| The transfer operator T(,(lamda))f(x): = e('(lamda)x)f(rx) + e('(lamda)(r/(1-r)-x))f(r + rx) has important applications in probability. In particular, the largest eigenvalue (nu)(,(lamda)) of T(,(lamda)) figures in the computation of fundamental quan- tities in two probabalistic models, which we present in the following paragraphs.; On the integers Z, a Gibbs random field can be constructed cor- responding to an interacting particle model. Suppose there are two; types of particles (0 and 1), and let (omega)(,i) denote the type of the particle situated at site i. Define an exponentially decaying pair potential: (UNFORMATTED TABLE FOLLOWS); r('(VBAR)i-j(VBAR)) if (omega)(,i) (NOT=) (omega)(,j),; U(i,j) =; 0 if (omega)(,i) = (omega)(,j),(TABLE ENDS); where 0 (LESSTHEQ) r ) 0,1 having an invariant measure (PI) gives rise to a sequence of random variables as follows. Let X be distributed according to (PI) and consider the sequence of iterates of f: X, f(X), f('(2))(X),... The average ('S)n/n where S(,n): = X + f(X)+...+f('(n))(x) con- verges a.e. to E X if f is ergodic, and we can ask about the asymp-; totic probabilities of large deviations of ('S)n/n from the mean. It turns out that for the tent map (UNFORMATTED TABLE FOLLOWS); 2x x (LESSTHEQ) .5; f(x) =; 2(1 - x) x > .5(TABLE ENDS); the asymptotic probabilities are given by; (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI); The value (nu)(,(lamda)) can be shown to be a real analytic function of (lamda). This thesis presents a recursive method for finding exactly the coefficients of the Taylor series for (nu)(,(lamda)) about (lamda) = 0. Connections are drawn between (nu)(,(lamda)) and eigenvalues of other transfer operators, some of which have been previously studied in the physics literature. A transfer operator for maps f: 0,1 (--->) 0,1 in a class which includes the tent map is derived. |
