k ψ }) is uniformly distributed modulo one for each irrational ψ p-good. In n n≥1 [12], the following theorem is proved. Theorem 4.1 If (k ) is p-good for any p > 1 and the dynamical system n n≥1 (X,β,μ, T ) is weak-mixing, then f (x ) = f dμμ almost everywhere. the following result. Note that transformations that have natural extensions that are Bernoulli are also weak-mixing [31]. Theorem 4.1 has a number of applications. 123 On Schneider’s Continued Fraction Map... Theorem 4.2 Suppose (k ) is an p-good and suppose F : R → R is a continu- n n≥1 ≥0 ous increasing function with |F (a (x ))| dx < ∞. For each n ∈ N and arbitrary real numbers d ,..., d , we deﬁne 1 n F (d ) + ··· + F (d ) 1 n −1 M (d ,..., d ) = F . F ,n 1 n Then we have −1 lim M (a (x),..., a (x )) = F F (a (x ))dx F ,n k k 1 1 n n→∞ almost everywhere with respect to Haar measure on M. Proof Apply Theorem 4.1 with f (x ) = F (a (x )). Theorem 4.3 For an p-good (k ) and a function H : N → R, suppose that n n≥1 |H (a (x),..., a (x ))| dx < ∞. 1 m Then we have × m 1 #(k ) (m) lim H (a (x ), a (x ), . . . , a (x )) = H (i ) k k +1 k +m n n n i +···+i 1 n N →∞ N #(k) (m) m n=1 i ∈N almost everywhere with respect to Haar measure on M. Proof Apply Theorem 4.1 with f (x ) = H (a (x ), . . . , a (x )). 1 m Theorem 4.4 For any p-good (k ) , we have n n≥1 1 #(k) lim a = , N →∞ N #(k ) n=1 and 1 #(k ) lim b = , N →∞ N 2 n=1 almost everywhere with respect to Haar measure on M. Proof Apply Theorem 4.1 with f (x ) = v(x ) and f (x ) = b(x ). 123 A. Haddley, R. Nair In the case k = n (n = 1, 2,...) and K = Q , the ﬁrst part of this result is from n p [14]. Unlike the natural numbers, however, most examples of p-good sequences are of zero density. We also have the following additional consequences. Theorem 4.5 For any p-good (k ) , we have n n≥1 1 #(k ) lim #{1 ≤ n ≤ N : a = i}= , N →∞ N #(k) 1 1 lim #{1 ≤ n ≤ N : a ≥ i}= , N →∞ N #(k) and 1 1 1 lim #{1 ≤ n ≤ N : i ≤ a < j}= 1 − ; i +1 j −i −1 N →∞ N #(k) #(k) almost everywhere with respect to Haar measure on M. Proof Apply Theorem 4.1 with f (x ) = I (x)(i = 1, 2, 3), where I is the charac- B B teristic function of the set B in the cases B ={x ∈ M : a (x ) = i }, 1 i B ={x ∈ M : a (x ) ≥ i } 2 i and B ={x ∈ M : i ≤ a (x)< j }. 3 i 5 Application of the Moving Average Pointwise Ergodic Theorem We begin by introducing some notation. Let Z be a collection of points in Z × N and let Z ={(n, k) : (n, k) ∈ Zand k ≥ h}, h 2 h Z ={(z, s) ∈ Z :|z − y| <α(s − r )forsome (y, r ) ∈ Z } and h h Z (λ) ={n : (n,λ) ∈ Z }.(λ ∈ N) α α Geometrically we can think of Z as the lattice points contained in the union of all solid cones with aperture α and vertex contained in Z = Z. We say a sequence of pairs of natural numbers (n , k ) is Stoltz if there exists a collection of points Z in Z×N, and l l l=1 123 On Schneider’s Continued Fraction Map... ∞ h(t ) a function h = h(t ) tending to inﬁnity with t such that (n , k ) ∈ Z and there l l l=t exist h , α and A > 0 such that for all integers λ> 0we have |Z (λ)|≤ Aλ. 0 0 α This technical condition is interesting because of the following theorem from [4]. Theorem 5.1 Let (X,β,μ, T ) denote a dynamical system, on set X, with a σ -algebra of its subsets β, a measure μ deﬁned on the measurable space (X,β) such that μ(X ) = 1 and a measurable, measure-preserving map T from X to itself. Suppose 1 ∞ fis in L (X,β,μ) and that the sequence of pairs of natural numbers (n , k ) is l l l=1 Stoltz then if (X,β,μ, T ) is ergodic, the limit n +i m (x ) = lim f (T x ), l→∞ k i =1 exists almost everywhere with respect to Lebesgue measure. Note that if we set n +i m (x ) = f (T x ) l, f i =1 then −1 n +k +1 n +1 l l l m (Tx ) − m (x ) = k ( f (T ) − f (T x )). l, f l, f This means that m (Tx ) = m (x)μ almost everywhere. A standard fact in ergodic f f theory is that if (X,β,μ, T ) is ergodic and m (Tx ) = m (x ) almost everywhere, then f f m (x ) = f dμμ almost everywhere [31]. The term Stoltz is used here because the condition on (k , n ) is analogous to the condition required in the classical l l l=1 non-radial limit theorem for harmonic functions also called a Stoltz condition, which suggested the above theorem to the authors of [4]. Averages where k = 1 for all l will be called non-moving. Moving averages satisfying the above hypothesis can be l l−1 2 2 constructed by taking for instance n = 2 and k = 2 . l l In this section, we state moving average variants of the results in the previous section. The proofs, which are very similar to those in the previous section, are forgone. Theorem 5.2 Suppose that (n , k ) is Stoltz. Suppose also that we have F : R → l l l≥1 ≥0 R which is continuous increasing and such that |F (a (x ))|dx < ∞. Suppose M (d ,..., d ) is deﬁned as in the previous section. Then F ,n 1 n −1 lim M (a (x),..., a (x )) = F F (a (x ))dx F ,l k k +n 1 l l l l→∞ almost everywhere with respect to Haar measure on M. 123 A. Haddley, R. Nair Theorem 5.3 Suppose (n , k ) is Stoltz and H : N → R is such that l l l≥1 |H (a (x ), . . . , a (x ))|dx < ∞. 1 m Then we have × m 1 #(k ) (m) lim H (a , a ,..., a )(x ) = H (i ) k +1+ j k +2+ j k +m+ j l l l i +···+i 1 m l→∞ #(k) j =1 (m) m i ∈N almost everywhere with respect to Haar measure on M. Theorem 5.4 Suppose (k , n ) is Stoltz then we have l l n≥1 1 #(k) lim a = , k + j l→∞ n #(k ) j =1 and 1 #(k ) lim b = , k + j l→∞ n 2 j =1 almost everywhere with respect to Haar measure on M. Theorem 5.5 For Stoltz (n , k ) , we have l l l≥1 1 #(k ) lim #{1 ≤ j ≤ n : a = i}= , l k + j l→∞ n #(k) 1 1 lim #{1 ≤ j ≤ n : a ≥ i}= , l k + j l→∞ n #(k) and 1 1 1 lim #{1 ≤ t ≤ n : i ≤ a < j}= 1 − , l k +t i +1 j −i −1 l→∞ n #(k) #(k) almost everywhere with respect to Haar measure on M. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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Arnold Mathematical Journal – Springer Journals
Published: Oct 15, 2021
Keywords: Schneider’s continued fraction map; Non-Archimedean fields; Bernoulli processes; Entropy; Primary 11K50; Secondary 11A55; 11J70
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