LYAPUNOV EXPONENTS FOR HIGHER DIMENSIONAL RANDOM MAPS

A random map is a discrete time dynamical system in which one of a number of transformations is selected randomly and implemented. Random 
maps have been used recently to model interference effects in quantum physics. The main results of this paper deal with the Lyapunov exponents for 
higher dimensional random maps, where the individual maps are Jablonski 
maps on the n -dimensional cube.


Introduction
Ergodic theory of dynamical systems deals with the qualitative analysis of iterations of a single transformation.Ulam and von Neuman [12] suggested the study of more general systems where one applies at each iteration a different transformation chosen at random from a set of transformations.In this setting one could consider a single transformation, where parameters defining the transformation are allowed to vary dis- cretely or even continuously.
The importance of studying higher dimensional random maps is, in part, inspired by fractals that are fixed points of iterated functions systems [1].Iterated function systems can be viewed as random maps, where the individual transformations are con- tractions.Recently, random maps were used in modeling interference effects such as those that occur in the two-slit experiment of quantum physics [2].For a general study of ergodic theory of random maps, the reader is referred to the text by Kifer   [8].Additional ergodic properties of random maps can be found in [4, 5, 10] and [11].
One of the most important ways of quantifying the complexity of a dynamical system is by means of the Lyapunov exponent.This quantifier of chaos can be defined for random maps.In this paper, we develop formulas for the individual Lyapunov exponents for higher dimensional maps, where the basic maps are Jabtofiski maps on the n-dimensional cube [7].
1Research supported by NSERC and FCAR grants.

Lyapunov Exponents
Our considerations are based on Oseledec's Multiplicative Ergodic Theorem [9].Let (X,%,m) be a probability space and let r be a measurable transformation -: X--.X preserving an invariant measure #, absolutely continuous with respect to m.Let A: X--,GL(n, ) be a measurable map with f log + I I A(. )11d# < + c.Then, in particular, the limit X A v lim ,1-_log [I A( vl-ix)A( vk-2x) .(A(vx))A(x)vI I exists for any vER n and # almost any xEX.The number A vcan have one of at most n values A1,..., A n.
In this note, X-I n-[0, 1] n and m is the Lebesgue measure on In. v is a piece- wise expanding C 2 transformation and A(x) is the derivative matrix of r, where it is well defined (it is not defined on a set of measure 0).In this case, the numbers A1,..., A n are called Lyapunov exponents.
For the Jabofiski transformation v, the derivative matrix ) there exists a measure # invariant under v (with density f) with respect to Lebesgue measure.
All measures considered in this paper are assumed to be probability measures.
The transformations v we consider have a finite number of ergodic absolutely contin- uous measures.To simplify our considerations, we assume that the absolutely contin- uous invariant measure # is unique.Maps, which are piecewise onto, will satisfy this condition as will maps which are Markov and for which the matrix A is irreducible.In the general case we would consider each ergodic absolutely continuous measure 1,..., k, separately and our formulas would hold for each of them #i-a.e.

J=IDj
We do not assume that the Ai's are numbered in increasing order nor that they are all different.
For the quasi-Jabtofiski transformation , the derivative matrix is given by A-Aj-o (.) xGDj.
(.) 0 If there exists a constant s > 1 such that inf inf I}1 >, i, j [aij bij then r 2 is an expanding Jabtofiski transformation and there exists a measure # invar- iant under r 2 with density f with respect to Lebesgue measure.

/ \ 0
If we take v 2 -1 )then, as above, we have "2 lim I I A( rk-Ix) .A(vx)A(x)v2 j (lg ]J (x2)] + log l'2j(Xl)])f(x)dx Ai" D. 3 It is easy to see that no other value is possible as a limit v" 5. Random Maps c be a sequence of transformations from X into X.A random map Let {rt} 1 3-{{rt}= 1, {Pt}tc= 1} is a discrete dynamical system, where at each iteration, r is chosen with probability Pt, Pt > O, _, = l Pt-1.
If n-1, then (6.1) reduces to 1 " Y Pt log r(x)I f (x)dx.t=l 0 o To Proof: Let fi(x,w)-log o.()1.Since the shift (a,P) is ergodic (even exact), and we assume uniqueness of the absolutely continuous rt-invariant measures, there exists a unique t invariant absolutely continuous measures ([10]), and it gives a T-ergodic measure #-P.We have: In the general case, we can have a finite number of such measures, not more then minimal number of absolutely continuous invariant measures for vt, t 1,2, In particular, if a least one of v has a unique absolutely continuous measure, then the 3- invariant measure # is unique ([5]).In the case of more then one invariant measure, the above formula holds for x in the support of any fixed , -a.e.

A1--A2 t=l 12
Proof: The first part of the theorem follows from Theorem1 and the observation that A(3)-1/2A(32).The last equality follows from the definition of Lyapunov expo- nent and the fact that there are two iterations of 3 for each iterate of 32.
If is 32 invariant, then the measure where rt.to r 1, is 3-invariant.Since any 3-invariant measure is 32-invariant, if is unique, then #]51 and is also 3-invariant.This implies that for any integrable function f E (X, %, m), slP s f o vsd f d, 12 12 which simplifies formulas for )1 and A 2.
8 Random Composition of Jabtofiski and Quasi-Jabto,Sski Transformations Let r 1 be a Jabtofiski transformation and r 0 a quasi-Jabtofiski transformation on 12.
In this section we will find Lyapunov exponents for a random map {rl, Vo, p,q},   wherep, q>_0, p+q=l.Consider a sequence of Jabtofiski transformations: T 1 T 1 T 3 -T O 0 T 1 0 T O T T OoT-2oTO, pt-2q2, and let Pl-P, Pt t-2,3, Instead of :], we will consider the random Jabtofiski transformation 1-{{7t}__ 1, {Pt}t--}" Lemma 1: Ai( A il (1), i-1,2. Proof: Ai( -_,li log I[ A(k)(x)vi I I lim k+oo For Ai(tl) we have the same expression under the norm sign, but the averaging, factor k is different.It is enough to prove that, on average, there are two iterations of for each interaction of The average number of iterations of in each iterate of 1 is: Pt {number of iterates of in vt} p + pt-2q2t 2. t=l t=2 Lemma 2: If # is 5-invariant, then # is 5l-invariant.