The Ergodic Behavior of a Class of Transformations including the a(tan x)+b FunctionAuthor(s): J. H. B. KempermanSource: Advances in Applied Probability, Vol. 7, No. 2 (Jun., 1975), p. 254Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1426068 .

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4TH CONFERENCE ON STOCHASTIC PROCESSES AND APPLICATIONS 4TH CONFERENCE ON STOCHASTIC PROCESSES AND APPLICATIONS

The ergodic behavior of a class of transformations including the a (tan x) + b

function

J. H. B. KEMPERMAN, University of Rochester

The function f(x) = a(tan x) + b on R (with a and b as real constants), as well as each member of a large class of meromorphic functions with only real and

simple poles, has the following property. Either f possesses a real fixed point x0 with If'(xo) I < 1, or the transformation

x' = f(x) of R has a unique invariant probability measure v which is equivalent to the Lebesgue measure A on R. In the second case, f is even mixing and thus

ergodic relative to v, while h = dv/dA is a Cauchy density

h(x) = (d/){(x - c)2 + d2}, (x E R, d > 0),

with z0 = c + id as the unique complex fixed point with Im(zo) > 0 of either z = f(z) or z' = f(z).

If inf If'(x) > 1 then f is even exact and thus mixing of all orders relative to v. We also present results for more general transformations f: D ->R with

D c R open such that A(DC) = 0. One further assumes that D is the disjoint union of open intervals I, such that fl Is is of class C1 and strictly monotone.

Energy of Markov chains

R. SYSKI, University of Maryland

A right-continuous, non-negative supermartingale (Ut) = (Ut,,,, 0 < t < oo),

adapted to an increasing, right-continuous family (St) of a-fields on a probability space, is called a potential supermartingale if limt,,EUt = 0. For terminology used here, see [2], [4]. If (Ut) belongs to the class (D), then there exists an integ- rable, natural, increasing process (At) such that (Ut) is a potential generated by (At), and Doob decomposition holds:

Ut = M,-At, 0 < t < oo

where (Mr) is an uniformly integrable martingale with M, = E(Aoo 1t); see [4]. The energy of U = (Ut) is the (possibly infinite) number en(U) defined by:

en(U) = E J(Ut, + U_)dA, = EA2

the Lebesgue-Stieltjes integral; cf. [3], [4]. If U and V are two potentials generated by increasing processes A and B, respectively, then their mutual energy is defined by:

The ergodic behavior of a class of transformations including the a (tan x) + b

function

J. H. B. KEMPERMAN, University of Rochester

The function f(x) = a(tan x) + b on R (with a and b as real constants), as well as each member of a large class of meromorphic functions with only real and

simple poles, has the following property. Either f possesses a real fixed point x0 with If'(xo) I < 1, or the transformation

x' = f(x) of R has a unique invariant probability measure v which is equivalent to the Lebesgue measure A on R. In the second case, f is even mixing and thus

ergodic relative to v, while h = dv/dA is a Cauchy density

h(x) = (d/){(x - c)2 + d2}, (x E R, d > 0),

with z0 = c + id as the unique complex fixed point with Im(zo) > 0 of either z = f(z) or z' = f(z).

If inf If'(x) > 1 then f is even exact and thus mixing of all orders relative to v. We also present results for more general transformations f: D ->R with

D c R open such that A(DC) = 0. One further assumes that D is the disjoint union of open intervals I, such that fl Is is of class C1 and strictly monotone.

Energy of Markov chains

R. SYSKI, University of Maryland

A right-continuous, non-negative supermartingale (Ut) = (Ut,,,, 0 < t < oo),

adapted to an increasing, right-continuous family (St) of a-fields on a probability space, is called a potential supermartingale if limt,,EUt = 0. For terminology used here, see [2], [4]. If (Ut) belongs to the class (D), then there exists an integ- rable, natural, increasing process (At) such that (Ut) is a potential generated by (At), and Doob decomposition holds:

Ut = M,-At, 0 < t < oo

where (Mr) is an uniformly integrable martingale with M, = E(Aoo 1t); see [4]. The energy of U = (Ut) is the (possibly infinite) number en(U) defined by:

en(U) = E J(Ut, + U_)dA, = EA2

the Lebesgue-Stieltjes integral; cf. [3], [4]. If U and V are two potentials generated by increasing processes A and B, respectively, then their mutual energy is defined by:

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