C Correia Ramos

Nuno Martins
email: nmartins@math.ist.utl.pt

A Nascimento
email: alexandra.nascimento@ipleiria.pt

A Nascimento Baptista

C Correia Ramos

C Correia

Ramos Cima

Dynamics on certain sets of stochastic matrices

Keywords: Matrix dynamics, Stochastic matrices, Iterated interval maps, Reversible Markov chains

HAL is

Introduction and preliminaries

Iteration on M n (R), the set of n × n real matrices, can be viewed as a natural generalization of iteration on R n or C n , concerning the algebraic structure of matrix algebras. In [START_REF] Nascimento Baptista | Difference equations on matrix algebras[END_REF] we have analyzed the iteration of the quadratic polynomial z → z 2 + c on M 2 (R), see also [START_REF] Serenevy | Dynamics of polynomial maps of 2 × 2 real matrices[END_REF]. Note that this example itself contains, as a subcase, the complex iteration and the related well-known Mandelbrot set, with all its dynamical, analytical and combinatorial complexity, see [START_REF] Milnor | Dynamics in one complex variable[END_REF] for details.

In the following we define the assumptions needed in order to obtain the iteration on the set of symmetric stochastic matrices. The iteration is made under a quadratic map (a one parameter family), conjugated to the logistic map, for which the one dimensional version exhibits complex behavior, in particular, the coexistence of an infinite number of periodic points and the sensitivity to initial conditions or, in other words, exhibits chaotic behavior, see [START_REF] Guckenheimer | Sensitive dependence to initial conditions for one-dimensional maps[END_REF].

Moreover, the symmetric stochastic matrices determine reversible Markov chains, which are used in many applications and also have great interest from a theoretical point of view.

The stochastic matrices are matrices X = (x ij )

n i,j=1
with non-negative entries, i.e., x ij ≥ 0 for which n j=1 x ij = 1. In this case, the vector u := (1, 1, ..., 1) ∈ R n is a right Perron eigenvector of X with Perron eigenvalue equal to 1 (spectral radius ρ (X) = 1). Now, consider the polynomial map g λ (x) := 1λx (1x), which is a modified logistic map so that x = 1 is a fixed point. This fact is essential in order to the iteration of a stochastic matrix still be a stochastic matrix. We denote by G λ the induced matrix map G λ (X) := 1 -λX (1 -X), where 1 denotes the identity matrix, with λ ∈ [0, 4] .

In the present paper we consider the restriction of the matrix set to the set of n × n positive definite symmetric stochastic matrices (necessarily doubly-stochastic since they are symmetric and stochastic), which we denote by Ω n . Therefore, a matrix X in Ω n has nonnegative entries, the Perron eigenvalue 1 and spectrum contained in ]0; 1].

To deal with Markov chains we would like to have G λ (X) ∈ Ω n whenever X ∈ Ω n . However, this is false, in general. Assuming X ∈ Ω n , then G λ (X) is necessarily positive definite, symmetric and will have u as Perron eigenvector associated with the Perron eigenvalue 1 (see Lemma below). However G λ (X) may have negative entries. The alternative is to consider the minimal invariant subset of Ω n which, under iteration of G λ , maintains the property of non-negativeness of the entries, so that the iterates are stochastic. This is not a new situation, since it is similar to the case of interval maps dynamics, when the invariant set, instead of the whole interval, is a Cantor set, a self-similar fractal subset of the interval, see for example [START_REF] Falconer | Techniques in fractal geometry[END_REF].

The paper is organized as follows: In section 2 we give preliminary general results on Ω n , relating the periodic points of the dynamics on Ω n with those on onedimensional dynamics. We also study two particular cases, for n = 2 and n = 3. Finally, in section 3 we focus on the dynamical behavior on the Markov chain set, considering a reducible case and a primitive case.

Discrete dynamics on the set of positive definite matrices

General results

Consider X ∈ M n (R). Let sp (X) denote the spectrum of X and X T denote the transpose matrix of X. The symmetric positive definite matrices are those which satisfy X = Y 2 , for some matrix Y ∈ M n (R); a symmetric matrix is positive definite if and only if all its eigenvalues are positive.

We say that X, Y ∈ M n (R) are equivalent if and only if they have the same spectrum, counting multiplicities. Since every symmetric matrix is diagonalizable, two matrices X, Y are equivalent if and only if there is an invertible P such that Y = P XP -1 .

Consider the modified logistic map

G λ : Ω n -→ M n (R) X → 1 -λX (1 -X)
, where 1 denotes the identity matrix, with the real parameter λ ∈ [0, 4] . Lemma 1 Let X ∈ Ω n , then G λ (X) is symmetric, positive definite and u := (1, ..., 1) ∈ R n is the Perron eigenvector associated with the Perron eigenvalue 1.

Proof We first show that G λ (X) is symmetric; as X T = X, then (G λ (X)) T = 1 -λ 1 -X T X T = 1 -λX T 1 -X T = 1 -λX (1 -X) = G λ (X) .
Now, since X is diagonalizable, there is an invertible matrix P such that D = P -1 XP is diagonal and

G λ (X) = G λ P DP -1 = P G λ (D) P -1 . Since D = diag (x 1 , ..., x n ) is diagonal and x i ∈ [0, 1], i = 1, ..., n, we have G λ (D) = diag (g λ (x 1 ) , ..., g λ (x n )). Therefore the spectrum of G λ (X) is contained in [0, 1] , and G λ (X) is positive definite. Finally, G λ (X) u = u -λX (1 -X) u = u -λX (u -Xu) = u,
since u is the Perron eigenvector associated with the Perron eigenvalue 1 of X.

Notice that, in order to G λ (X) ∈ Ω n , the only necessary condition that might fail is that some of the entries of G λ (X) might be negative. Now, let

Λ n (λ) := X ∈ Ω n : G k λ (X) ∈ Ω n , for all k ∈ N . This is an invariant set, that is G λ (Λ n (λ)) = Λ n (λ),
and it is the analogue of the invariant Cantor set for the iteration on the interval, see [START_REF] Falconer | Techniques in fractal geometry[END_REF].

In the following we analyze how the matrix dynamics in some aspects can be reduced to the onedimensional dynamics on the spectrum. For every X ∈ Ω n there is an invertible matrix P such that D = P -1 XP is diagonal, in particular the diagonal entries, which correspond to the spectrum of X, belong to the interval [0, 1]. We thus have

G λ (X) = G λ P DP -1 = P G λ (D) P -1 . Since D = diag (x 1 , ..., x n ) is diagonal and x i ∈ [0, 1], i = 1, ..., n, we have G λ (D) = diag (g λ (x 1 ) , ..., g λ (x n )). Moreover, G k λ (X) = G k λ P DP -1 = P G k λ (D) P -1 = = P diag g k λ (x 1 ) , ..., g k λ (x n ) P -1
. This last expression allows us to use the results of iteration on interval maps in order to obtain results on the matrix dynamics.

Proposition 1

The matrix X ∈ Ω n is a periodic point, with respect to G λ , if and only if the sp (X) is a set of periodic points with respect to g λ . Moreover the period of X is the minimum common multiple of the periods of the eigenvalues of X.

Proof Let X ∈ Ω n , which is diagonalizable with a certain P such that P -1 XP = diag (sp (X)). If sp (X) is a set of periodic points of g λ , then there is a positive integer m (minimum common multiple of the periods) for which

diag (g m λ (x 1 ) , ..., g m λ (x n )) = diag (x 1 , ..., x n ) , therefore G m λ (X) = P diag (x 1 , ..., x n ) P -1 = X. Con- versely, if there is a matrix X ∈ Ω n such that G m λ (X) = X, for some positive integer m, then this implies that diag (g m λ (x 1 ) , ..., g m λ (x n )) = diag (x 1 , ..., x n )
and that the spectrum of X consists of periodic points with respect to g λ .

Some results in low dimension

The n = 2 case

In this situation we have

G λ (Ω 2 ) = Ω 2 . A symmetric stochastic matrix X ∈ M 2 (R) is X = x 1 -x 1 -x x , x ∈ [0, 1] .
The eigenvalues of X are 1 and 2x -1. Therefore, to ensure that X is positive definite, we must consider x ∈ ]1/2, 1]. On the other hand, we have

G λ (X) = 1 + λ -3xλ + 2x 2 λ -1 + 3x -2x 2 λ -1 + 3x -2x 2 λ 1 + λ -3xλ + 2x 2 λ . Since λ ∈ [0, 4], for x ∈ ]1/2, 1], the matrix G k λ (X) has non-negative entries for every k ∈ N. Thus the equality G λ (Ω 2 ) = Ω 2 follows, that is, Ω 2 = Λ 2 (λ), for every λ ∈ [0, 4].
For each cycle {a 1 , ..., a k } of g λ , with period k, we have the orbit

a1+1 2 1-a1 2 1-a1 2 a1+1 2 , ..., a k +1 2 1-a k 2 1-a k 2 a k +1 2
, of G λ , with the same period. Note that, after diagonalization, we have

a i 0 0 1 , i = 1, ..., k;
nevertheless, this matrix is not stochastic, since a i = 1 for all i = 1, ..., k, otherwise we would have the trivial case of the fixed point. When n = 2 there is no Y ∈ Ω 2 equivalent to a given X ∈ Ω 2 , that is, there is no nontrivial P ∈ GL n such that P XP -1 ∈ Ω 2 . Moreover, there is a one-to-one correspondence between orbits in Ω 2 , under G λ , and orbits in [0, 1], under g λ . Therefore the matrix dynamics in Ω 2 is essentially equivalent to the one-dimensional dynamics.

The n = 3 case

In this case, it is not true that Ω 3 = Λ 3 (λ), for every λ. For example, consider the matrix X ∈ Ω 3 , The eigenvalues of X are 1, 0.50132 and 0.191458 (roundoff to the 7 th digit). For λ = 3.9 the second iterate

X = ⎛ ⎝ 0.
G 2 λ (X) = G λ (G λ (X)) has two negative entries (al- though G λ (X) does not have it), G 2 λ (X) = ⎛ ⎝ 0.79695 -0.107963 0.311013 -0.107963 0.79695 0.311013 0.311013 0.311013 0.377974 ⎞ ⎠ , and therefore G 2 λ (X) / ∈ Ω 3 .
In this case we have that X, given above, does not belong to Λ 3 (λ), for λ = 3.9.

For n = 3, a symmetric stochastic matrix is given by

X = ⎛ ⎝ x y 1 -x -y y z 1 -y -z 1 -x -y 1 -y -z x + 2y + z -1 ⎞ ⎠ , with x, y, z ∈ [0, 1].
In this case, both the explicit positivity condition on x, y, z and the condition for the nonnegativity of the entries are more complicated than the ones for the n = 2 case. Let us analyze the subfamily that is characterized by z = x. In this case, G λ (X) also belongs to this subfamily and we obtain an invariant subset of Λ 3 (λ). This leads to an interesting theoretical question which is to know what are the minimal invariant subsets of Λ n (λ), given n. The set Λ n (λ) itself is not minimal as we see here for n = 3. As we mentioned, let us now consider the family of matrices

X = ⎛ ⎝ x y 1 -x -y y x 1 -x -y 1 -x -y 1 -x -y 2y + 2x -1 ⎞ ⎠ , (1) with x, y ∈ [0, 1].
We must impose additional conditions on x, y to guarantee that X ∈ Ω 3 . The eigenvalues of X are 1, x-y and 3x+3y -2. Therefore, to ensure that X is positive definite, we must consider x > y and x > 2/3y.

Let Λ (λ) be the subset of Λ 3 (λ) of matrices in the form (1). In the Figure 1 we show the invariant subset Λ (λ), for λ = 3.9, which illustrates how different can be the sets Ω 3 and Λ 3 (λ). In the Figure 2 we show a detail of the Figure 1. The example given in the beginning of this section was taken from this picture, searching for pairs (x, y) for which the color is close to white (in the example given we have chosen x = 0.615903 and y = 0.114583). Note that is much more difficult to give, with the same method, a pair (x, y) for which G k λ (X) has non-negative entries for every positive integer k. This difficulty arises because the invariant set Λ (λ) is a set with nontrivial fractal structure. A more efficient way to give X ∈ Λ (λ), that is, X in the form [START_REF] Berman | Nonnegative matrices in the mathematical sciences[END_REF] for which G k λ (X) has non-negative entries for every positive integer k, is to give the periodic points of G λ . Nevertheless, some careful is needed, since even in this case we must assure the non-negativity of the entries. Let a = xy and b = 3x + 3y -2 be the eigenvalues of X in terms of the entries x, y. Determining x, y, we can explicitly write

X = 1 6 ⎛ ⎝ 3a + b + 2 -3a + b + 2 2 -2b -3a + b + 2 3a + b + 2 2 -2b 2 -2b 2 -2b 2 + 4b ⎞ ⎠ .
The conditions on the eigenvalues a, b, in order to x, y ∈ [0, 1] are 1 6 (3a + b + 2) ≥ 0 and 1 6 (-3a + b + 2) ≥ 0. To ensure that X is positive definite we have also 0 < a, b ≤ 1, therefore there is a domain, in the plane (a, b), of possible eigenvalues of a matrix in Ω 3 , see the Figure 3. Let Δ be this domain, which is the region defined by Let {a 1 , ..., a k } be a cycle of period k, with respect to g λ , that is, g λ (a i ) = a i+1 , i = 1, 2, ..., k -1 and g λ (a k ) = a 1 . To obtain a periodic matrix orbit for G λ it is necessary to choose a pair of points from {a 1 , ..., a k }. In the considered set, this pair (a i , a j ) will determine a periodic orbit for G λ if and only if (a i+r , a j+r ) ∈ Δ for every positive integer r (reminding that g λ (a k ) = a 1 ). Since (a, a) ∈ Δ, for which a ∈ [0, 1], a possible choice is the pair (a i , a i ); in this case we say that the matrix orbit is in phase. Otherwise we call it out of phase. The pair (a i , a j ) gives the matrix

Δ = (a, b) ∈ R 2 : 0 < a ≤ 1, 0 < b, 3a -2 ≤ b ≤ 1 . Note that (a, b) ∈ Δ does not imply that (b, a) ∈ Δ.
1 6 ⎛ ⎝ 3a i + a j + 2 -3a i + a j + 2 2 -2a j -3a i + a j + 2 3a i + a j + 2 2 -2a j 2 -2a j 2 -2a j 2 + 4a j ⎞ ⎠ with i, j = 1, ..., k.
We can implement an algorithm to obtain all the periodic points of G λ . Let λ ∈ [0, 4]. Each cycle, with respect to g λ , can be determined using numerical routines or using symbolic dynamics and combinatorial arguments, see [START_REF] Lampreia | Subtrees of the unimodal maps tree[END_REF] and [START_REF] Ramos | Finite dimensional representations of * -algebras arising from a quadratic map[END_REF]. For the unidimensional orbit {a 1 , ..., a k } of least period k we have a periodic matrix orbit in phase, generated by the pair (a 1 , a 1 ) . To obtain the other matrix orbits, the out of phase orbits, we only need to check if (a 1+r , a i+r ) ∈ Δ, with r = 1, ..., k -1, for each pair (a 1 , a i ), with i = 2, ..., k.

Example 1 Consider

X = 1 6 ⎛ ⎝ 3a i + a j + 2 -3a i + a j + 2 2 -2a j -3a i + a j + 2 3a i + a j + 2 2 -2a j 2 -2a j 2 -2a j 2 + 4a j ⎞ ⎠ .
Let λ = 4. The two cycles of period 3, with respect to g λ , are {a 1 , a 2 , a 3 } = {0.0301537, 0.883022, 0.586824} and {a 4 , a 5 , a 6 } = {0.0495156, 0.811745, 0.38874} (obviously we exclude the fixed points and present approximate values). Remark that not all the pairs (a i , a j ) generated by the elements of the cycles of g λ , give a matrix orbit. In the table below, the symbol × represent the pairs (a i , a j ) that generate periodic orbits for G λ .

The symbol ∅ represent the other cases.

a 1 a 2 a 3 a 4 a 5 a 6 a 1 × ∅ ∅ × ∅ ∅ a 2 ∅ × ∅ ∅ × ∅ a 3 ∅ ∅ × ∅ ∅ × a 4 × × ∅ × ∅ ∅ a 5 ∅ × × ∅ × ∅ a 6 × ∅ × ∅ ∅ ×
This means that there exists 2 distinct matrix orbits of G λ of period three that are in phase,

Discrete dynamics on the Markov chain set

As we discussed in the introduction, each stochastic matrix determines a Markov chain. Therefore, the dynamics on Λ n (λ) generated by G λ corresponds to a certain dynamical system on the set of reversible Markov chains (reversible since we deal with symmetric matrices). Since the set Λ n (λ) is too large, we will study families of matrices belonging to Λ n (λ), for a certain positive integer n, which are invariant under iteration of G λ with potential interest for Markov chains analysis. We stress that many different Markov chains can be obtained, although the same iterative map G λ is considered.

Let us explain the general idea. Let λ ∈ [0, 4]. We start with a particular Markov chain characterized by a stochastic matrix X ∈ Λ n (λ). This Markov chain can be seen as a model of a certain system in a regular regime. The periodic structural changes in the system are characterized by iteration of the map G λ . After a singular regime (corresponding to an iteration under G λ ), in the next regular regime, the system is modeled again by a Markov chain characterized by the stochastic matrix G λ (X). Therefore, we obtain an evolutionary process for Markov chains and we use some useful techniques from iterated maps on the interval, that allow us to analyze this evolution.

A reducible case

A n × n matrix X is reducible if there is a permutation matrix P such that

P XP -1 = A B 0 C , ( 2
)
where A, B, C are non necessarily square matrices, see [START_REF] Falconer | Techniques in fractal geometry[END_REF]. A reducible matrix X has the property that the powers of X, X k , have entries which are equal to 0 for every positive integer k (it must have at least n -1 zero entries). Consider the following reducible matrix

Y = ⎛ ⎜ ⎜ ⎝ y 0 0 z 0 0 0 y 0 0 z 0 z 0 0 y 0 0 0 z 0 0 y 0 ⎞ ⎟ ⎟ ⎠ .
In order to Y be stochastic, we must have 0 ≤ y 0 , z 0 ≤ 1 and y 0 + z 0 = 1. The eigenvalues of Y are y 0z 0 and y 0 + z 0 = 1, both with multiplicity 2. Therefore, if y 0 > z 0 we have that Y is positive definite and in that case Y ∈ Ω 4 . The k th -iterate of Y will have the form

G k λ (Y ) = ⎛ ⎜ ⎜ ⎝ y k 0 z k 0 0 y k 0 z k z k 0 y k 0 0 z k 0 y k ⎞ ⎟ ⎟ ⎠ ,
with

y k+1 = 1 -λy k + λy 2 k + λz 2 k and z k+1 = (2y k -1) λz k . ( 3
)
We conclude that G λ preserves the initial matrix form and the eigenvalues of G λ (Y ) are y kz k and y k + z k = 1. Therefore, we have in fact a one-dimensional system with

y k+1 = 1-λy k +λy 2 k +λ (1 -y k ) 2 = 1+λ-3λy k +2λy 2 k . ( 4
)
Now we are in position to analyze the dynamical behavior of G λ directly from the matrix entries and, in particular, to analyze if any entry will eventually be negative. The map x → h λ (x) := 1 + λ -3λx + 2λx 2 , restricted to x ∈ [1/2, 1] (in order to satisfy the previous conditions, given in paragraph 2.2.1) remains in [1/2, 1] . This means that, for every λ ∈ [0; 4], all the entries of G k λ (Y ) are non-negative, and that, for every k, the set of considered matrices is invariant under G λ , in particular

y k ∈ [1/2, 1].
We can see that we have two Markov chains, with underlying vertices {1, 3} and {2, 4}, each one associated with a 2-full-shift, coupled by the recursion (3). For a given k we obtain a Markov chain for which the probability of maintaining the same state is y k and to change the state is z k = 1y k . In the next iteration (generation), the probability of maintaining the same state is

y k+1 = h λ (y k ) = 1+λ -3λy k + 2λy 2 k .
In Figure 4 we exemplify how this probability evolves, starting from y 0 = 0.7 and considering different values of λ.

It is interesting to note that the map h λ , with λ ∈ [0, 4] , reproduces the different features of the quadratic maps dynamics, as we can see in the bifurcation diagram in Figure 5.

If λ = 2 (or λ < 3), every initial condition in [1/2, 1[ is attracted, under h λ , to the fixed point. If λ = 3.56995... we have the Feigenbaum point, the beginning of "chaos". If λ > 3.56995... we have positive topological entropy and an infinity of repulsive periodic points, see [START_REF] Sharkovsky | Dynamics of one-dimensional maps[END_REF].

A natural generalization is considered in the next result:

Theorem 1 Let Y ∈ Ω 2n+2 , n ≥ 1, be a stochastic matrix of the form Y = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ y 0 z 0 z • • • 0 0 y 0 z 0 • • • z z 0 y 0 z • • • 0 0 z 0 y 0 • • • z z 0 z 0 y 0 • • • • • • • • • 0 z 0 z 0 • • • y ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 5
)
Then

Y k := G k λ (Y ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ y k 0 z k 0 z k • • • 0 0 y k 0 z k 0 • • • z k z k 0 y k 0 z k • • • 0 0 z k 0 y k 0 • • • z k z k 0 z k 0 y k • • • 0 • • • • • • • • • 0 z k 0 z k 0 • • • y k ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , with y k+1 = 1 -λy k + λy 2 k + λnz 2 k , z k+1 = (2y k -1) λz k + λ (n -1) z 2 k (6)
and

y 0 = y, z 0 = z, y k + nz k = 1. ( 7
)
In particular

G k λ (Y ) ∈ Ω 2n+2 for every positive integer k, with λ ∈ [0, 4].
Proof If Y is in the form (5) and stochastic, then y = 1nz and z ∈ [0, 1/n]. The eigenvalues of a matrix in the form (5) are y + nz and yz. As Y is positive definite, we have y > z and G λ (Y k ) belongs to Ω 2n+2 . Moreover, by direct computation of G λ (Y k ) , we obtain the claimed relation for y k+1 and the matrix Y k+1 also in the form [START_REF] Milnor | Dynamics in one complex variable[END_REF]. Therefore, y k+1 + nz k+1 = 1. By induction, the result follows for every positive integer k.

A primitive case

A matrix X is primitive if there is a positive integer k such that every entry of X k is positive. If X is a nonnegative primitive matrix then, by the Perron Frobenius theorem, one of its eigenvalues is positive (a simple root of the characteristic equation of X) and greater (in absolute value) than all the other eigenvalues, see [START_REF] Berman | Nonnegative matrices in the mathematical sciences[END_REF].

Let

Y = ⎛ ⎝ x 0 y 0 y 0 y 0 x 0 y 0 y 0 y 0 x 0 ⎞ ⎠ .
This matrix is primitive iff y 0 = 0. In order to Y be stochastic, we must have 0 ≤ y 0 , x 0 ≤ 1 and 2y 0 + x 0 = 1. The eigenvalues of Y are 1 and 1 -3y 0 with multiplicity 2. In this case, we have to consider y 0 < 1/3 to ensure that Y is positive definite so that Y ∈ Ω 3 . Now, it can be easily seen that G k λ (Y ) has the form

G k λ (Y ) = ⎛ ⎝ x k y k y k y k x k y k y k y k x k ⎞ ⎠ ,
and consequently, we have x k = 1 -2y k . Moreover, a simple calculation shows that y k = λy k-1 (1 -3y k-1 ).

This means that in this case the dynamics of the matrix entries can be given explicitly (and also using the spectrum). Note that the quadratic f λ,3 (y) = λy (1 -3y) maps the interval [0, 1/3] into itself and reproduces the dynamics of the logistic equation on the unit interval. Furthermore, this shows that all the entries of G k λ (Y ) are non-negative for every y 0 ∈ [0, 1/3[ and every k.

As a generalization of this case, we can consider a family of n × n matrices Y for which the entries outside the diagonal are equal to y and in the diagonal are equal to x = 1 -(n -1) y. Let f λ,n (y) = λy (1ny). The map f λ,n (y) sends the interval [0, 1/n] into itself and reproduces the dynamics of the logistic equation on the unit interval. Now, we have:

Theorem 2 Let Y ∈ Ω n be a stochastic matrix of the form Y = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ x y • • • y y x . . . . . . . . . . . . . . . y y • • • y x ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ . ( 8
)
Then

Y k := G k λ (Y ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ x k y k • • • y k y k x k . . . . . . . . . . . . . . . y k y k • • • y k x k ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , with x k+1 = 1 -(n -

Conclusions

In this paper we have analyzed the dynamical behavior of iterated matrices under quadratic maps. Choosing an appropriate one-parameter family of quadratic maps it was possible to preserve the stochasticity of a matrix. Therefore, we have obtained a discrete dynamical system on certain invariant subsets of n×n matrices. These invariant sets depend on the size and the internal structure (relations between certain entries, fixed zero entries, etc.) of the initial matrix. In the case of a general 3 × 3 stochastic matrix the invariant subset reveals a fractal structure resulting from discarding the matrices which have eventually negative entries. Different classes of n × n stochastic matrices, with certain given relations between the entries, such as those in 3.1 and 3.2, have as invariant sets domains in R 2 (more generally R n ). Since the behavior of quadratic maps is well known and studied we are now able to study and analyze in detail the dynamical behavior of reversible Markov chains, arising from the symmetric stochastic matrices, which evolve under the iteration of the quadratic map G λ . This study can be made regarding the choice of the initial conditions (an initial Markov chain) and the dependence on the parameter λ ∈ [0, 4]. For instance, if λ ≤ 2 for every initial condition the correspondent orbit is attracted to the fixed point. If λ ≤ 3.56995... (Feigenbaum point) the occurrence of periodic points is limited to those with period 2 k , for certain positive integer k. If λ > 3.56995... then there are infinite repulsive periodic points with periods conditioned by the Sharkovsky theorem. This is reflected directly in the dynamics of the stochastic matrices and the evolving Markov chains. As an example we have considered in section 3.2 a family of n × n stochastic matrices, characterized by two variables: x (associated with the probability of staying in a certain state) and y (associated with probability of changing the state). This family is preserved by the map G λ and the dynamics is, regarding the matrix entries, determined by a one-dimensional map f λ,n , where n is the matrix size.

Starting with λ = 0 and increasing its value we follow the route to chaos by period doubling (and more generally by bifurcation phenomena). We obtain, therefore, successive oscillating Markov chains, first a fixed point, then period two, period four, and so on. In this setting, and choosing the appropriate parameter we obtained orbits of Markov chains which are aperiodic (as in example 2).

Fig. 1

 1 Fig.1Graph of the invariant set Λ (λ) for λ = 3.9.

Fig. 2

 2 Fig. 2 Detail of the invariant set Λ (λ) for λ = 3.9.

Fig. 3

 3 Fig. 3 Domain Δ in the plane (a, b).

(a 1 , a 1

 11 [START_REF] Lampreia | Subtrees of the unimodal maps tree[END_REF] , a 4 ) ←→ ⎛ ⎝ 0.874497 0.0627517 0.0627517 0.0627517 0.874497 0.0627517 0.0627517 0.0627517 0.874497 ⎞ ⎠ , and 3 distinct matrix orbits of period three that are out of phase:(a 1 , a 4 )

Fig. 4 Fig. 5

 45 Fig. 4 Evolution of the transition probability y k of maintaining the same state, with the initial condition y 0 = 0.7. The probability of transition to a different state is, for each iterate k, z k = 1y k . (a) with λ = 3. (b) with λ = 4.

Example 2

 2 1)y k+1 , y k+1 = λy k (1ny k ) , and y 0 = y, x 0 = x. In particular G k λ (Y ) ∈ Ω n for every positive integer k, with λ ∈ [0, 4]. Proof If Y is in the form (8) and belongs to Ω n , then x = 1 -(n -1) y and y ∈ [0, 1/n]. The eigenvalues of a matrix in the form (8) are x + (n -1) y and xy. As Y is positive definite, we have x > y and G λ (Y k ) belongs to Ω n . Moreover, by direct computation of G λ (Y k ) , we obtain the claimed relation for y k+1 and a matrix Y k+1 also in the form (8). Therefore, x k+1 + (n -1) y k+1 = 1. The result follows by induction, for every positive integer k. Let us consider n = 5 and λ = 4. In the Figure 6 we can see the iterates of the Markov chains for different initial conditions y 0 = 0.1850217...(period 4), y 0 = 0.1889865... (aperiodic).

Fig. 6

 6 Fig. 6 Evolution of the transition probability y k of changing the state, with λ = 4. The probability of transition to the same state is for each iterate k, x k = 1 -(n -1) y k . (a) with y 0 = 0.1850217... (period 4). (b) with y 0 = 0.1889865... (aperiodic).


A NEW ROLE FOR LOW GERMAN? LANGUAGE INSERTION AS BILINGUAL PRACTICE IN THE PROCESS OF LANGUAGE SHIFT

Keywords: Language contact, language shift, Low German, language mixing, language insertion, bilingual advertising. Ausgehend von einer funktionalen, sprecherzentrierten soziolinguistischen Herangehensweise werden im folgenden Artikel bilinguale Einfügungen als Publikationen untersucht, um die sprachlichen Techniken heraus zu arbeiten, mit denen niederdeutsche Elemente kommunikativen Funktionen mit ihnen verbunden sind. Der Artikel kommt zu dem Ergebnis, dass während eines Prozesses von fortschreitendem

This article analyses language insertion as a bilingual communicative practice, applying a functional, speaker-focused approach to the study of sociolinguistics and language contact. The article is based on a study of contact phenomena in a formerly diglossic region in Northern Germany, where the previously spoken language -Low German -is in the process of being replaced by the dominant standard variety, German. It examines regional publications in order to establish the linguistic techniques by which Low German elements are incorporated into the Standard German texts and the communicative purposes that they serve. The paper concludes that in the process of language shift an emblematic repertoire from Low German is created which can be applied into the dominant contact language, German, for specific communicative purposes.

INTRODUCTION AND SOCIOLINGUISTIC BACKGROUND

This article investigates language shift among Low German speakers in Northern Germany. It raises the question of how the individual and the speech community react to the fact that one of their languages is continuously losing importance as means of communication in day-to-day interaction.

Less than 150 years ago, the Northern German area was host to clusters of multilingualism. The Low German varieties 1 served as a spoken and the emerging German Standard language as a written and high variety in wellestablished diglossic speech communities. Other languages also played a role in day-to-day communication and in the formation of national and regional identities and allegiances, both politically and culturally. In the north of the region, Frisian, Danish, Low German and Standard German were used alongside each other. In the east, Low German and Standard German were in contact with Slavic languages. In the west, for example in the region of East Frisia and in parts of Westphalia, Dutch, Low German and Standard German served as means of communication in different domains, as did a Jewish variety based on Western Yiddish [START_REF] Reershemius | Die Sprache der Auricher Juden. Zur Rekonstruktion westjiddischer Sprachreste in Ostfriesland[END_REF]2004). Since then, the previously multilingual landscape of Northern Germany has changed dramatically in an accelerating process of language shift and language change, in which smaller varieties have been lost and Low German is currently threatened by decline. Two representative linguistic surveys conducted in 1984 [START_REF] Stellmacher | Niedersächsischer Dialektzensus. Statistisches zum Sprachgebrauch im Bundesland Niedersachsen[END_REF] and 2007 [START_REF] Möller | Plattdeutsch im 21. Jahrhundert. Bestandsaufnahmen und Perspektiven[END_REF] show that Low German has lost more than half of its active speakers in less than thirty years, mainly due to the fact that speakers no longer raise their children with Low German. If these trends continue it is foreseeable that within a relatively short time Northern Germany will have become monolingual, in the sense that it will be dominated by regional varieties of spoken Standard German (see, e.g., [START_REF] Herrgen | Die Dynamik der modernen Regionalsprachen[END_REF].

At the same time, recent studies [START_REF] Arendt | Niederdeutschdiskurse. Spracheinstellungen im Kontext von Laien, Printmedien und Politik[END_REF][START_REF] Möller | Plattdeutsch im 21. Jahrhundert. Bestandsaufnahmen und Perspektiven[END_REF][START_REF] Reershemius | Bilingualismus oder Sprachverlust? Zur Lage und zur Verwendung des Niederdeutschen in Ostfriesland am Beispiel einer Dorfgemeinschaft[END_REF] show that attitudes towards Low German have become overwhelmingly positive. What used to be seen as a stigma is now considered an important part of regional culture and heritage, and the regional administration supports the use of Low German in education and the workplace [START_REF] Nath | Die Zukunft ist mehrsprachig. Vorteile der mehrsprachigen Erziehung in Familie, Kindergarten und Grundschule[END_REF]2004). Low German has become increasingly popular and visible (Bundesraat för Nedderdütsch 2009; Reershemius in print b), apparently serving new, symbolic functions. At the moment, speakers of Low German find themselves in a complex situation. The surveys suggest that they cherish Low German. However, the old fear that Low German could be an obstacle to education and prevent success in a rapidly modernizing society is deeply rooted. 2 Thus, they may not speak the language with their children any more, but an increasing number of Northern Germans, not necessarily Low German speakers, engage in what [START_REF] Shandler | Adventures in Yiddishland: Postvernacular language and culture[END_REF] calls postvernacular cultural practices: 3 Low German amateur theatre in the region is extremely popular, primary schools organize Low German reading competitions on a regular basis, and local newspapers and magazines present columns in Low German. Folk music with Low German lyrics is very popular. There are a few, but regular, broadcasts in Low German by some of the regional radio stations. Recently, too, the Internet has begun to play a role in the Low German speech communities not only in East Frisia but across Germany, Europe and in diaspora communities in the United States (Zurawski 2007; Reershemius in print a). All these efforts and activities have in common the fact that they fall mainly into a very specific part of an individual's or a community's activities: it is the domain of culture, entertainment and leisure, and has little connection with day-to-day life (see, e.g., [START_REF] Schröder | Niederdeutsch im Kontext der Sprachpolitik[END_REF][START_REF] Peters | Zur Sprachgeschichte des niederdeutschen Raumes[END_REF]; in fact, there is a tendency for speakers to reject the use of Low German when it is not connected with these domains or used as a vernacular among friends, family and neighbourhood [START_REF] Reershemius | Bilingualismus oder Sprachverlust? Zur Lage und zur Verwendung des Niederdeutschen in Ostfriesland am Beispiel einer Dorfgemeinschaft[END_REF].

This article examines one of the linguistic solutions speakers of Low German find to this dilemma as they try both to live in their language and to keep up with a society in which that language is rapidly losing its significance as a day-to-day means of communication. It asks how Low German is used and applied at a time when it is being abandoned as a vernacular, and how these practices might influence the regional varieties of spoken standard German.

The theoretical background of the study is a functionalist approach to sociolinguistics and the study of language contact, focusing on the bilingual or multilingual speaker who engages with language as social activity and uses language to achieve communicative goals [START_REF] Matras | Language contact[END_REF]).

The analysis focuses on regional publications from East Frisia, a peninsula in the most north-westerly part of Germany bordering the Netherlands, which has approximately half a million inhabitants. The region has been chosen for three reasons. Firstly, East Frisia is one of the northern German areas where Low German varieties, although threatened by language shift, still play a considerable role in day-to-day life [START_REF] Stellmacher | Niedersächsischer Dialektzensus. Statistisches zum Sprachgebrauch im Bundesland Niedersachsen[END_REF].

Secondly, in East Frisia more than 600 years of language contact led to the development of a form of Low German distinct from neighbouring varieties, with Frisian and Dutch components, mainly in the lexicon [START_REF] Versloot | Grundzüge ostfriesischer Sprachgeschichte[END_REF]Ebeling 2001a;Ebeling 2001b;[START_REF] Reershemius | Niederdeutsch in Ostfriesland. Zwischen Sprachkontakt, Sprachveränderung und Sprachwechsel[END_REF]). Thirdly, people living in the region see it as a cultural, if not political, unit. Linguistic changes correspond with social and economic transformation in the region, which was formerly almost entirely dependent on agriculture but now relies heavily on service industries and the public sector, with manufacturing industries accounting for only one quarter of the gross domestic product. Since the 1980s tourism has gradually become one of the most important pillars of the region's economy. This has led to specific forms of commodification [START_REF] Heller | Bilingualism and identity in the post-modern world[END_REF]2003) of Low German, which has begun to play a role in the design of a regional image constructed for purposes of tourism marketing and identity building alike. Low German has also become more visible in public space, for example in street signs, advertisements or house names (Reershemius in print b). Thus, while declining dramatically in terms of the number of active speakers and communicative domains, Low German is becoming more visible, more popular, an economic commodity and increasingly connected with issues of regional identity construction.

The present article focuses on the incorporation of Low German in otherwise Standard German East Frisian publications, in order to analyse the new role of the formerly spoken variety. The corpus of data analysed consists of twelve issues of a glossy regional magazine, written by professional journalists and writers, which means that the study analyses forms of bilingual writing. According to the definition given by Adams (2003: 30-31), a bilingual text is a text written in two languages. Adams distinguishes between mixedlanguage texts, in which two languages are merged to form a single product, and those texts "in which some sort of juxtaposition of two separate versions is the norm". 4 Whereas oral bilingual practices occur on a semi-conscious, intuitive level in interaction, the process of writing differs considerably from speaking inasmuch as writing texts, especially when intended for publication, involves a high level of conscious planning. The conscious and strategic insertion of elements of a minoritized variety into a dominant written standard language can thus be categorized as a bilingual linguistic practice comparable with, yet distinct from, practices such as borrowing and oral code-switching.

The incorporation of, for example, Low German into otherwise Standard

German texts is a deliberate and carefully planned act. As a linguistic practice, language insertion -written and oral -is part of the overall bilingual activity that can be summarized as language mixing. In his analysis of mixed languages, Matras (2009: 288-307) underlines that the emergence of such varieties is due to conscious, indeed deliberate, acts by speakers who apply parts of the lexicon of a minoritized variety into the dominant matrix language.

Matras also stresses that mixed languages appear on a continuum, or functional cline. At one extreme there are mixed languages which have developed into fully fledged community languages, for example Michif in Canada [START_REF] Bakker | A language of our own. The genesis of Michif -the mixed Cree-French language of the Canadian Métis[END_REF] or Ma'a in Tanzania [START_REF] Mous | The making of a mixed language. The case of Ma'a / Mbugu[END_REF]. At the other end there are varieties "whose use is limited to a number of narrow and specialised communicative functions" (Matras 2009: 291), for example cryptolects like Jenisch (Yenish) in Austria, Switzerland and south-west Germany, or Angloromani, "a style or register that consists of the occasional insertion of a Romani-derived word into English" (Matras et al. 2007: 142). The practice of bilingual language insertion can be compared with the processes leading to the emergence of mixed varieties, albeit on a different scale, since only a limited set of lexical items, phrases, greeting formulae, sayings and proverbs are embedded into the matrix language, as will be shown in the following section. However, it remains to be asked whether the practice of language insertion can also contribute to the emergence of new regional varieties or ethnolects when an iconic repertoire [START_REF] Coupland | Home truths: globalisation and the iconising of Welsh in a Welsh-American newspaper[END_REF] drawn from a minority or lesser used language is integrated into a dominant matrix language for purposes to be specified.

LOW GERMAN IN A REGIONAL MAGAZINE

In a situation of language contact between an autochthonous minority language (Gardner-Chloros 2009) and a dominant standard variety, hybrid bilingual texts tend to be stigmatized due to normative and puristic attitudes prevailing both towards written forms of the standard and towards the minority language.

Therefore, bilingual texts tend to be highly marked and do not occur across all genres of texts in equal measure. In the following section, insertion of Low German elements into standard German texts will be analysed in twelve issues of a regional monthly glossy magazine Ostfriesland Magazin 'East Frisia Magazine' (OM). OM has been published since 1984, a period that roughly coincides with the development of tourism in mainland East Frisia. A statement by the editor of OM in the jubilee edition for the magazine's twenty-fifth anniversary indicates that the aim of OM is to report on the region for visitors and locals alike (OM 4/2009: 46). Articles in OM are displayed among glossy photographs, which show the region from the most positive perspective. OM provides a vehicle for the regional economy by contributing to tourism marketing and giving local businesses advertising space. It also covers cultural events and developments or recent achievements in the region, for example wind-power technology or a modern shipyard building luxury cruise ships. The magazine is published in Standard German, but Low German is applied consciously to very specific journalistic genres and discursive domains: in twelve issues between July 2008 and June 2009 seventeen reports were dedicated to Low German or Low German literature and theatre. In addition, every issue contains two to three texts written in Low German, and normally includes a Low German cartoon and a Low German column, confirming the findings of the 2007 survey which state that the most popular literary text genres in Low German are newspaper columns and short humorous texts (Möller 2008: 39). Other Low German texts in the twelve issues examined include three letters to the editor (two of them in a jubilee edition), several poems, the winning texts of a Low German literary competition and an interview with a language activist who responds in Low German to questions put to her in Standard German. Thus, on average, every issue contains two to three texts in Low German and one to two articles on topics related to Low German, adding up to approximately five pages of each issue of the approximately 130-page magazine.

INSERTION OF LOW GERMAN IN EDITORIALS AND TOPICAL ARTICLES

A distinction must be made here between Low German for readers of Low German and Low German for readers of Standard German. The Low German column is clearly addressed to readers of Low German. The majority of Low German occurrences in OM, however, are aimed at readers of Standard German with some, little or no knowledge of Low German. This apparent paradox can be explained by looking at the application of Low German elements in Standard German texts in more detail. Low German elements tend to be used only for certain text genres, mainly editorials and those articles which deal with a distinct complex of subjects, especially local folklore, regional history of the last hundred years and entertainment. Eight out of twelve editorials contain Low German elements, normally a single word or phrase. These words and phrases belong to a set of frequently used Low German items, for example Krintstut 'raisin bread', Kluntje 'rock candy' or schöfeln 'ice-skating', which keep recurring. Since OM addresses not only locals but also visitors, the lexical items are normally placed in quotation marks to indicate their distinctiveness and interrupt the flow of reading, as shown in example 1, where the Low German element is underlined:

1

Die lieben Kleinen buddeln im Sand, backen Kuchen, bauen

The dear little ones dig about in the sand, make sand cakes, build

Burgen und manschen mit "Puttjeklei".

sand castles and slosh about with mud.

Example 1 is an excerpt from an editorial in the August issue (OM 8/2008: 3), which deals with the pleasures of life on the beach. The Low German element Puttjeklei 'mud' would not be an obstacle for monolingual readers of Standard German since its meaning can easily be derived from the context. For bilingual readers it adds a further dimension to the sentence about children playing on the beach because it semantically belongs to Low German children's talk (Doornkaat-Koolmann 1882: 780). The Low German word is presented as a quote. The quotation marks indicate that the element is interrupting the flow of reading but also convey the additional connotation of 'they say' or rather 'they used to say', in the old days when everybody spoke Low German as a matter of course. 5 In three out of twelve issues, the editorial contains a Low German proverb or aphorism, also a form of quoting the accumulated wisdom of a distinct culture, with a tendency to appear at the end of the text, summing up the content without adding new information. Thus, the language of the editorials manages to draw on the regional language in a way that avoids incomprehensibility for monolingual readers but claims a certain degree of distinctiveness by hinting at the regional language.

In OM, Low German elements can be found in topical articles which deal with a specific cluster of issues such as local history and folklore, local cuisine, art and entertainment. In their study of two Irish-language newspapers, Kelly-Homes and Atkinson (2007: 39) make a similar observation: the use of the Irish language is mainly restricted to what the authors call traditional domains: "Traditional domains were considered to be music, arts events, literature, language industry and education. [START_REF] Piller | Identity constructions in multilingual advertising[END_REF] points out that the vast majority of instances where a language other than the dominant one is applied are in the names of products.

Based on a study of public signage and advertising in the city of Amsterdam, [START_REF] Edelman | What's in a name? Classification of proper names by language[END_REF] confirms these findings and underlines the fact that the readers of names on advertisements do not necessarily have to understand their meaning. The names' main purpose is to evoke certain connotations, which are associated with a specific language. The Low German elements in advertisements therefore do not need to convey factual information, a task that is normally fulfilled by those parts of the ads written in Standard German. This also applies to advertisements in OM where Low German elements are involved. While the factual information about the products is given in Standard German, the products are named in Low German, often with reference to bygone times. This is no coincidence but part of a pattern: whereas English elements in German advertising as a whole indicate a future orientation (Piller 2001: 165), the use of Low German is oriented on the past and evokes the return to a traditional, almost pre-modern society. Studies of multilingual advertising confirm that the main role of inserted language elements is to evoke specific connotations and stereotypes which are normally connected with a specific language and culture (Kelly-Holmes 2005, Luna and Peracchio 2005a;2005b).

Slogans in Low German do not play a major role in the corpus analysed here, although they are the main focus in experimental studies into bilingual consumer behaviour (Luna and Peracchio 2005a;2005b). In a few cases, however, Low German slogans are applied in the corpus, for example in the otherwise Standard German advertisement by an estate agent Dat is mien Tohus anne See! 'This is my home at the seaside' (OM 7/2008: 117). The slogan is superimposed onto one of the photographs, very appropriately, since its effect is not only based on the content of the sentence but is also pictorial, visualizing the distinctiveness of the region by depicting its regional language in addition to the landscape, very much in an ornamental way.

A number of advertisements in the magazine use the Low German greeting formula Moin! (e.g., OM 9/2008: 81) or the adapted version Moin, moin! (e.g., OM 10/2008: 26). The Low German greeting moin!, short for moien dach 'beautiful day', is evidently distinct from the otherwise common formulae used for greetings in Northern Germany Guten Tag! 'good day' or, more informally, Hallo! 'hello'. 6 To summarize, bilingual advertising in the magazine using Low German applies three linguistic techniques in order to implement Low German elements into Standard German texts: quoting, naming and greeting. These techniques are basic bilingual techniques which serve multiple purposes.

Firstly, they help to present the region as bilingual, thus adding another distinct feature to the image created. In their studies of British television holiday programmes, Jaworski, Ylänne-McEwen, Thurlow and Lawson (2003a;2003b) emphasize the role of linguascapes in presentations of tourist destinations:

concentrating on the most basic phatic and instrumental formulae in the language other than English, such presentations provide viewers with a flavour of the local language, thus accentuating another distinctive aspect of the region.

Clearly, such presentations of distinctive linguascapes will need to avoid any impression of potential communicative problems between hosts and tourists, since that might discourage visitors from choosing to travel to this particular area. But it is not only tourists who might find this image of a multifaceted, bilingual culture attractive; it also appeals to the sense of pride that local readers -monolingual or bilingual -may feel. The insertion of Low German elements into the Standard German texts of OM reinforces constructions of regional identity.

Secondly, bilingual readers will appreciate Low German references, since they add a stylistic dimension to the otherwise Standard German texts, as in example 1, and may trigger what Matras, Gardener, Jones and Schulman (2007) in their analysis of emblematic language use call the emotive mode.

Because the use of minority in-group languages is normally restricted to the more private communicative domains such as family, friends or neighbourhood, the application of elements from the minority language in the dominant contact languages triggers the special, intimate knowledge connected with these domains and forms a special bond between speaker and hearer, in this case between writer and reader, who then communicate in an emotive mode (Matras et al. 2007: 143).

CONCLUSION: LANGUAGE INSERTION AND THE ENREGISTERMENT OF AN EMBLEMATIC REPOSITORY

The analysis of regional publications from East Frisia has shown that the now minoritized former spoken variety, Low German, is frequently inserted consciously and deliberately into certain journalistic and literary genres, mainly editorials, articles dealing with traditional domains, and advertisements. These genres have in common that they either address their readers directly (editorials, advertisements) or have a strong narrative component (articles on traditional domains). It is important to note that the phenomenon is less likely to occur in genres related to more factual content, for example articles dealing with contemporary affairs, the economy and other aspects of modern life.

Three linguistic techniques were identified by which Low German elements are inserted into the otherwise Standard German texts: quoting, naming and greeting. These techniques are basic forms of bilingual practices which allow speakers, semi-speakers and even non-speakers of Low German to consider themselves part of the regional community (or regional speech-community).

Thus, the functions of language insertion as shown above are threefold. Firstly, it enables the region to be presented to the outside world and to prospective visitors with the additional feature of a bilingual culture, by creating a linguascape that reinforces the attractions of the landscape. Secondly, the image created by insertion of Low German, providing distinctiveness without risking Otherness, is adapted by locals in order to construct a form of regional identity. Thirdly, language insertion as shown in the examples discussed above allows members of a formerly bilingual speech community to preserve elements of the variety they are in the process of losing.

As shown in the textual analysis above, language insertion draws on a limited set of single nouns, some verbs, phrases, sayings, greeting formulae, proverbs and -in more narrative contexts -short passages of quoted speech or consciously transferred grammatical structures. In his study of the use of Welsh in a Welsh-American community newspaper, [START_REF] Coupland | Home truths: globalisation and the iconising of Welsh in a Welsh-American newspaper[END_REF] makes similar observations, referring to an "iconic repertoire". Matras et al. (2007: 149) analyse the emergence of Angloromani and state that a limited set of "core vocabulary, a few productive rules of vocabulary formation and a few fossilized expressions" are taken from Romani and incorporated into English as the matrix language. This basic vocabulary forms a pool of synonyms which is applied to the matrix language when speakers attempt to evoke the emotive mode -a special bond between speaker and hearer which refers back to a previous stage in the process of language shift, when the minoritized language was reduced to serving as a vernacular in more private domains of communication like family, friends and neighbourhood.

These observations also apply to the analysis of Low German-German contact phenomena. Recent surveys show that Low German is in decline even as the spoken variety of the private domains. However, a certain limited repertoire of Low German elements is being preserved and inserted deliberately into Low German texts. The question arises, then, whether language insertion of a Low German repertoire could have the potential to establish a new regional variety of German in the formerly bilingual speech communities.

The present article focuses on written language. A larger-scale study looking at spoken varieties of Standard German in Northern Germany would be desirable in order to establish whether language insertion as part of language mixing could have the potential to trigger new regional varieties of spoken German or provide Northern German speakers of German with a specific register [START_REF] Matras | Angloromani: A different kind of language?[END_REF]) to be applied for the specific communicative purpose of evoking the emotive mode.

Secondly, the discussion of the textual analysis in this article has highlighted the fact that the phenomenon dealt with is by no means restricted to Low German-Standard German language contact: Angloromani, Irish, Welsh and Yiddish have been indicated as examples of languages involved in similar or identical processes. A comparison of case studies covering a range of minority languages in the process of language shift towards a dominant contact language would help to answer the question of whether language insertion as part of language mixing needs to be established as a distinctive component of the process of language shift and change.

NOTES 1 An ongoing argument among linguists, language activists and speakers of Low German centres on the question of whether Low German is a language or a dialect. Considering the well-established problems within the linguistic discipline in defining the terms language and dialect satisfactorily, the argument seems somewhat superfluous. Nevertheless, because the debate is ideologically highly loaded in the Low German speaking communities, it keeps resurfacing. Schröder (2004: 35) distinguishes between the perception of most speakers, that Low German is a language, and the prevalent view among linguists that the Low German varieties are part of the diasystem of the German language in the widest sense. However, when the European Charter for Regional or Minority Languages was signed by the German government in 1999, Low German was recognized as a regional language worth preserving, supporting and promoting, explicitly not as a dialect (Goltz, Lesle and Möller 2008).

2 The discourse of Low German as an obstacle in the education especially of disadvantaged social groups goes back well into the nineteenth century, and was expressed for example by Ludolf [START_REF] Wienbarg | Soll die plattdeutsche Sprache gepflegt oder ausgerottet werden? Gegen Ersteres und für Letzteres[END_REF] in his pamphlet "Soll die plattdeutsche Sprache gepflegt oder ausgerottet werden? Gegen Ersteres und für Letzteres".

3 Shandler claims that a variety which is no longer used as a vernacular may nevertheless have certain important purposes: it can gain in symbolic value what it has lost in communicative functions. Individuals may not be able to speak or fully understand a language, but they still consider themselves to be members of a certain speech community and engage in postvernacular cultural practices like performing in amateur theatre, engaging in discourses about the language, using or making translations, learning the basics of the language in evening classes or surrounding themselves with objects related to the language.

All it takes to become part of a postvernacular language community is the decision of an individual to participate.

4 [START_REF] Reh | Multilingual writing: a reader-oriented typology -with examples from Lira municipality (Uganda)[END_REF] points out that sociolinguistic research has focused on oral bilingualism and widely ignored its written forms. This claim is confirmed by a look at recent handbooks, textbooks and general introductions to bilingualism, multilingualism and language contact: apart from the analysis of borrowing, bilingual writing hardly features at all or is subsumed under bilingual practices in general. 5 The presentation of elements of the minoritized language in quotation marks was also observed by Coupland (2003: 171). In his analysis of Welsh elements in the Welsh-American community newspaper Y Drych, Coupland stresses that this practice indicates that the Welsh language element is not simply used but deployed.

Figure 1 :

 1 Figure 1: Map of East Frisia

  Low German, as in example 2, from a report on contemporary local art, where one of the pictures introduced has a Low German proverb as a title (OM The majority, 53%, of Low German elements applied in the Standard German texts in the twelve issues of the magazine between July 2008 and June 2009, however, are namings: amateur theatre groups, musicians, museums,

		Journal of Sociolinguistics
	Standard German. The Low German quote does, however, add a certain
	element of authenticity to the story; it functions as a contextualization cue
	4/2009: 194): (Gumperz 1982). The availability of a second language -Low German -to the
	author of the story, and the fact that a considerable number of the intended hotels, restaurants, cafes, events, societies and associations in the region have
	2 readers of OM will be able to comprehend it in a certain context, mean Low Well neet will dieken, de mutt wieken. given themselves Low German names, for example Dörpmuseum 'village
	'He who does not help to maintain the dykes will have to leave' German is used here as a stylistic device (see, e.g., Auer 2007). This technique museum', Theatergruppe "Antjemöh" 'theatre society "Old Antje"',
	is applied frequently in the narrative sections of articles in the twelve issues "Strandlooper" '"beach runner"' (a monthly calendar of events) and so on,
	This saying echoes one of the best-known stereotypes of the region, familiar to analysed. with a preponderance of compound nouns consisting of one Low German and
	all locals and most tourists and usually referred to as a way of explaining the Another technique observed is fictitious quoted speech in Low German, one Standard German element. Most of the cases of namings add quotation
	regional mentality. for example in an article about a gardener who succeeded in planting a marks to their Low German name.
	beautiful garden against the odds in a challenging environment, observed
	critically by her neighbours (OM 6/2009: 118):
	4	Die gab's zwar in Parkanlagen, aber in einem Privatgarten?
		'Those could be found in parks, but in a private garden?
		Unvorstellbar! SO'N TÜDELKRAM!
		Impossible! Such a nonsense!'
	By adding the Low German element to the fictitious neighbours' voices, the
	author contrasts the enterprising gardener who tries to do new things with her
	more conventional, old-fashioned (and Low German speaking) neighbours.
	Again, Low German is used as a stylistic device in this example. In OM, for
	editorials and articles dealing with traditional domains the quoting of single

" Articles in OM which belong to such traditional domains also draw on the linguistic technique of quoting elements of Low German, for example by applying proverbs and sayings in Far more often than sayings, however, the topical clusters of articles use quoted speech in Low German, for example in a report about a faith-healing eighty years ago

(OM 7/2009: 54-57)

. In this story, a girl with severe walking problems is taken to a healer after various failed attempts by the medical profession to find out what is wrong with her. The healer examines the child, smiles and then delivers her diagnosis in Low German: 3 "Dat sitt nich in de Kneei und ok nich in de Foot! Dat sitt hooger, '"It [the problem] doesn't sit in the knees and not in the foot. It sits higher up, dat Kind hett de Hüft ut't Lidd!" the child has a dislocated hip!"' Thus, the surprising twist of the story is delivered in Low German. The article is written in Standard German and would be perfectly comprehensible without the element of quoted speech since the main facts of the story are explained in words, phrases, proverbs, sayings and elements of reported speech is one of the bilingual techniques by which Low German is implemented into the Standard German text. Page 13 of 25 For review only. Confidential. Should not be cited. INSERTION OF LOW GERMAN IN ADVERTISING The average issue of OM contains 169 advertisements, of which 7% entail one or more Low German elements. The above-mentioned linguistic technique of naming in Low German also plays an important role in advertising in the magazine. The object of advertising is given a Low German name, such as Café Störmhuus 'Café Storm House' (OM 9/2008: 81) or Marianne's Schapp 'Marianne's Wardrobe' (OM 7/2008: 2). In her analysis of multilingual advertising,

For review only. Confidential. Should not be cited.Journal of Sociolinguistics

See also[START_REF] Coupland | Home truths: globalisation and the iconising of Welsh in a Welsh-American newspaper[END_REF], who observes that one of the ways to incorporate Welsh elements into the otherwise American English community paper Y Drych is the use of salutations.

Page 25 of 25

For review only. Confidential. Should not be cited.

Journal of Sociolinguistics