Dynamical Systems on Vector Bundles

Sergiu Corbu and Mihai Postolache

Abstract

This paper contains our current research on techniques for the construction of fractal objects using random transformations. Special emphasis is put on the study of invariant probability measures on vector bundle.

Mathematics Subject Classification: 60A10; 65C20.

Key words: random dynamical system, invariance.

1 Introduction

We consider W and X be two measurable spaces, W and X be the Borelian spaces of W and X respectively. In this paper we shall refer to W as the state space and to X as the parameters space. If f is an (W ⊗ X , W)-measurable application and q is a probability measure on X then the quadruple (W, X, f, q) is called random dynamical system ([?], [?]). To a random dynamical system one associates a Markov chain ξ, with an arbitrary set of states that is the space W. The asymptotic behaviour of random dynamical system is closely related to the asymptotic behaviour of the Markov chain ξ, hence to the study of ξ-invariant measures. In this note, we shall define another Markov chain on R^{k} which will be a repartee of ξ.

2 Random dynamical systems on vector bundles

Let W be a measurable space and E = W × R^{k} be the cartesian product between W and the k-dimensional euclidean space.

Definition 1 Suppose f : W → W is a measurable mapping on W. The pair F = (f, A_{f}) is called vector bundle mapping (with respect to f) if

F : E → E, F (w, s) = (f(w), A_{f}(w)s); w ∈ W, s ∈ R^{k},

where A_{f}(w) is an k × k real matrix whose elements are measurable functions of W.

Balkan Journal of Geometry and Its Applications, Vol.3, No.2, 1998, pp. 47-56 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

We shall denote by T the space of vector bundle mappings on E. We remark that T is endowed with a measurable structure given by the measurable product structure on E,

(F, u) → F u, F ∈ T , u ∈ E.

In the following, we consider the discrete set F = {f_{1}, f_{2}, …, f_{r}} of mappings W → W and the set A = {A_{1}, A_{2}, …, A_{r}} of corresponding k × k matrices. Let q be a probability measure on F

q(B) = ∑_{i∈B} ρ_{i}, ρ_{i} ≥ 0, ∑_{i=1}^{r} ρ_{i} = 1, B ∈ X ,

where X = {1, 2, …, r}, and X = P(X) is the set of all parts of X.

Definition 2 Let ν be a probability measure on T . A T -random variable F whose distribution is ν is called random bundle mapping.

By definition, any random bundle mapping F = (f, A_{f}) is generated by a random transformation f : W × X → W. Let us introduce the well-known notations

f^{(n)} = f_{x_{n}} ∘ ⋯ ∘ f_{x_{1}}, f_{x_{i}} ∈ F, x_{i} ∈ X, i ∈ {1, 2, …, n},
F^{(n)} = F_{x_{n}} ∘ ⋯ ∘ F_{x_{1}}, F_{x_{i}} ∈ T , x_{i} ∈ X, i ∈ {1, 2, …, n},
A^{(n)} = A_{x_{n}} ⋯ A_{x_{1}}.

If F_{x}(w, s) = (f_{x}(w), A_{x}s), where w ∈ W, s ∈ R^{k}, then we obtain F^{(2)}(w, s) = (f_{x_{2}} ∘ f_{x_{1}}w, A_{x_{2}}A_{x_{1}}s) = (f^{(2)}(w), A^{(2)}s) and, for all n ≥ 1, we have

F^{(n)}(w, s) = (f_{x_{n}} ∘ ⋯ ∘ f_{x_{1}}w, A_{x_{n}} ⋯ A_{x_{1}}s)
= (f^{(n)}(w), A^{(n)}s).

We now consider Ω = T × T × ⋯, K be the σ-algebra of Ω and p = ν^{N} with ν a probability measure on T . Then (Ω, K, p) is a probability space. An element of Ω is a sequence ω = (F_{1}, F_{2}, …), F_{i} ∈ T while the probability measure p is given by

p{ω : F_{l}(ω) ∈ ψ_{0}, …, F_{l+n}(ω) ∈ ψ_{n}} = ∏_{i=0}^{n} ν(ψ_{l+i}).

If σ : Ω → Ω is a shift on Ω and

F_{n}(σω) = F_{n+1}(ω), f_{n}(σω) = f_{n+1}(ω)

are shift mappings, then we can define the skew products T and τ as follows:

T : E × Ω → E × Ω, T (u, ω) = (F_{1}(ω)u, σω),
τ : W × Ω → W × Ω, τ(w, ω) = (f_{1}(ω)w, σω).

If π : W × R^{k} → W is the natural projection from E on W then from F = (f, A_{f}) we have f = πF π^{−1}. Therefore

q(B) = ν{F ∈ T : πF π^{−1} ∈ B}

As in [?] we consider the Markov chain ξ given by

w_{n} = f^{(n)}(w_{0}) = πF^{(n)}π^{−1}w_{0}

and the transition probability

P (w, A) = ∑_{x∈X} ρ_{x}I_{A}(f_{x}(w)). (1)

If A_{f_{1}(ω)}(w) = A(w, ω) then A ∘ τ(w, ω) = A(τ(w, ω)) and

A^{(n)}(w, ω) = A ∘ τ^{n−1}(w, ω) ⋯ A ∘ τ(w, ω) · A(w, ω).

Remark 1 . On the space T of random vector bundles we built the Markov chain ξ on W with transition probabilities (??). Also, we can build a Markov chain on R^{k} given by the product of random matrices

s_{n} = A_{n}(w, ω) ⋯ A_{1}(w, ω)s_{0},

where s_{0} ∈ R^{k} is fixed. This may be written as

s_{n} = A_{(n)}s_{n−1}, n ≥ 1,

where A_{(n)} = A_{f_{n}(w)} ∈ A and transition probabilities

Q(s, G) = ∫∫ I_{G}(A_{x}(w)s)dρ(x)dη(w),

where ρ is a probability measure on X and η is ξ-invariant probability measure. In the discrete case, we have

Q(s, G) = ∑_{x∈X} ρ_{x} ∫ I_{G}(A_{x}(w)s)dη(w),

with G ⊂ R^{k} and s ∈ R^{k}. We shall denote by ξ^{∗} this Markov chain on R^{k}. This one will be a repartee, on R^{k}, of ξ

Definition 3 . Let μ and α be probability measures on W and R_{1}^{k} = {s ∈ R^{k}/∥s∥ = 1}. We say that μ × α is F_{x}-invariant if

E_{x}(μ × α)(F_{x}^{−1}(E_{1} × E_{2})) = (μ × α)(E_{1} × E_{2}),

where we denoted by E_{x} the expectation over the probability measure ρ on X, F_{x}(w, s) = (f_{x}(w), A_{x}(w)s), E_{1} ∈ W, E_{2} ∈ K, K being the Borelian space of R_{1}^{k}.

In the following, we shall generalize Lemma 3.1 of Morita given in [?]. In this Lemma, Morita states that a measure μ on W is ξ-invariant if and only if μ × P is T -invariant, where P is a product measure on X^{N} and T.

Theorem 1 . Let μ, ρ be probability measures on W and X respectively. Then μ is ξ-invariant if and only if μ × α is F_{x}-invariant, where α is a probability measure on R_{1}^{k}, (α(R_{1}^{k}) = 1).

Proof. Let μ be a ξ-invariant probability measure, E_{1} ∈ W, E_{2} ∈ K. Then we have

E_{x}(μ × α)(F_{x}^{−1}(E_{1} × E_{2})) = ∫_{X} [∫∫ I_{E_{1}×E_{2}}(F_{x}(w, s))dμ(w)dα(s)] dρ(x)
= ∫_{X} [∫∫ I_{E_{1}}(wx)I_{E_{2}}(A_{x}(w)s)dμ(w)dα(s)] dρ(x)
= ∫_{X} [∫∫ I_{E_{1}}(wx)I_{A_{x}^{−1}E_{2}}(s)dμ(w)dα(s)] dρ(x)
= ∫_{X} [∫∫ I_{E_{1}}(wx)α (A_{x}^{−1}E_{2}) dμ(w)dα(s)] dρ(x)
= α(E_{2}) ∫∫ [∫_{X} I_{E_{1}}(wx)dρ(x)] dμ(w)dα(s)
= α(E_{2}) ∫∫ P (w, E_{1})dμ(w)dα(s)
= α(E_{2}) ∫_{W} P (w, E_{1})dμ(w) ∫_{R_{1}^{k}} dα(s)
= α(E_{2})μ(E_{1})α (R_{1}^{k})
= (μ × α)(E_{1} × E_{2})

that is μ × α is F_{x}-invariant.

To prove the converse, we suppose μ × α is F_{x}-invariant. Then

μ(E_{1}) = (μ × α)(E_{1} × R_{1}^{k}) = (μ × α)E_{x}F_{x}^{−1}(E_{1} × R_{1}^{k})
= ∫∫ [∫_{X} I_{E_{1}}(wx)I_{R_{1}^{k}}(A_{x}(w)s)dμ(w)dα(s)] dρ(x)
= ∫∫ [∫_{X} I_{E_{1}}(wx)dρ(x)] dμ(w)dα(s) =∫∫ P (w, E_{1})dμ(w)dα(s)
= ∫_{R_{1}^{k}} dα(s) ∫_{W} P (w, E_{1})dμ(w) =∫ P (w, E_{1})dμ(w)

hence μ is ξ-invariant.

An interesting particular case is that when the matrices from A are triangular

A_{f}(w) = ( \begin{matrix}a_{f}^{(1)}(w) & ∗ & ∗ & … & ∗ \\ 0 & a_{f}^{(2)}(w) & ∗ & … & ∗ \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & a_{f}^{(k)}(w) \end{matrix} )

with zeros under diagonal and a_{f}^{(1)} … a_{f}^{(k)} ≠ 0. In this case the diagonal elements are eigenvalues and the dominant eigenvalue is the spectral radius. For k = 2 we have

A_{f}(w) = ( \begin{matrix}a_{f}(w) & b_{f}(w) \\ 0 & c_{f}(w) \end{matrix} )

We consider μ as ξ-invariant measure and we put

a = ∫∫ log |a_{f}(w)|dμ(w)dν(f) = ∑_{x∈X} ∫ ρ_{x} log |a_{f_{x}}(w)|dμ(w)

b = ∫∫ log |b_{f}(w)|dμ(w)dν(f) = ∑_{x∈X} ∫ ρ_{x} log |a_{f_{x}}(w)|dμ(w).

Then they are characteristic exponents.

The product matrix is

A_{f}^{(n)} = ( \begin{matrix}a_{1} … a_{n} & ∑_{i=1}^{n} a_{n} … a_{i+1}b_{i}c_{i−1} … c_{1} \\ 0 & c_{1} … c_{n} \end{matrix} )

where a_{i} = a_{f_{x_{1}}} … a_{f_{x_{i}}}, b_{i} = b_{f_{x_{1}}} … b_{f_{x_{i}}}, c_{i} = c_{f_{x_{1}}} … c_{f_{x_{i}}}.

If a < b then for μ-almost any initial point w_{0} and any non-null vector from the Ox-axis the increasing rate of the eigenvector of product matrix is of the order e^{na} and any non-null vector from the real plane is linearly independent and has the increasing rate e^{nb}, [?].

From the product matrix A_{f}^{(n)} we deduce that the second vector has the form

v = (∑_{i=1}^{n} a_{n} … a_{i+1}b_{i}c_{i−1} … c_{1}, a_{1} … a_{n} − c_{1} … c_{n})

Kaijser [?], defines the sequence of random variables on the unit sphere R^{k} as follows

s_{n}(s) = \frac{A^{(n)} · s}{∥A^{(n)}s∥}, n ≥ 1

and proves that this sequence is a Markov chain whose distribution at the moment n, μ_{n}, converges to an invariant measure of process. Moreover, there exist

lim_{n→∞} \frac{1}{n} log ∥A^{(n)}s∥ and lim_{n→∞} \frac{1}{n} log ∥A^{(n)}∥ a.s.

3 Illustrations

The snowflake in Fig. 1 is obtained as invariant of just three Möbius transformations. We remark that any Möbius transformation may be written as

f(z) = \frac{αz + β}{z + γ} , α, β, γ ∈ C.

It generates a random number q. If q ∈ (0, 0.8) then α = −3 + 5i, β = 0.8 − i, and γ = −4 + 5i. If not, for q ∈ [0.8, 0.86) choose α = 2 + 0.4i, β = −0.7 − i, γ = 1 − 2i else α = −4 + 4i, β = 0.7 − i, γ = 5 + 2i. Compute and plot a random orbit of z_{0} under f. In Fig. 1 the result is displayed after 150, 000 iterations (for details, see [?]).

Now we shall transform the points of the snowflake using some affine transformations.

To state our computer algorithm, in the following we deal with the set of affine mappings

F = {f/f(w) = T w + b, T is k × k matrix, b ∈ R^{k}}.

By all means, the pair {R^{k}, f(w)} represents a dynamical system.

Let us consider the set T = {T_{0}, T_{1}, …, T_{r}} of (r + 1) non-singular k × k matrices which correspond to some affine transformations of F. If we use the method studied in [?] we obtain the representation of this sequence

w_{0} given,
w_{k+1} = T^{(k)}w_{0} + T^{(k)} [(T_{k} ⋯ T_{0})^{−1}b_{k} + ⋯ + (T_{1}T_{0})^{−1}b_{1} + T_{0}^{−1}b_{0}] ,

where T^{(k)} = T_{k}T_{k−1} ⋯ T_{0} and b_{i} ∈ R^{k}, i = 0, … n.

For practical reasons, let us now consider W = R^{2}, X = T , W the Borelians of

W, and X the set of all parts of X. If T_{n} = ( \begin{matrix}a_{n} & b_{n} \\ c_{n} & d_{n} \end{matrix} ) ∈ T , n ≥ 1 then we build

the sequence (w_{n})_{n≥0} as follows:

w_{0} given, w_{1} = f_{1}(z_{0}), w_{2} = (f_{2} ∘ f_{1})(w_{0}), ….

Let us suppose, without loss of generality, that the matrices T_{k} are positive (the general case is similar). For practical reasons we have to study the asymptotic behaviour of the following product of random matrices

T^{(k)} = T_{k} ⋯ T_{1}T_{0}.

For this purpose, we consider a positive matrix T = ( \begin{matrix}a & b \\ c & d \end{matrix} ) and its decomposition

as a product of a diagonal matrix and a stochastic one as follows:

D = ( \begin{matrix}a + b & 0 \\ 0 & c + d \end{matrix} ) , P = ( \begin{matrix}\frac{a}{a+b} & \frac{b}{a+b} \\ \frac{c}{c+d} & \frac{d}{c+d} \end{matrix} ) .

Therefore we have T = D · P. Using this decomposition for T^{(k)} we have

T^{(k)} = D_{k}P_{k}T_{k−1} ⋯ T_{0}
= D_{k}T̄_{k−1}T_{k−2} ⋯ T_{0}
= D_{k}D_{k−1}P_{k−2}T_{k−3} ⋯ T_{0}
⋮
= D_{0} ⋯ D_{k}P_{k}.

Here D_{0}, …, D_{k} are diagonal matrices, P_{k} is a stochastic one and T̄_{k−1} = P_{k}T_{k−1}. In general, if we denote by T̄ the product between a general matrix and a stochastic one, that is, T̄ = T P, thus from the componentwise interpretation we get

(2)  min_{i,j} t_{ij} < min_{i,j} t̄_{ij} < max_{i,j} t̄_{ij} < max_{i,j} t_{ij}.

If we denote

T^{(k)} = ( \begin{matrix}a^{(k)} & b^{(k)} \\ c^{(k)} & d^{(k)} \end{matrix} ) , D_{0} ⋯ D_{k} = ( \begin{matrix}h^{(k)} & 0 \\ 0 & g^{(k)} \end{matrix} ) ,

P_{k} = ( \begin{matrix}p_{1k} & 1 − p_{1k} \\ 1 − p_{2k} & p_{2k} \end{matrix} ) ,

then we have

T^{(k)} = ( \begin{matrix}h^{(k)}p_{1k} & h^{(k)} (1 − p_{1k}) \\ g^{(k)} (1 − p_{2k}) & g^{(k)}p_{2k} \end{matrix} ) .

Based on the inequalities (??) and taking into account the positivity of T_{k}, it follows that δ > 0 and k_{0} > 1 exist such that p_{1k} > δ, p_{2k} > δ for all k > k_{0}. Then we obtain

Proposition 1 In the conditions described above, there exists the limit

lim_{k→∞} \frac{1}{k} log a^{(k)}.

By Proposition ?? it follows that if k is large enough then T^{(k)} approaches to

a matrix with proportional rows T^{(k)} = ( \begin{matrix}e^{kh} & e^{kg} \\ e^{kh} & e^{kg} \end{matrix} ) , where h and g are positive

random variables.

The theoretical approach given above, suggest the following procedure for finding self-similar sets:

Algorithm 1 Let X be a general compact metric space and w_{i}: X → X be contraction mappings with

d(w_{i}(x), w_{i}(y)) ≤ rd(x, y), for all x, y ∈ X,

for i = 1, 2, …, N, where 0 ≤ r < 1. Let {p_{1}, p_{2}, …, p_{N}} be probabilities with p_{i} > 0 and ∑ p_{i} = 1. Choose x_{0} ∈ X and pick recursively

x_{n} ∈ {w_{1}(x_{n−1}), w_{2}(x_{n−1}), …, w_{N}(x_{n−1}), }, for n = 1, 2, …, M,

where M is a large integer and p_{i} = P (x_{n} = w_{i}(x_{n−1})).

In this respect, the two shadows in Fig. 2 are obtained as invariant set of the following three affine transformations w_{i}: R^{2} → R^{2} (i = 1, 2, 3)

w_{1}(u, v) = ((1.1x^{2} + 0.1y)u + 3.5yv + 0.2; (0.3x + 0.7y)v + 1.2y^{2}u + 0.3)
w_{2}(u, v) = (0.12xv + (1.78y + 0.5)u + 1; (0.8x + 1.2y)u + 0.5yv + 0.5)
w_{3}(u, v) = ((x + 0.2y^{2})u + yv; yv + (0.1x + 0.9xy)u) .

We generated v_{0} = (x_{0}, y_{0}) ∈ R^{2} as random point, and we defined a set of points {v_{n} ∈ R^{2}/n = 0, 1, …, 10^{5}} recursively according to

v_{n+1} = \{ \begin{matrix}w_{1}v_{n} & with & p_{1} = 0.86 \\ w_{2}v_{n} & with & p_{2} = 0.13 \\ w_{3}v_{n} & with & p_{3} = 0.01 \end{matrix}

54 S.Corbu and M.Postolache

If those points from the set which lie in the square {(x, y)/ − 1 ≤ x, y ≤ 1} are plotted the result will be similar to Fig. 2.

The resulting picture in Fig. 2 appears to be the same no matter which initial point v_{0} is chosen. Also, it is obvious that these sets share the self-similarity property.

The theoretical presentation in this paper is not exhaustive, so the reader is encouraged to consult another research works ([?], [?], [?], [?], [?], [?], [?], [?]).

References

[1] K.B. Athreya and P.E. Ney: Branching Processes, Springer-Verlag, Heidelberg and New York, 1972.

[2] M. Barnsley and S. Demko: Iterated functions system and the global construction of fractals, Proc. R. Soc., London, A 399 (1985), 243-275.

[3] M. Barnsley and J. Elton: A new class of Markov processes for image encoding, Adv. Appl. Prob., 20 (1988), 14-32.

[4] S. Corbu and M. Postolache: Fractals generated by Möbius transformations and applications, in New Frontiers in Algebra, Groups and Geometries, (Gr. Tsagas (Ed.)), Hadronic Press, 1996, 407.

[5] P. Gora: Random comparing mappings, small stochastic perturbations and attractors, Z. wahr. verw. Gabiete., 69 (1985), 137-160.

[6] M. Iosifescu and S. Grigorescu: Dependence with Complete Connections and Its Applications, Cambridge Univ. Press, 1991.

[7] T. Kaijser: A limit theorem for Markov chains in compact metric spaces with applications to products of random matrices, Duke Math. J., 45 (1978), 311-349.

[8] N. Kaplan: Some result about multidimensional branching processes with random environments, Ann. Prob., 2 (1974), 411-455.

[9] Y. Kifer: Ergodic Theory of Random Transformations, Birkhäuser, 1986.

[10] T. Morita: Asymptotic behaviour of one-dimensional random dynamical systems, J. Math. Soc. Japan, 37 (1985), 651-663.

[11] T. Morita: Hiroshima Math. J., 18 (1988), 15.

[12] T. Ohno: Asymptopic behaviour of dynamical systems with random parameters, Publ. RIMS Kyoto Univ., 19 (1983), 83-98.

[13] B. Oksendal: Stochastic Differential Equations, Springer-Verlag, Heidelberg and New York, 1995.

[14] V.I. Oseledec: A multiplicative ergodic theorem, Trudîi Moscov Mat. Obsc., 19 (1968), 179-210.

[15] M. Postolache and T. Kawaguchi: Random affine transformations and fractal structures, Tensor, N.S., 58 (1997), to appear.

[16] M. Postolache: Numerical Methods (Second Revised Edition), Sirius Publishers, Bucharest, 1994.

[17] W. Smith and W. Wilkinson: On branching processes in random environment, Ann. Math. Statistics, 40 (1969), 814-827.

University Politehnica of Bucharest Department of Mathematics I Splaiul Independenţei 313 77206 Bucharest, Romania E-mail: mihai@mathem.pub.ro