Content-Type: multipart/mixed; boundary="-------------0009010838161" This is a multi-part message in MIME format. ---------------0009010838161 Content-Type: text/plain; name="00-331.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-331.keywords" Integrated density of states, random {Schr\"{o}dinger operators}, Riemannian manifold with compact quotient, amenable groups, Templeman's ergodic theorem ---------------0009010838161 Content-Type: application/postscript; name="manif816.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="manif816.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.85 Copyright 1999 Radical Eye Software %%Title: manif816.dvi %%Pages: 11 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: /net/prod/appl/tex/bin/alpha-OSF1/dvips manif816.dvi %+ -o manif816.ps %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2000.08.16:1800 %%BeginProcSet: tex.pro %! 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y(as)i(the)h(ergo)r(dicit)n(y) e(prop)r(erties)h(of)g(the)h(random)f(p)r(oten)n(tial.)639 2128 y(If)g(w)n(e)g(de\014ne)g Fo(U)1139 2140 y Fn(\015)1181 2128 y Fo(f)9 b Fq(\()p Fo(x)p Fq(\))24 b(:=)e Fo(f)9 b Fq(\()p Fo(\015)c(x)p Fq(\))27 b(as)f(the)h(unitary)f(op)r(erator)f (on)i Fo(L)2753 2098 y Fh(2)2789 2128 y Fq(\()p Fo(X)7 b Fq(\))27 b(corresp)r(ond-)515 2228 y(ing)21 b(to)g(the)g(isometry)g Fo(\015)27 b Fq(:)c Fo(X)30 b Fp(!)23 b Fo(X)7 b Fq(,)22 b(\(4\))f(can)g(b)r(e)h(written)f(in)g(the)h(form)f Fo(U)2792 2240 y Fn(\015)2834 2228 y Fo(V)2901 2198 y Fn(!)2949 2228 y Fo(U)3015 2198 y Fj(\003)3006 2248 y Fn(\015)3076 2228 y Fq(=)i Fo(V)3231 2198 y Fn(T)3270 2206 y Fi(\015)3308 2198 y Fn(!)3356 2228 y Fq(.)515 2327 y(Since)28 b Fp(\000)p Fq(\001)f(is)g(in)n(v)-5 b(arian)n(t)27 b(under)g(the)h(op)r(erators)e Fo(U)2128 2339 y Fn(\015)2198 2327 y Fq(one)h(gets)1616 2495 y Fo(U)1673 2507 y Fn(\015)1715 2495 y Fo(H)1791 2460 y Fn(!)1839 2495 y Fo(U)1905 2460 y Fj(\003)1896 2515 y Fn(\015)1966 2495 y Fq(=)c Fo(H)2130 2460 y Fn(T)2169 2468 y Fi(\015)2207 2460 y Fn(!)2255 2495 y Fo(:)995 b Fq(\(5\))639 2662 y(There)31 b(are)f(t)n(w)n(o)g(approac)n(hes)f(at)i (hand)h(to)f(pro)n(v)n(e)e(the)j(con)n(v)n(ergence)c(in)k(\(1\).)48 b(Either)515 2761 y(one)26 b(uses)g(the)i(Laplace)d(transform)h(of)h (the)g(normalised)e(eigen)n(v)-5 b(alue)26 b(coun)n(ting)g(function,) 515 2861 y(whic)n(h)k(can)f(b)r(e)h(written)h(in)f(terms)f(of)h(the)h (trace)e(of)h(the)g(corresp)r(onding)e(heat)i(equation)515 2961 y(semigroup.)58 b(In)35 b(this)h(case)e(one)h(applies)g(a)f(p)r (oin)n(t)n(wise)h(ergo)r(dic)f(theorem)h(for)f(additiv)n(e)515 3060 y(\(w.r.t.)59 b(domains\))24 b(pro)r(cesses.)34 b(Or)23 b(one)g(uses)h(Diric)n(hlet-Neumann)f(brac)n(k)n(eting)f(to)i (sho)n(w)515 3160 y(that)g(the)g(eigen)n(v)-5 b(alue)23 b(coun)n(ting)g(functions)i(form)e(themselv)n(es)g(a)h(sup)r(eradditiv) n(e)f(pro)r(cess.)515 3260 y(F)-7 b(or)33 b(suc)n(h)g(pro)r(cesses)f (there)h(are)g(also)f(ergo)r(dic)h(theorems)g(a)n(v)-5 b(aliable)32 b([3)o(,)i(10)o(].)55 b(Ho)n(w)n(ev)n(er)515 3359 y(only)31 b(in)g(the)h(ab)r(elian)f(case)f(they)i(assure)e(the)i (p)r(oin)n(t)n(wise)e(existence)h(of)h(a)f(limit.)48 b(In)32 b(our)515 3459 y(non-ab)r(elian)26 b(setting)i(w)n(e)f(w)n (ould)g(get)h(just)g(a)f(limit)h(in)g(mean.)639 3558 y(Th)n(us)e(w)n(e)f(rely)g(on)g(the)h(\014rst)f(alternativ)n(e)g(and)g (therefore)g(need)g(some)g(to)r(ols)g(from)h(the)515 3658 y(theory)h(of)g(heat)h(equation)f(semigroups.)515 3930 y Fr(Heat)46 b(k)l(ernel)g(b)t(ounds)515 4112 y Fq(Under)33 b(the)g(ab)r(o)n(v)n(e)e(assumptions)h Fo(H)1713 4082 y Fn(!)1794 4112 y Fq(is)h(a)f(densely)h(de\014ned)g(selfadjoin)n (t)f(op)r(erator)f(on)515 4211 y Fo(L)572 4181 y Fh(2)608 4211 y Fq(\()p Fo(X)7 b Fq(\).)43 b(By)29 b(the)g(sp)r(ectral)g (theorem)g(w)n(e)g(can)g(de\014ne)h(the)f(op)r(erator)f(exp\()p Fp(\000)p Fo(tH)3060 4181 y Fn(!)3107 4211 y Fq(\))i(whic)n(h)515 4311 y(has)i(an)g(in)n(tegral)f(k)n(ernel)h Fo(k)1391 4281 y Fn(!)1438 4311 y Fq(\()p Fo(t;)14 b Fp(\001)p Fo(;)g Fp(\001)p Fq(\).)53 b(Let)32 b Fo(D)h Fp(\032)e Fo(X)39 b Fq(b)r(e)33 b(a)f(regular)f(domain,)i(i.e.,)h(a)e(con-)515 4411 y(nected)f(subset)g(with)g(compact)g(closure)f(and)g(smo)r(oth)h (b)r(oundary)-7 b(.)46 b(W)-7 b(e)32 b(denote)f(the)g(re-)515 4510 y(striction)c(of)h Fo(H)1014 4480 y Fn(!)1090 4510 y Fq(to)f Fo(D)j Fq(with)e(Diric)n(hlet)g(b)r(oundary)f(conditions)h(b) n(y)f Fo(H)2785 4480 y Fn(!)2778 4533 y(D)2866 4510 y Fq(and)h(the)g(corre-)515 4610 y(sp)r(onding)f(heat)h(k)n(ernel)e(of)i (exp\()p Fp(\000)p Fo(tH)1722 4580 y Fn(!)1715 4633 y(D)1775 4610 y Fq(\))g(b)n(y)f Fo(k)1996 4580 y Fn(!)1993 4633 y(D)2053 4610 y Fq(\()p Fo(t;)14 b Fp(\001)p Fo(;)g Fp(\001)p Fq(\).)639 4710 y(Since)26 b(the)g(heat)g(k)n(ernel)f(dep)r(ends)h (monotonously)e(on)h(the)i(domain)e(\(see)g([6,)h(18)o(]\),)g(w)n(e)515 4809 y(ha)n(v)n(e)1121 4976 y(0)d Fp(\024)g Fo(k)1320 4942 y Fn(!)1317 4997 y(D)1377 4976 y Fq(\()p Fo(t;)14 b(x;)g(y)s Fq(\))24 b Fp(\024)e Fo(k)1793 4942 y Fn(!)1841 4976 y Fq(\()p Fo(t;)14 b(x;)g(y)s Fq(\))28 b Fp(8)f Fo(t)c(>)g Fq(0;)14 b Fo(x;)g(y)25 b Fp(2)f Fo(D)29 b(:)501 b Fq(\(6\))1926 5255 y(3)p eop %%Page: 4 4 4 3 bop 515 523 a Fq(W)-7 b(e)38 b(need,)j(ho)n(w)n(ev)n(er,)d(more)g (information)f(ab)r(out)h(the)g(in)n(tegral)f(k)n(ernels.)67 b(Firstly)-7 b(,)40 b(w)n(e)515 623 y(w)n(ould)32 b(lik)n(e)g(to)g(b)r (ound)h(the)g(rhs)f(of)g(\(6\))h(b)n(y)f(some)g(constan)n(t)g(indep)r (enden)n(t)h(on)g Fo(x;)14 b(y)35 b Fq(and)515 722 y Fo(!)s Fq(.)h(F)-7 b(urthermore)26 b(w)n(e)g(w)n(ould)h(lik)n(e)f(to)h (con)n(trol)e(the)j(di\013erence)e(b)r(et)n(w)n(een)h(the)g(heat)g(k)n (ernel)515 822 y(on)g(the)g(whole)g(manifold)g(and)g(the)h(one)f (restricted)f(to)h(the)h(domain)e Fo(D)r Fq(.)37 b(W)-7 b(e)28 b(collect)f(this)515 922 y(information)g(in)h(the)g(follo)n (wing)e(lemma.)515 1104 y Fg(Lemma)j(1)42 b Ff(L)l(et)29 b Fo(k)1132 1074 y Fn(!)1180 1104 y Fo(;)14 b(k)1263 1074 y Fn(!)1260 1127 y(D)1350 1104 y Ff(b)l(e)30 b(the)g(he)l(at)g (kernels)g(as)g(ab)l(ove.)40 b(Then)30 b(we)g(have)639 1204 y(\(a\))g Fo(k)825 1174 y Fn(!)873 1204 y Fq(\()p Fo(t;)14 b(x;)g(y)s Fq(\))24 b Fp(\024)e Fo(C)6 b Fq(\()p Fo(t)p Fq(\))p Fo(;)45 b Fp(8)p Fo(x;)14 b(y)s Fq(;)g Fo(!)s Ff(,)639 1303 y(\(b\))32 b Fp(8)p Fo(t;)14 b(\017)25 b(>)h Fq(0)31 b Fp(9)p Fo(h)26 b Fq(=)g Fo(h)p Fq(\()p Fo(t;)14 b(\017)p Fq(\))27 b Fo(<)f Fp(1)31 b Ff(such)h(that,)h(for)f (al)t(l)h(r)l(e)l(gular)e(domains)i Fo(D)h Ff(and)e(al)t(l)515 1403 y Fo(!)25 b Fp(2)f Fq(\012)p Ff(,)30 b(we)g(have)1050 1586 y Fq(0)22 b Fp(\024)h Fo(k)1248 1551 y Fn(!)1296 1586 y Fq(\()p Fo(t;)14 b(x;)g(y)s Fq(\))19 b Fp(\000)f Fo(k)1703 1551 y Fn(!)1700 1606 y(D)1760 1586 y Fq(\()p Fo(t;)c(x;)g(y)s Fq(\))24 b Fp(\024)f Fo(\017)59 b Fp(8)p Fo(x;)14 b(y)25 b Fp(2)e Fo(D)e Fp(n)d Fo(@)2693 1598 y Fn(h)2736 1586 y Fo(D)r(;)443 b Fq(\(7\))515 1768 y Ff(wher)l(e)30 b Fo(@)793 1780 y Fn(h)836 1768 y Fo(D)25 b Fq(:=)e Fp(f)p Fo(x)g Fp(2)h Fo(D)r Fp(j)29 b Fo(d)p Fq(\()p Fo(x;)14 b(@)5 b(D)r Fq(\))24 b Fp(\024)f Fo(h)p Fp(g)p Ff(.)639 1951 y Fq(The)f(pro)r(of)f(of)g(\(a\))h(for)f(a)g (\014xed)g(p)r(oten)n(tial)h Fo(V)2021 1921 y Fn(!)2090 1951 y Fq(can)f(b)r(e)h(infered)g(from)f(section)g(3)g(in)h([11)o(].) 515 2051 y(Actually)j(the)h(dep)r(endence)g(of)f(the)h(heat)g(k)n (ernel)e(on)h(the)h(p)r(oten)n(tial)g(can)f(b)r(e)h(b)r(ounded)f(b)n(y) 515 2150 y(a)i(function)h(of)f(the)h(sup-norm)f(of)g(the)h(p)r(oten)n (tial)f(only)-7 b(.)37 b(Since)28 b(our)e(family)i(of)f(p)r(oten)n (tials)515 2250 y(has)c(an)h(uniform)g(norm)g(b)r(ound)h(w)n(e)f(can)f (c)n(ho)r(ose)g(the)i(constan)n(t)e Fo(C)6 b Fq(\()p Fo(t)p Fq(\))26 b(to)e(b)r(e)g(indep)r(enden)n(t)515 2350 y(of)30 b Fo(!)s Fq(.)44 b(The)30 b(pro)r(of)f(of)h(\(b\))h(uses)f (a)g(result)f(ab)r(out)h(unit)h(propagation)d(sp)r(eed)i(of)g(the)h(w)n (a)n(v)n(e)515 2449 y(equation)i(with)h(p)r(oten)n(tial,)h(as)e(it)h (can)g(b)r(e)g(found,)h(for)f(example,)h(in)e([18].)55 b(The)34 b(b)r(ound)515 2549 y(on)g(the)h(di\013erence)f(of)g(the)h(t)n (w)n(o)f(heat)g(k)n(ernels)f(is)i(then)f(obtained)h(b)n(y)f(an)g (argumen)n(t)f(of)515 2648 y(U.)28 b(Bunk)n(e)f(as)g(outlined)h(in)g (Theorem)e(2.26)h(of)g([12)o(].)37 b(See)28 b(also)e([14)o(].)515 2923 y Fr(Assumptions)44 b(on)h(the)g(group)g Fe(\000)515 3105 y Fq(In)36 b(the)h(Euclidean)e(case,)j(an)n(y)e(monotone)f (exhaustion)h(of)g Fm(R)2522 3075 y Fn(d)2603 3105 y Fq(b)n(y)g(cub)r(es)g(\003)3022 3117 y Fn(l)3043 3125 y Fd(1)3079 3105 y Fo(;)14 b Fq(\003)3174 3117 y Fn(l)3195 3125 y Fd(2)3231 3105 y Fo(;)g(:)g(:)g(:)515 3204 y Fq(guaran)n(teed)29 b(the)j(existence)e(of)h(a)g(nonrandom)f(IDS)i(b)n(y)e(an)h (application)g(of)g(an)f(ab)r(elian)515 3304 y(ergo)r(dic)j(theorem.)55 b(Actually)34 b(there)g(is)f(m)n(uc)n(h)h(more)f(freedom.)56 b(One)33 b(is)h(ev)n(en)f(allo)n(w)n(ed)515 3404 y(to)g(shift)i(the)f (cub)r(es)g(a)g(little)g(bit)h(on)e(eac)n(h)g(scale,)i(so)e(that)h(the) g(cub)r(es)g(do)g(not)g(con)n(tain)515 3503 y(the)f(preceeding)g(ones)f (an)n(y)g(more;)k(one)c(can)h(c)n(ho)r(ose)f(balls)h(instead)g(of)g (cub)r(es)g(or)g(other)515 3603 y(con)n(v)n(ex)23 b(domains)h(as)f (long)h(as)g(they)h(satisfy)f(some)g(v)n(ery)f(reasonable)f(gro)n(wth)h (conditions,)515 3703 y(cf.)37 b([9,)27 b(10)o(].)639 3802 y(On)38 b(a)g(Riemannian)g(manifold)h(one)e(has)h(to)g(deal)g (with)h(the)g(problem)e(to)i(\014nd)f(an)515 3902 y(adequate)27 b(substitute)i(for)e(the)i(cub)r(es)f(\003)1815 3914 y Fn(l)1840 3902 y Fq(.)38 b(This)28 b(problem)g(already)e(app)r(eared) i(in)g([2)o(])h(for)515 4001 y(a)k(single)h(op)r(erator)e(with)j(p)r (erio)r(dic)f(p)r(oten)n(tial.)56 b(If)34 b(one)g(considers)f(a)g (\000-ergo)r(dic)g(family)515 4101 y(of)e(p)r(oten)n(tials)g(and)h(w)n (an)n(ts)e(to)i(pro)n(v)n(e)e(existence)h(of)g(the)h(IDS)g(there)f(is)h (need)f(for)g(a)h(non-)515 4201 y(ab)r(elian)h(ergo)r(dic)g(theorem.)54 b(W)-7 b(e)34 b(will)g(use)g(an)f(ergo)r(dic)g(theorem)g(due)h(to)g(T) -7 b(emp)r(elman)515 4300 y([19)o(,)26 b(20)o(].)37 b(This)26 b(theorem)g(is)g(only)g(applicable)g(for)g(exhaustions)f(or)h(summing)g (sequences)515 4400 y(\(the)c(analoga)d(of)i(\003)1128 4412 y Fn(l)1149 4420 y Fd(1)1185 4400 y Fo(;)14 b Fq(\003)1280 4412 y Fn(l)1301 4420 y Fd(2)1338 4400 y Fo(;)g(:)g(:)g(:)f Fq(\))22 b(whic)n(h)f(satisfy)g(m)n(uc)n(h)g(more)g(restrictiv)n(e)f (conditions)h(than)515 4500 y(those)34 b(imp)r(osed)h(in)g(ab)r(elian)g (ergo)r(dic)f(theorems.)58 b(The)35 b(follo)n(wing)f(de\014nition)h(la) n(ys)f(the)515 4599 y(grounds)26 b(to)i(apply)f(T)-7 b(emp)r(elman's)28 b(ergo)r(dic)e(theorem)h(\(in)h(the)g(v)n(ersion)e (giv)n(en)h(in)h([10)o(]\).)515 4782 y Fg(De\014nition)j(2)41 b Ff(A)30 b(se)l(quenc)l(e)g(of)h(subsets)e Fp(f)p Fo(I)1936 4794 y Fn(n)1981 4782 y Fp(g)2023 4794 y Fn(n)p Fj(2)p Fc(N)2185 4782 y Ff(of)i Fq(\000)f Ff(is)h(c)l(al)t(le)l(d)g(an)g (admissible)h(sum-)1926 5255 y Fq(4)p eop %%Page: 5 5 5 4 bop 515 523 a Ff(ming)30 b(se)l(quenc)l(e)f(if)1078 706 y Fq(\()p Fo(P)12 b Fq(0\))196 b Fo(I)1481 718 y Fn(n)1550 706 y Fp(\032)22 b Fo(I)1673 718 y Fn(n)p Fh(+1)1803 706 y Fo(;)44 b(;)14 b Fq(0)22 b Fo(<)h Fp(j)p Fo(I)2118 718 y Fn(n)2163 706 y Fp(j)h Fo(<)e Fp(1)30 b(8)p Fo(n)22 b Fp(2)h Fm(N)t Fo(;)1078 830 y Fq(\()p Fo(P)12 b Fq(1\))196 b Fp(8)p Fo(\015)27 b Fq(:)52 b Fp(j)p Fo(I)1696 842 y Fn(n)1742 830 y Fq(\001)p Fo(I)1847 842 y Fn(n)1893 830 y Fo(\015)5 b Fp(j)p Fo(=)p Fp(j)p Fo(I)2065 842 y Fn(n)2110 830 y Fp(j)23 b(\000)-14 b(!)23 b Fq(0)29 b Ff(for)i Fo(n)23 b Fp(!)g(1)p Fo(;)1078 955 y Fq(\()p Fo(P)12 b Fq(2\))196 b Fp(9)p Fo(K)28 b(<)23 b Fp(1)g Fq(:)53 b Fp(j)p Fo(I)1926 920 y Fj(\000)p Fh(1)1919 975 y Fn(n)2015 955 y Fo(I)2051 967 y Fn(n)2097 955 y Fp(j)23 b(\024)g Fo(K)6 b Fp(j)p Fo(I)2367 967 y Fn(n)2412 955 y Fp(j)30 b Fo(:)515 1137 y Ff(Her)l(e)f Fp(j)p Fo(A)p Fp(j)h Ff(denotes)h(the)e(c)l(ar)l(dinality)j(of)e(a)h(\014nite)e(set)g Fo(A)p Ff(.)639 1320 y Fq(The)19 b(condition)f(\(P1\))g(is)g(an)g(form) n(ulation)g(of)g(F\034lner's)g(condition)g(for)g(discrete)f(groups.)515 1420 y(\(P2\))25 b(is)g(a)f(gro)n(wth)g(condition)h(whic)n(h)g(can)g(b) r(e)g(understo)r(o)r(d)g(b)r(est,)h(e.g.,)f(in)h(the)f(particular)515 1519 y(case)31 b(where)g(the)i(sets)e Fo(I)1289 1531 y Fn(n)1367 1519 y Fq(are)g(com)n(binatorial)f(balls)h(of)h(radius)f Fo(n)h Fq(\(w.r.t.)h(a)e(w)n(ord)g(met-)515 1619 y(ric\):)j(Doubling)24 b(the)g(radius)f(increases)f(the)i(cardinalit)n(y)e(of)i(the)g(ball)f (b)n(y)g(only)h(a)f(constan)n(t)515 1719 y(m)n(ultiple.)55 b(Admissible)33 b(summing)h(sequences)e(are)h(the)h(main)f(input)h(in)g (T)-7 b(emp)r(elman's)515 1818 y(ergo)r(dic)31 b(theorem)h(applied)g (to)h(the)g(group)e Fo(T)1975 1830 y Fn(\015)2017 1818 y Fo(;)14 b(\015)36 b Fp(2)31 b Fq(\000.)51 b(In)33 b(the)g(next)f (section,)i(w)n(e)e(con-)515 1918 y(struct,)k(to)f(a)f(giv)n(en)g (admissible)g(summing)h(sequence,)h(a)e(corresp)r(onding)f(sequence)h (of)515 2017 y(subsets)e(of)g(the)h(Riemannian)f(manifold)h(to)f(whic)n (h)h(w)n(e)f(will)g(restrict)g(our)g(Sc)n(hr\177)-42 b(odinger)515 2117 y(op)r(erators.)515 2392 y Fr(Admissible)45 b(exhaustions)h(and)f(the)g(main)g(result)515 2573 y Fq(As)c(explained)g(in)h(section)f(3)g(of)g([2)o(],)k(there)c(exists)g (a)g(smo)r(oth)g(triangulation)f(of)i(the)515 2673 y(compact)31 b(manifold)h Fo(M)41 b Fq(and,)33 b(b)n(y)f(lifting)h(the)f(simplices)g (to)g Fo(X)39 b Fq(in)32 b(a)g(suitable)g(manner,)515 2773 y(w)n(e)37 b(obtain)g(a)g(connected)g(p)r(olyhedral)g (\000-fundamen)n(tal)g(domain)g Fp(F)8 b Fq(.)66 b(P)n(articularly)35 b Fp(F)515 2872 y Fq(consists)d(of)h(\014nitely)h(man)n(y)f(smo)r(oth)g (images)f(of)h Fo(d)p Fq(-dimensional)g(simplices)g(whic)n(h)g(can)515 2972 y(o)n(v)n(erlap)25 b(only)i(at)h(their)f(b)r(oundaries.)515 3155 y Fg(De\014nition)k(3)41 b Ff(F)-6 b(or)30 b(an)g(admissible)h (summing)f(se)l(quenc)l(e)f Fp(f)p Fo(I)2492 3167 y Fn(n)2537 3155 y Fp(g)2579 3167 y Fn(n)p Fj(2)p Fc(N)2740 3155 y Ff(we)i(de\014ne)1693 3354 y Fo(D)1762 3366 y Fn(n)1830 3354 y Fq(=)1948 3275 y Fb([)1918 3453 y Fn(\015)t Fj(2)p Fn(I)2031 3461 y Fi(n)2085 3354 y Fo(\015)5 b Fp(F)1080 b Fq(\(8\))515 3626 y Ff(wher)l(e)39 b Fp(F)46 b Ff(is)39 b(a)g(fundamental)g(domain)g(as)g(explaine)l(d)h(ab)l(ove.)66 b(We)38 b(c)l(al)t(l)i Fp(f)p Fo(D)3034 3638 y Fn(n)3078 3626 y Fp(g)3120 3638 y Fn(n)p Fj(2)p Fc(N)3290 3626 y Ff(an)515 3726 y(admissible)32 b(exhaustion)e(if,)h(additional)t(ly,) 946 3952 y Fq(\()p Fo(P)12 b Fq(\))208 b Ff(F)-6 b(or)30 b(al)t(l)h Fo(h)23 b(>)f Fq(0)h(:)1843 3896 y Ff(vol)p Fq(\()p Fo(@)2020 3908 y Fn(h)2064 3896 y Fo(D)2133 3908 y Fn(n)2178 3896 y Fq(\))p 1843 3933 368 4 v 1886 4009 a Ff(vol)q Fq(\()p Fo(D)2089 4021 y Fn(n)2134 4009 y Fq(\))2243 3952 y Fp(\000)-14 b(!)23 b Fq(0)59 b Ff(for)30 b Fo(n)23 b Fp(!)g(1)30 b Fo(:)325 b Fq(\(9\))639 4183 y(It)40 b(w)n(as)e(sho)n(wn)h(in)g([1,)g(2])g(that)h(the)f(geometric)g (condition)g(\(P\))g(is)g(automatically)515 4283 y(satis\014ed)g(if)i Fp(j)p Fo(I)994 4295 y Fn(n)1039 4283 y Fo(A)1101 4253 y Fn(n)1173 4283 y Fp(\000)27 b Fo(I)1301 4295 y Fn(n)1346 4283 y Fp(j)44 b(\024)f(j)p Fo(I)1580 4295 y Fn(n)1626 4283 y Fp(j)p Fo(=)p Fq(\()p Fo(n)p Fp(j)p Fo(A)1858 4253 y Fn(n)1903 4283 y Fp(j)p Fq(\),)g(for)d(all)f Fo(n)44 b Fp(2)g Fm(N)t Fq(,)49 b(where)39 b Fo(A)44 b Fp(\032)f Fq(\000)d(is)g(the)515 4382 y(follo)n(wing)26 b(particular)h(set)g(of)h (generators:)34 b Fo(A)24 b Fq(=)e Fp(f)p Fo(\015)27 b Fq(:)p 2238 4316 116 4 v 23 w Fo(\015)5 b Fp(F)26 b(\\)p 2446 4316 68 4 v 19 w(F)31 b(6)p Fq(=)23 b Fp(;g)p Fq(.)515 4565 y Fg(Lemma)29 b(4)42 b Ff(F)-6 b(or)33 b(an)h(admissible)h (exhaustion)f Fp(f)p Fo(D)2153 4577 y Fn(n)2197 4565 y Fp(g)2239 4577 y Fn(n)p Fj(2)p Fc(N)2404 4565 y Ff(and)g(any)g Fo(\016)f(>)c Fq(0)k Ff(ther)l(e)g(exists)515 4665 y(a)d(se)l(quenc)l (e)f Fp(f)p Fo(M)1047 4677 y Fn(n)1091 4665 y Fp(g)1133 4677 y Fn(n)p Fj(2)p Fc(N)1295 4665 y Ff(of)h(r)l(e)l(gular)g(domains)h (such)f(that:)1136 4847 y Fo(D)1205 4859 y Fn(n)1269 4847 y Fp(n)18 b Fo(@)1373 4859 y Fn(\016)1409 4847 y Fo(D)1478 4859 y Fn(n)1546 4847 y Fp(\032)23 b Fo(M)1715 4859 y Fn(n)1783 4847 y Fp(\032)f Fo(D)1939 4859 y Fn(n)1998 4847 y Fo(;)44 b(@)5 b(M)2195 4859 y Fn(n)2262 4847 y Fp(\032)23 b Fo(@)2394 4859 y Fn(\016)2430 4847 y Fo(D)2499 4859 y Fn(n)2558 4847 y Fo(;)44 b Fp(8)p Fo(n;)486 b Fq(\(10\))1926 5255 y(5)p eop %%Page: 6 6 6 5 bop 515 523 a Ff(and,)30 b(for)h Fo(n)23 b Fp(!)g(1)p Ff(,)30 b(we)g(have)822 629 y Fb(\014)822 679 y(\014)822 729 y(\014)822 779 y(\014)850 750 y Fp(j)p Fo(D)942 762 y Fn(n)987 750 y Fp(j)1010 715 y Fj(\000)p Fh(1)1113 637 y Fb(Z)1159 825 y Fn(D)1213 833 y Fi(n)1272 750 y Fo(k)1318 715 y Fn(!)1366 750 y Fq(\()p Fo(t;)14 b(x;)g(x)p Fq(\))p Fo(dx)20 b Fp(\000)e(j)p Fo(M)1925 762 y Fn(n)1970 750 y Fp(j)1993 715 y Fj(\000)p Fh(1)2096 637 y Fb(Z)2142 825 y Fn(M)2205 833 y Fi(n)2264 750 y Fo(k)2310 715 y Fn(!)2307 770 y(M)2370 778 y Fi(n)2416 750 y Fq(\()p Fo(t;)c(x;)g(x)p Fq(\))p Fo(dx)2768 629 y Fb(\014)2768 679 y(\014)2768 729 y(\014)2768 779 y(\014)2820 750 y Fp(\000)-14 b(!)23 b Fq(0)29 b Fo(:)160 b Fq(\(11\))515 986 y Ff(We)30 b(c)l(al)t(l)g Fp(f)p Fo(M)941 998 y Fn(n)986 986 y Fp(g)1028 998 y Fn(n)p Fj(2)p Fc(N)1189 986 y Ff(a)g(r)l(e)l (gularisation)h(of)f Fp(f)p Fo(D)1989 998 y Fn(n)2034 986 y Fp(g)2076 998 y Fn(n)p Fj(2)p Fc(N)2207 986 y Ff(.)639 1167 y Fq(F)-7 b(or)33 b(the)g(pro)r(of)f(of)h(\(11\))g(one)f(uses)h (Lemma)f(1\(b\))i(to)e(b)r(ound)i(the)f(di\013erence)g(of)g(the)515 1267 y(k)n(ernels)39 b(b)n(y)i(some)f(arbitrary)f(small)h(n)n(um)n(b)r (er)h Fo(\017)k(>)f Fq(0)d(a)n(w)n(a)n(y)d(from)j(the)g(b)r(oundary)f (of)515 1366 y Fo(D)584 1378 y Fn(n)629 1366 y Fq(.)54 b(The)34 b(region)e(near)g(the)i(b)r(oundary)-7 b(,)34 b(where)f(this)h(estimate)f(is)h(not)f(v)-5 b(alid,)35 b(is)e(small)515 1466 y(b)n(y)h(prop)r(ert)n(y)g(\(P\).)h(There)f(w)n (e)g(use)h(Lemma)f(1\(a\))h(and)f(the)h(fact)g(that)g(the)g(symmetric) 515 1566 y(di\013erence)27 b(of)h Fo(M)1062 1578 y Fn(n)1134 1566 y Fq(and)g Fo(D)1365 1578 y Fn(n)1437 1566 y Fq(is)g(small,)f(as)g (w)n(ell.)639 1665 y(The)39 b(ergo)r(dicit)n(y)d(of)j Fo(V)1388 1635 y Fn(!)1475 1665 y Fq(implies)f(\(5\).)69 b(An)39 b(analogous)d(relation)h(is)h(v)-5 b(alid)38 b(for)g(the)515 1765 y(sp)r(ectral)29 b(pro)5 b(jections)28 b(of)h Fo(H)1430 1735 y Fn(!)1508 1765 y Fq(as)g(can)g(b)r(e)h (inferred)f(from)g([21)o(,)h(Theorem)f(7.15])f(and)i(the)515 1864 y(uniqueness)d(of)h(the)g(sp)r(ectral)f(pro)5 b(jection.)1448 2046 y Fo(U)1505 2058 y Fn(\015)1548 2046 y Fo(E)g Fq(\()p Fo(\025;)14 b(!)s Fq(\))p Fo(U)1884 2011 y Fj(\003)1875 2066 y Fn(\015)1945 2046 y Fq(=)23 b Fo(E)5 b Fq(\()p Fo(\025;)14 b(T)2265 2058 y Fn(\015)2308 2046 y Fo(!)s Fq(\))27 b Fo(:)786 b Fq(\(12\))515 2227 y(This)41 b(relation)f(also)h (holds)g(for)g(exp)o(\()p Fp(\000)p Fo(tH)1918 2197 y Fn(!)1966 2227 y Fq(\))h(and)f(th)n(us)g(its)h(k)n(ernel)e Fo(k)2845 2197 y Fn(!)2893 2227 y Fq(\()p Fo(t;)14 b(x;)g(y)s Fq(\))42 b(is)f(an)515 2327 y(ergo)r(dic)26 b(sto)r(c)n(hastic)h(pro)r (cess)f(in)i(the)g(follo)n(wing)f(sense)1423 2508 y Fo(k)1469 2474 y Fn(!)1517 2508 y Fq(\()p Fo(t;)14 b(\015)5 b(x;)14 b(\015)5 b(x)p Fq(\))23 b(=)g Fo(k)2032 2474 y Fn(T)2071 2482 y Fi(\015)2109 2474 y Fn(!)2157 2508 y Fq(\()p Fo(t;)14 b(x;)g(x)p Fq(\))29 b Fo(:)760 b Fq(\(13\))515 2689 y(This)27 b(enables)g(us)h(to)f(apply)h(later)f(the)h(next)f(lemma)h(to)f(the)h (heat)g(k)n(ernels.)515 2871 y Fg(Lemma)h(5)42 b Ff(L)l(et)30 b Fp(f)p Fo(D)1198 2883 y Fn(n)1243 2871 y Fp(g)1285 2883 y Fn(n)p Fj(2)p Fc(N)1447 2871 y Ff(b)l(e)h(an)g(admissible)j (exhaustion)d(and)g Fo(f)j Fq(:)26 b(\012)19 b Fp(\002)g Fo(X)32 b Fp(!)25 b Fm(R)37 b Ff(b)l(e)32 b(a)515 2970 y Fq(\000)p Ff(-er)l(go)l(dic)e(sto)l(chastic)h(pr)l(o)l(c)l(ess.)39 b(Then)993 3197 y Fq(lim)964 3246 y Fn(n)p Fj(!1)1151 3197 y Fp(j)p Fo(D)1243 3209 y Fn(n)1288 3197 y Fp(j)1311 3162 y Fj(\000)p Fh(1)1414 3084 y Fb(Z)1460 3272 y Fn(D)1514 3280 y Fi(n)1573 3197 y Fo(f)9 b Fq(\()p Fo(!)s(;)14 b(x)p Fq(\))p Fo(dx)24 b Fq(=)f Fp(jF)8 b(j)2142 3162 y Fj(\000)p Fh(1)2245 3197 y Fm(E)2328 3080 y Fb(\024)2371 3084 y(Z)2418 3272 y Fj(F)2489 3197 y Fo(f)h Fq(\()p Fp(\017)p Fo(;)14 b(x)p Fq(\))p Fo(dx)2819 3080 y Fb(\025)2907 3197 y Fo(:)301 b Fq(\(14\))515 3428 y(Here)41 b Fm(E)56 b Fq(denotes)27 b(the)h(exp)r(ectation)f(with)h(resp)r(ect)g(to)f(the)h (measure)41 b Fm(P)14 b Fq(.)515 3627 y Fg(Pro)s(of:)86 b Fq(Set)26 b Fo(F)12 b Fq(\()p Fo(!)s Fq(\))23 b(=)g Fp(jF)8 b(j)1410 3597 y Fj(\000)p Fh(1)1513 3561 y Fb(R)1552 3657 y Fj(F)1624 3627 y Fo(f)h Fq(\()p Fo(!)s(;)14 b(x)p Fq(\))p Fo(dx)p Fq(.)37 b(No)n(w)25 b(transform)f(the)i(term)f(on)h (the)g(lhs)f(of)515 3727 y(\(14\).)865 3951 y Fp(j)p Fo(D)957 3963 y Fn(n)1002 3951 y Fp(j)1025 3917 y Fj(\000)p Fh(1)1128 3838 y Fb(Z)1174 4026 y Fn(D)1228 4034 y Fi(n)1287 3951 y Fo(f)9 b Fq(\()p Fo(!)s(;)14 b(x)p Fq(\))p Fo(dx)84 b Fq(=)e Fp(j)p Fo(I)1920 3963 y Fn(n)1966 3951 y Fp(j)1989 3917 y Fj(\000)p Fh(1)2109 3872 y Fb(X)2092 4050 y Fn(\015)t Fj(2)p Fn(I)2205 4058 y Fi(n)2259 3951 y Fp(jF)8 b(j)2373 3917 y Fj(\000)p Fh(1)2476 3838 y Fb(Z)2523 4026 y Fn(\015)t Fj(F)2632 3951 y Fo(f)h Fq(\()p Fo(!)s(;)14 b(x)p Fq(\))p Fo(dx)1714 4216 y Fq(=)82 b Fp(j)p Fo(I)1920 4228 y Fn(n)1966 4216 y Fp(j)1989 4182 y Fj(\000)p Fh(1)2109 4137 y Fb(X)2092 4315 y Fn(\015)t Fj(2)p Fn(I)2205 4323 y Fi(n)2259 4216 y Fp(jF)8 b(j)2373 4182 y Fj(\000)p Fh(1)2476 4103 y Fb(Z)2523 4292 y Fj(F)2594 4216 y Fo(f)h Fq(\()p Fo(!)s(;)14 b(\015)5 b(x)p Fq(\))p Fo(dx)1714 4481 y Fq(=)82 b Fp(j)p Fo(I)1920 4493 y Fn(n)1966 4481 y Fp(j)1989 4447 y Fj(\000)p Fh(1)2109 4402 y Fb(X)2092 4580 y Fn(\015)t Fj(2)p Fn(I)2205 4588 y Fi(n)2259 4481 y Fp(jF)8 b(j)2373 4447 y Fj(\000)p Fh(1)2476 4368 y Fb(Z)2523 4557 y Fj(F)2594 4481 y Fo(f)h Fq(\()p Fo(T)2725 4493 y Fn(\015)2767 4481 y Fo(!)s(;)14 b(x)p Fq(\))p Fo(dx)1714 4720 y Fq(=)82 b Fp(j)p Fo(I)1920 4732 y Fn(n)1966 4720 y Fp(j)1989 4686 y Fj(\000)p Fh(1)2109 4641 y Fb(X)2092 4819 y Fn(\015)t Fj(2)p Fn(I)2205 4827 y Fi(n)2259 4720 y Fo(F)12 b Fq(\()p Fo(T)2405 4732 y Fn(\015)2448 4720 y Fo(!)s Fq(\))23 b Fp(\000)-15 b(!)2732 4698 y Fq(~)2715 4720 y Fo(f)8 b Fq(\()p Fo(!)s Fq(\))515 5006 y(for)27 b Fo(n)c Fp(!)g(1)k Fq(and)h(a)f(measurable)1614 4984 y(~)1596 5006 y Fo(f)32 b Fq(:)23 b(\012)g Fp(!)g Fm(R)6 b Fq(.)37 b(Here)27 b(w)n(e)h(used)f(that)1926 5255 y(6)p eop %%Page: 7 7 7 6 bop 616 523 a Fq(1.)41 b Fo(D)791 535 y Fn(n)868 523 y Fq(can)32 b(b)r(e)g(written)g(as)g(an)f(union)h(of)g(translates)f (of)h(the)g(fundamen)n(tal)g(domain)722 623 y Fp(F)8 b Fq(,)616 789 y(2.)41 b Fo(\015)33 b Fq(is)27 b(an)g(isometry)g(and)g (th)n(us)h(preserv)n(es)e(the)i(v)n(olume)f(elemen)n(t)g Fo(dx)p Fq(,)616 955 y(3.)41 b(w)n(e)27 b(can)f(transfer)f(the)i (action)g(of)f Fo(\015)31 b Fq(from)c(the)g(geometric)e(to)i(the)g (random)e(param-)722 1054 y(eter)j(b)n(y)f(the)h(ergo)r(dicit)n(y)e(of) i(the)g(pro)r(cess)e Fo(f)9 b Fq(,)616 1220 y(4.)41 b(admissible)30 b(summing)h(sequences)f(satisfy)g(the)h(conditions)f(of)g(T)-7 b(emp)r(elman's)31 b(er-)722 1320 y(go)r(dic)c(theorem.)515 1486 y(W)-7 b(e)23 b(additionally)g(kno)n(w)f(from)h(this)g(ergo)r(dic) f(theorem)h(that)g(the)h(function)2924 1464 y(~)2906 1486 y Fo(f)32 b Fq(is)23 b(in)n(v)-5 b(arian)n(t)515 1586 y(under)37 b(all)f Fo(T)933 1598 y Fn(\015)975 1586 y Fo(;)14 b(\015)44 b Fp(2)39 b Fq(\000.)65 b(By)36 b(assumption,)j (these)e(transformations)e(are)h(ergo)r(dic)g(and,)515 1695 y(th)n(us,)750 1673 y(~)732 1695 y Fo(f)46 b Fq(is)37 b(constan)n(t)f(almost)h(surely)-7 b(.)64 b(The)37 b(con)n(v)n(ergence) e(calculated)h(ab)r(o)n(v)n(e)g(implies)515 1795 y(that)28 b(this)g(constan)n(t)e(has)h(to)h(b)r(e)g(equal)f(to)41 b Fm(E)23 b Fq(\()p Fo(F)12 b Fq(\).)556 1894 y Fa(2)639 1994 y Fq(In)24 b(order)e(to)i(pro)n(v)n(e)e(our)h(main)g(result)g (\(Theorem)h(7\))f(w)n(e)g(need,)i(\014nally)-7 b(,)24 b(the)g(follo)n(wing)515 2094 y(lemma)40 b(\(see)g([15)o(,)j(Lemma)d (5.2]\).)74 b(It)41 b(will)f(b)r(e)h(used)f(to)g(deriv)n(e)f(con)n(v)n (ergence)f(of)i(the)515 2193 y(normalized)24 b(eigen)n(v)-5 b(alue)24 b(coun)n(ting)g(function)i(from)f(con)n(v)n(ergence)e(of)i (the)g(corresp)r(onding)515 2293 y(heat)i(k)n(ernels.)515 2476 y Fg(Lemma)i(6)j(\()973 2455 y(\024)970 2476 y(Subin\))41 b Ff(L)l(et)30 b Fo(N)1498 2488 y Fn(n)1567 2476 y Fq(:)25 b Fm(R)30 b Fp(!)24 b Fm(R)1860 2445 y Fh(+)1952 2476 y Ff(b)l(e)30 b(a)h(se)l(quenc)l(e)f(of)h(left-c)l(ontinuous,)g(mono-) 515 2575 y(tone)e(incr)l(e)l(asing)i(functions)e(such)h(that)613 2741 y(1.)42 b(ther)l(e)30 b(exists)f(a)i Fo(c)23 b Fp(2)g Fm(R)29 b Fq(:)53 b Fo(N)1592 2753 y Fn(n)1637 2741 y Fq(\()p Fo(\025)p Fq(\))24 b(=)e(0)89 b Fp(8)p Fo(\025)23 b Fp(\024)f Fo(c;)28 b(n)23 b Fp(2)g Fm(N)t Ff(,)613 2907 y(2.)42 b(ther)l(e)25 b(exists)f(a)g Fo(C)30 b Fq(:)23 b Fm(R)1401 2877 y Fh(+)1485 2907 y Fp(!)g Fm(R)29 b Fq(:)23 b Fo( )1774 2919 y Fn(n)1819 2907 y Fq(\()p Fo(t)p Fq(\))h(:=)2048 2840 y Fb(R)2117 2907 y Fo(e)2156 2877 y Fj(\000)p Fn(\025t)2276 2907 y Fo(dN)2386 2919 y Fn(n)2431 2907 y Fq(\()p Fo(\025)p Fq(\))g Fp(\024)f Fo(C)6 b Fq(\()p Fo(t)p Fq(\))49 b Fp(8)p Fo(n)22 b Fp(2)i Fm(N)t Fo(;)33 b(t)23 b(>)g Fq(0)p Ff(,)613 3073 y(3.)42 b Fq(lim)838 3085 y Fn(n)p Fj(!1)1029 3073 y Fo( )1083 3085 y Fn(n)1128 3073 y Fq(\()p Fo(t)p Fq(\))24 b(=:)f Fo( )s Fq(\()p Fo(t)p Fq(\))30 b Ff(exists)f(for)i(al)t(l)g Fo(t)23 b(>)f Fq(0)p Ff(.)515 3239 y(Then)30 b(the)g(limit)1580 3422 y Fo(N)9 b Fq(\()p Fo(\025)p Fq(\))23 b(:=)52 b(lim)1902 3472 y Fn(n)p Fj(!1)2089 3422 y Fo(N)2156 3434 y Fn(n)2201 3422 y Fq(\()p Fo(\025)p Fq(\))515 3629 y Ff(exists)31 b(at)h(al)t(l)h(c)l(ontinuity)f(p)l(oints.)45 b Fo(N)c Ff(is)32 b(a)g(non-ne)l(gative,)h(monotone)f(incr)l(e)l(asing,)i(left-) 515 3729 y(c)l(ontinuous)29 b(function)g(of)i Fo(\025)p Ff(,)g(and)f(its)g(L)l(aplac)l(e)h(tr)l(ansform)f(is)g Fo( )s Ff(.)515 3911 y Fg(Theorem)g(7)41 b Ff(L)l(et)36 b Fp(f)p Fo(D)1270 3923 y Fn(n)1314 3911 y Fp(g)1356 3923 y Fn(n)p Fj(2)p Fc(N)1524 3911 y Ff(b)l(e)g(an)g(admissible)i (exhaustion)e(and)g Fp(f)p Fo(M)2870 3923 y Fn(n)2914 3911 y Fp(g)2956 3923 y Fn(n)p Fj(2)p Fc(N)3124 3911 y Ff(a)g(r)l(e)l(gu-)515 4011 y(larisation)31 b(of)g Fp(f)p Fo(D)1096 4023 y Fn(n)1140 4011 y Fp(g)p Ff(.)38 b(De\014ne)893 4194 y Fo(\036)942 4159 y Fn(!)942 4214 y(n)991 4194 y Fq(\()p Fo(\025)p Fq(\))24 b(:=)e(#)p Fp(f)30 b Ff(eigenvalues)h(of)f Fo(H)1986 4159 y Fn(!)1979 4214 y(M)2042 4222 y Fi(n)2111 4194 y Fp(\024)22 b Fo(\025)p Fp(g)p Fo(;)44 b(N)2431 4159 y Fn(!)2422 4214 y(n)2502 4194 y Fq(=)22 b Fp(j)p Fo(M)2693 4206 y Fn(n)2738 4194 y Fp(j)2761 4159 y Fj(\000)p Fh(1)2850 4194 y Fo(\036)2899 4159 y Fn(!)2899 4214 y(n)2978 4194 y Fo(:)230 b Fq(\(15\))515 4376 y Ff(Then,)31 b(for)f(almost)g(al)t(l)h Fo(!)26 b Fp(2)d Fq(\012)p Ff(,)30 b(the)g(limit)1585 4559 y Fo(N)9 b Fq(\()p Fo(\025)p Fq(\))24 b(=)52 b(lim)1885 4609 y Fn(n)p Fj(!1)2072 4559 y Fo(N)2148 4525 y Fn(!)2139 4579 y(n)2196 4559 y Fq(\()p Fo(\025)p Fq(\))923 b(\(16\))515 4771 y Ff(exists)23 b(at)h(al)t(l)h(c)l(ontinuity)f(p)l(oints)g(of)h Fo(N)9 b Fq(\()p Fp(\001)p Fq(\))p Ff(.)37 b(Mor)l(e)l(over,)27 b Fo(N)32 b Ff(is)25 b(a)f(non-ne)l(gative,)i(monotone)515 4870 y(incr)l(e)l(asing,)31 b(left-c)l(ontinuous)e(function)g(\(indep)l (endent)i(of)f Fo(!)s Ff(\).)1926 5255 y Fq(7)p eop %%Page: 8 8 8 7 bop 515 623 a Fg(Pro)s(of:)91 b Fq(W)-7 b(e)28 b(c)n(hec)n(k)1234 602 y(\024)1232 623 y(Subin's)g(h)n(yp)r(otheses:)639 722 y(1.)37 b Fo(\036)790 692 y Fn(!)790 743 y(n)838 722 y Fq(\()p Fo(\025)p Fq(\))24 b(=)f(0)k(for)g(all)h Fo(!)d Fp(2)f Fq(\012)p Fo(;)14 b(n)22 b Fp(2)i Fm(N)t Fq(,)34 b(if)28 b Fo(\025)23 b(<)g Fp(\000k)p Fo(V)2297 692 y Fn(!)2345 722 y Fp(k)2387 734 y Fj(1)2457 722 y Fq(.)639 822 y(2.)965 1015 y Fp(j)p Fo(M)1069 1027 y Fn(n)1114 1015 y Fp(j)1137 981 y Fj(\000)p Fh(1)1240 902 y Fb(Z)1337 1015 y Fo(e)1376 981 y Fj(\000)p Fn(t\025)1496 1015 y Fo(d\036)1588 981 y Fn(!)1588 1036 y(n)1637 1015 y Fq(\()p Fo(\025)p Fq(\))84 b(=)e Fp(j)p Fo(M)2084 1027 y Fn(n)2129 1015 y Fp(j)2152 981 y Fj(\000)p Fh(1)2375 937 y Fb(X)2255 1118 y Fn(\025)2294 1126 y Fi(n)2335 1118 y Fj(2)p Fn(\033)r Fh(\()p Fn(H)2504 1098 y Fi(!)2500 1138 y(M)2553 1151 y(j)2589 1118 y Fh(\))2629 1015 y Fo(e)2668 981 y Fj(\000)p Fn(t\025)2784 989 y Fi(n)1833 1291 y Fq(=)g Fp(j)p Fo(M)2084 1303 y Fn(n)2129 1291 y Fp(j)2152 1257 y Fj(\000)p Fh(1)2269 1291 y Fq(T)-7 b(r)27 b Fo(e)2421 1244 y Fj(\000)p Fn(tH)2556 1219 y Fi(!)2552 1261 y(M)2605 1274 y(j)1833 1458 y Fq(=)82 b Fp(j)p Fo(M)2084 1470 y Fn(n)2129 1458 y Fp(j)2152 1424 y Fj(\000)p Fh(1)2255 1345 y Fb(Z)2301 1534 y Fn(M)2364 1542 y Fi(n)2423 1458 y Fo(k)2469 1424 y Fn(!)2466 1478 y(M)2529 1486 y Fi(n)2575 1458 y Fq(\()p Fo(t;)14 b(x;)g(x)p Fq(\))p Fo(dx)1833 1688 y Fp(\024)82 b(j)p Fo(M)2084 1700 y Fn(n)2129 1688 y Fp(j)2152 1654 y Fj(\000)p Fh(1)2255 1575 y Fb(Z)2301 1764 y Fn(M)2364 1772 y Fi(n)2423 1688 y Fo(k)2469 1654 y Fn(!)2517 1688 y Fq(\()p Fo(t;)14 b(x;)g(x)p Fq(\))p Fo(dx)1833 1868 y Fp(\024)82 b Fo(C)6 b Fq(\()p Fo(t)p Fq(\))p Fo(:)639 2039 y Fq(3.)807 2233 y(lim)778 2283 y Fn(n)p Fj(!1)965 2233 y Fp(j)p Fo(M)1069 2245 y Fn(n)1114 2233 y Fp(j)1137 2198 y Fj(\000)p Fh(1)1240 2120 y Fb(Z)1337 2233 y Fo(e)1376 2198 y Fj(\000)p Fn(t\025)1496 2233 y Fo(d\036)1588 2198 y Fn(!)1588 2253 y(n)1637 2233 y Fq(\()p Fo(\025)p Fq(\))84 b(=)111 b(lim)1980 2283 y Fn(n)p Fj(!1)2168 2233 y Fp(j)p Fo(M)2272 2245 y Fn(n)2316 2233 y Fp(j)2339 2198 y Fj(\000)p Fh(1)2442 2120 y Fb(Z)2489 2308 y Fn(M)2552 2316 y Fi(n)2611 2233 y Fo(k)2657 2198 y Fn(!)2654 2253 y(M)2717 2261 y Fi(n)2762 2233 y Fq(\()p Fo(t;)14 b(x;)g(x)p Fq(\))p Fo(dx)1833 2463 y Fq(=)111 b(lim)1980 2513 y Fn(n)p Fj(!1)2168 2463 y Fp(j)p Fo(D)2260 2475 y Fn(n)2305 2463 y Fp(j)2328 2429 y Fj(\000)p Fh(1)2431 2350 y Fb(Z)2477 2539 y Fn(D)2531 2547 y Fi(n)2590 2463 y Fo(k)2636 2429 y Fn(!)2684 2463 y Fq(\()p Fo(t;)14 b(x;)g(x)p Fq(\))p Fo(dx)1833 2700 y Fq(=)82 b Fp(jF)8 b(j)2094 2666 y Fj(\000)p Fh(1)2197 2700 y Fm(E)2280 2583 y Fb(\024)2324 2587 y(Z)2370 2776 y Fj(F)2442 2700 y Fo(k)2488 2666 y Fj(\017)2526 2700 y Fq(\()p Fo(t;)14 b(x;)g(x)p Fq(\))p Fo(dx)2878 2583 y Fb(\025)2965 2700 y Fo(:)515 2917 y Fq(Here)27 b(w)n(e)g(used)h(the)g(Lemmata)f(4)g(and)h (5.)556 3017 y Fa(2)515 3290 y Fr(An)44 b(Example)515 3472 y Fq(W)-7 b(e)30 b(supply)g(a)f(non-ab)r(elian)g(example)g(of)h(a) g(pair)f(\()p Fo(X)r(;)14 b Fq(\000\))30 b(whic)n(h)f(satis\014es)g (all)h(necessary)515 3571 y(conditions)d(to)g(de\014ne)h(an)f(IDS)i(in) f(the)g(ab)r(o)n(v)n(e)e(manner.)639 3671 y(F)-7 b(or)26 b(an)n(y)g(Riemannian)h(manifold)g(with)g(a)f(p)r(olynomially)g(b)r (ounded)h(co)r(compact)f(dis-)515 3770 y(crete)21 b(group)g(of)g (isometries)g(an)h(admissible)f(exhaustion)g Fp(f)p Fo(D)2442 3782 y Fn(n)2487 3770 y Fp(g)2529 3782 y Fn(n)p Fj(2)p Fc(N)2682 3770 y Fq(can)g(b)r(e)i(constructed.)515 3870 y(In)31 b(particular,)h(this)g(is)f(the)h(case)e(for)h(nilp)r(oten)n(t) i(groups.)47 b(Th)n(us,)32 b(an)f(example)g(is)h(giv)n(en)515 3970 y(b)n(y)27 b(the)h(Heisen)n(b)r(erg)f(group)1295 4237 y Fo(X)i Fq(=)1481 4066 y Fb(8)1481 4141 y(<)1481 4290 y(:)1555 4070 y(0)1555 4219 y(@)1669 4136 y Fq(1)82 b Fo(x)i(y)1669 4236 y Fq(0)h(1)i Fo(z)1669 4335 y Fq(0)e(0)i(1)2009 4070 y Fb(1)2009 4219 y(A)2096 4237 y Fp(j)23 b Fo(x;)14 b(y)s(;)g(z)27 b Fp(2)c Fm(R)2505 4066 y Fb(9)2505 4141 y(=)2505 4290 y(;)3231 4237 y Fq(\(17\))515 4513 y(with)32 b(a)f(left-in)n(v)-5 b(arian)n(t)30 b(metric,)i(and)g(\000)d(=)g Fo(X)e Fp(\\)21 b Fo(M)9 b Fq(\(3)p Fo(;)14 b Fm(Z)o Fq(\).)43 b Fo(X)37 b Fq(is)32 b(di\013eomorphic)e(to)i Fm(R)3313 4483 y Fh(3)3356 4513 y Fq(,)515 4613 y(but)g(non-ab)r (elian.)47 b(Note)31 b(that)h(the)f(discrete)g(group)f(\000)i(acts,)f (b)n(y)g(isometries,)h(from)f(the)515 4712 y(left)d(on)f Fo(X)34 b Fq(and)28 b(that)g(the)g(quotien)n(t)f(\000)p Fp(n)p Fo(X)33 b Fq(is)28 b(compact.)639 4907 y(An)33 b(example)f(for)g(a)g(\000-ergo)r(dic)f(p)r(oten)n(tial)h(on)g(\()p Fo(X)r(;)14 b Fq(\000\))33 b(is)f(the)h(analogue)d(of)j(the)f(one)515 5006 y(whic)n(h)27 b(is)h(called)f(allo)n(y-t)n(yp)r(e)f(p)r(oten)n (tial)i(in)g(the)f(Euclidean)g(setting.)1926 5255 y(8)p eop %%Page: 9 9 9 8 bop 639 523 a Fq(Let)38 b Fo(u)h Fq(:)h Fo(X)46 b Fp(!)39 b Fm(R)44 b Fq(b)r(e)38 b(a)f(smo)r(oth)g(function)i(supp)r (orted)e(in)h Fp(F)8 b Fq(.)67 b(W)-7 b(e)37 b(c)n(ho)r(ose)g(\012)i(=) 515 560 y Fb(N)607 648 y Fn(\015)t Fj(2)p Fh(\000)749 623 y Fm(R)p Fq(,)33 b(equipp)r(ed)26 b(with)h(the)g(pro)r(duct)f (measure)39 b Fm(P)c Fq(:=)2386 560 y Fb(N)2478 648 y Fn(\015)t Fj(2)p Fh(\000)2621 623 y Fo(\026)p Fq(,)26 b(where)g Fo(\026)g Fq(is)g(a)g(prob-)515 722 y(abilit)n(y)c(measure)g (on)g Fm(R)p Fq(.)42 b(Consider)22 b(the)h(indep)r(enden)n(t,)h(iden)n (tically)f(distributed)g(random)515 822 y(v)-5 b(ariables)28 b Fo(\031)908 834 y Fn(\015)976 822 y Fq(:)e(\012)g Fp(!)g Fm(R)p Fo(;)14 b(\031)1358 834 y Fn(\015)1407 822 y Fq(\()p Fo(!)s Fq(\))26 b(:=)f Fo(!)1717 834 y Fn(\015)1760 822 y Fo(;)14 b(\015)30 b Fp(2)c Fq(\000)j(and)g(the)h(measure)e (preserving)g(transfor-)515 922 y(mations)f(\()p Fo(T)910 934 y Fn(\015)952 922 y Fq(\()p Fo(!)s Fq(\)\))1104 949 y Fn(\014)1172 922 y Fq(:=)22 b Fo(!)1334 934 y Fn(\015)t(\014)1417 922 y Fq(.)37 b(Then)28 b(the)g(p)r(oten)n(tial,)g(giv)n(en)e(b)n(y) 1528 1139 y Fo(V)1595 1105 y Fn(!)1643 1139 y Fq(\()p Fo(x)p Fq(\))e(:=)1891 1060 y Fb(X)1889 1238 y Fn(\015)t Fj(2)p Fh(\000)2027 1139 y Fo(!)2079 1151 y Fn(\015)2121 1139 y Fo(u)p Fq(\()p Fo(\015)5 b(x)p Fq(\))p Fo(;)880 b Fq(\(18\))515 1412 y(is)30 b(\000-ergo)r(dic)g(and)h(satis\014es)f (the)h(required)f(regularit)n(y)f(assumptions)h(\(3\))h(to)g(de\014ne)g (the)515 1511 y(IDS.)21 b(Actually)-7 b(,)23 b(it)e(seems)g(that)g(b)n (y)f(using)h(probabilistic)f(estimates)h(for)f(the)h(heat)g(k)n (ernels,)515 1611 y(instead,)f(these)f(regularit)n(y)d(assumptions)i (can)g(b)r(e)h(relaxed)f(to)g(the)h("natural"ones.)31 b(These)515 1711 y(are)23 b(form)n(ulated)h(in)h(terms)g(of)f(the)h (exp)r(ectation)g(of)f(some)g(lo)r(cal)h Fo(L)2613 1680 y Fn(p)2650 1711 y Fq(-norm)f(of)h Fo(V)3051 1680 y Fn(!)3099 1711 y Fq(,)g(cf.)36 b([9].)515 1985 y Fr(Discussion)515 2167 y Fq(The)g(IDS)h(is)f(a)g(quan)n(tit)n(y)g(related)f(to)h(the)h (restrictions)e(of)h(Sc)n(hr\177)-42 b(odinger)35 b(op)r(erators)f(in) 515 2267 y(a)k(familiy)g Fo(H)958 2236 y Fn(!)1006 2267 y Fo(;)14 b(!)44 b Fp(2)e Fq(\012,)f(to)d(suitable)h(exhaustions)e(of)i (a)f(non-compact)f(Riemannian)515 2366 y(manifold)21 b Fo(X)7 b Fq(.)34 b(It)22 b(should)f(b)r(e)h(p)r(ossible)f(to)g(use)g (the)h(same)e(functional)i(analytic)e(argumen)n(ts)515 2466 y(as)33 b(in)h(the)g(Euclidean)f(case)g(to)g(establish)h(the)g (equalit)n(y)f(of)g(its)h(p)r(oin)n(ts)g(of)f(increase)g(\(2\))515 2565 y(with)28 b(the)g(sp)r(ectrum)g(of)f Fo(H)1380 2535 y Fn(!)1428 2565 y Fq(.)639 2665 y(The)d(de\014nition)h(of)e(an)h (admissible)g(exhaustion)f(con)n(tains)g(assumptions)g(ab)r(out)h(b)r (oth)515 2765 y(the)f(sequence)f(of)g(subsets)g Fo(I)1399 2777 y Fn(n)1467 2765 y Fq(of)h(the)g(group)e(\000)i(as)e(w)n(ell)i(as) e(the)i(geometry)e(of)i(the)g(domains)515 2864 y Fo(D)584 2876 y Fn(n)629 2864 y Fq(.)36 b(It)27 b(w)n(ould)g(b)r(e)g(desirable)f (to)g(form)n(ulate)g(the)h(conditions)g(in)g(terms)f(of)h(either)g (group-)515 2964 y(theoretic)35 b(or)g(geometric)g(prop)r(erties)g (only)-7 b(.)61 b(Recen)n(tly)-7 b(,)38 b(w)n(e)d(observ)n(ed,)i(in)f (the)g(case)f(of)515 3064 y(a)c(p)r(olynomially)h(b)r(ounded)g(group)f (\000,)i(that)g(Theorem)e(7)g(is)h(also)f(v)-5 b(alid)32 b(for)g(a)g(sequence)515 3163 y Fo(M)596 3175 y Fn(n)665 3163 y Fp(\032)23 b Fo(X)35 b Fq(of)28 b(regularized)f(geo)r(desic)g (balls)h(with)g(increasing)f(radii)h(satisfying)f(condition)515 3263 y(\(P\).)j(Th)n(us)f(for)g(this)h(c)n(hoice)f(one)h(obtains)f(via) g(\(16\))g(a)h(non-random)e(IDS.)i(It)g(is)g(natural)515 3362 y(to)i(ask)g(for)g(more)g(general)f(classes)g(of)i(exhaustions)e (of)i Fo(X)7 b Fq(,)33 b(for)f(whic)n(h)g(the)h(normalised)515 3462 y(eigen)n(v)-5 b(alue)28 b(coun)n(ting)g(functions)i(con)n(v)n (erge)d(and)h(whether)h(they)h(giv)n(e)e(rise)g(to)h(the)h(same)515 3562 y(IDS.)639 3661 y(The)38 b(remark)e(in)i(Krengel)e([10)o(])i (after)f(the)h(statemen)n(t)f(of)h(T)-7 b(emp)r(elman's)38 b(theorem)515 3761 y(implies)c(the)h(uniqueness)e(of)i(the)f(IDS)h(in)f (the)h(follo)n(wing)e(sense.)56 b(Let)34 b Fo(I)2868 3773 y Fn(n)2914 3761 y Fo(;)14 b(n)33 b Fp(2)i Fm(N)44 b Fq(b)r(e)34 b(a)515 3861 y(sequence)19 b(of)g(subsets)g(of)g(\000,)i (not)f(necessarily)d(admissible)i(in)h(our)e(sense)h(and)g Fo(D)2992 3873 y Fn(n)3037 3861 y Fo(;)14 b(n)23 b Fp(2)g Fm(N)30 b Fq(as)515 3960 y(in)19 b(\(8\).)35 b(Assume)19 b(that)g(it)h(satis\014es)e(condition)h(\(P\))h(and)f(that)g(a)g (regularisation)e Fo(M)3086 3972 y Fn(n)3131 3960 y 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y(exhaustions)k(dep)r(ending)i(on)f(the)h(sto)r(c) n(hastic)f(pro)r(cess)f Fo(V)2362 4727 y Fn(!)2410 4757 y Fq(,)i(to)r(o,)f(the)h(con)n(v)n(ergence)d(\(16\))515 4857 y(of)i(the)g(normalised)e(eigen)n(v)-5 b(alue)26 b(coun)n(ting)h(functions)g(to)f(an)h(IDS)g(could)g(b)r(e)g(ac)n(hiev)n (ed)f(in)515 4956 y(more)h(general)f(geometric)g(situations.)1926 5255 y(9)p eop %%Page: 10 10 10 9 bop 639 523 a Fq(Finally)-7 b(,)29 b(it)f(should)g(b)r(e)h(p)r (ossible)f(to)g(study)g(an)g(ergo)r(dic)f(family)h(of)g(op)r(erators)f (where)515 623 y(higher)36 b(order)f(parts)h(of)h(the)g(op)r(erators)e (dep)r(end)j(on)e(the)h(random)f(parameter)g Fo(!)j Fq(and)515 722 y(to)30 b(carry)f(through)g(the)i(same)f(pro)r(cedure)f(as)g(in)i (this)g(pap)r(er.)44 b(Suc)n(h)31 b(a)e(situation)i(w)n(ould)515 822 y(o)r(ccur)25 b(if)h Fo(!)i Fq(w)n(ere)d(related)f(to)i(the)g(c)n (hange)e(of)i(a)f(magnetic)g(\014eld)h(or)e(to)i(the)g(c)n(hange)e(of)i (the)515 922 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b(R)l(eine)f(A)n(ngew.)h(Math.)p Fq(,)g(323:53{67,)23 b(1981.)556 2557 y([4])41 b(V.)f(Bonc)n(h-Bruevic) n(h,)g(R.)f(Enderlein,)i(B.)e(Esser,)h(R.)g(Keip)r(er,)h(A.)e(Mirno)n (v,)i(and)685 2656 y(I.)23 b(Zyv)-5 b(agin.)27 b Ff(Elektr)l(onenthe)l (orie)f(unge)l(or)l(dneter)e(Halbleiter)p Fq(.)29 b(VEB)22 b(Deutsc)n(her)g(V)-7 b(er-)685 2756 y(lag)27 b(der)g(Wissensc)n (haften,)h(1984.)35 b(Russisc)n(hes)26 b(Original:)36 b(Mosk)-5 b(au,)26 b(Nauk)-5 b(a,1981.)556 2922 y([5])41 b(R.)30 b(Carmona)e(and)h(J.)h(Lacroix.)41 b Ff(Sp)l(e)l(ctr)l(al)31 b(The)l(ory)i(of)f(R)l(andom)g(Schr\177)-42 b(odinger)33 b(Op-)685 3022 y(er)l(ators)p Fq(.)38 b(Birkh\177)-42 b(auser,)25 b(Boston,)i(1990.)556 3188 y([6])41 b(J.)22 b(Do)r(dziuk.)28 b(Maxim)n(um)22 b(principle)g(for)g(parab)r(olic)f (inequalities)g(and)h(the)h(heat)f(\015o)n(w)685 3287 y(on)35 b(op)r(en)g(manifolds.)57 b Ff(Indiana)38 b(University)e (Mathematics)i(Journal)p Fq(,)f(32\(5\):704{)685 3387 y(716,)27 b(1983.)556 3553 y([7])41 b(H.)e(Englisc)n(h)e(and)i(K.-D.)f (K)r(\177)-44 b(ursten.)69 b(In\014nite)39 b(represen)n(tabilit)n(y)d (of)j(Sc)n(hr\177)-42 b(odinger)685 3653 y(op)r(erators)37 b(with)h(ergo)r(dic)g(p)r(oten)n(tial.)68 b Ff(Z.)40 b(A)n(nal.)g(A)n(nwendungen)p Fq(,)g(3\(4\):357{366,)685 3752 y(1984.)556 3918 y([8])h(W.)31 b(Kirsc)n(h.)43 b(Random)30 b(Sc)n(hr\177)-42 b(odinger)29 b(op)r(erators.)42 b(In)31 b(H.)f(Holden)h(and)f(A.)g(Jensen,)685 4018 y(editors,)20 b Ff(Schr\177)-42 b(odinger)23 b(Op)l(er)l(ators)p Fq(,)e(Lecture)d (Notes)g(in)h(Ph)n(ysics,)g Fg(345)p Fq(,)h(Berlin,)g(1989.)685 4118 y(Springer.)556 4284 y([9])41 b(W.)27 b(Kirsc)n(h)d(and)i(F.)g (Martinelli.)34 b(On)25 b(the)h(densit)n(y)g(of)g(states)f(of)h(Sc)n (hr\177)-42 b(odinger)24 b(op)r(er-)685 4383 y(ators)j(with)h(a)f (random)g(p)r(oten)n(tial.)36 b Ff(J.)30 b(Phys.)p Fq(,)f(A)f (15:2139{2156,)23 b(1982.)515 4549 y([10])40 b(U.)21 b(Krengel.)j Ff(Er)l(go)l(dic)h(The)l(or)l(ems)p Fq(.)i(Studies)21 b(in)f(Mathematics.)h(W)-7 b(alter)20 b(de)h(Gruyter,)685 4649 y(1985.)35 b(With)29 b(a)e(Supplemen)n(t)h(b)n(y)g(An)n(toine)f (Brunel.)515 4815 y([11])40 b(P)-7 b(.)28 b(Li)g(and)g(S.)h(T.)f(Y)-7 b(au.)38 b(On)28 b(the)h(parab)r(olic)e(k)n(ernel)g(of)h(the)g(Sc)n (hr\177)-42 b(odinger)27 b(op)r(erator.)685 4915 y Ff(A)l(cta)j (Mathematic)l(a)p Fq(,)f(156:153{201,)23 b(1986.)1905 5255 y(10)p eop %%Page: 11 11 11 10 bop 515 523 a Fq([12])40 b(W.)21 b(L)r(\177)-44 b(uc)n(k)19 b(and)g(T.)h(Sc)n(hic)n(k.)k Fo(L)1599 493 y Fh(2)1635 523 y Fq(-T)-7 b(orsion)18 b(of)i(h)n(yp)r(erb)r(olic)f (manifolds)h(of)f(\014nite)i(v)n(olume.)685 623 y(preprin)n(t)27 b(do)n(wnloaded)g(from)g(M)r(\177)-44 b(unster)27 b(homepage,)g(1999.) 515 789 y([13])40 b(L.)32 b(A.)h(P)n(astur)d(and)i(A.)h(L.)f(Figotin.) 50 b Ff(Sp)l(e)l(ctr)l(a)33 b(of)i(R)l(andom)f(and)g(A)n(lmost-Perio)l (dic)685 888 y(Op)l(er)l(ators)p Fq(.)j(Springer)27 b(V)-7 b(erlag,)27 b(Berlin,)g(1992.)515 1054 y([14])40 b(D.)30 b(Ra)n(y)f(and)g(I.)h(Singer.)41 b(R-T)-7 b(orsion)28 b(and)h(the)h(Laplacian)e(on)h(Riemannian)g(mani-)685 1154 y(folds.)37 b Ff(A)l(dv.Math.)p Fq(,)30 b(7:145{210,)24 b(1971.)515 1320 y([15])40 b(M.)21 b(A.)914 1299 y(\024)912 1320 y(Subin.)26 b(Densit)n(y)21 b(of)g(states)f(of)h(self)g(adjoin)n (t)g(op)r(erators)e(with)i(almost)f(p)r(erio)r(dic)685 1420 y(co)r(e\016cien)n(ts.)37 b Ff(A)n(mer.)30 b(Math.)h(So)l(c.)f(T) -6 b(r)l(anslations)p Fq(,)28 b(118:307{339,)23 b(1982.)515 1586 y([16])40 b(A.-S.)e(Sznitman.)68 b(Lifsc)n(hitz)38 b(tail)g(and)f(Wiener)h(sausage)e(on)h(h)n(yp)r(erb)r(olic)h(space.)685 1685 y Ff(Comm.)31 b(Pur)l(e)f(Appl.)h(Math.)p Fq(,)e (42\(8\):1033{1065,)22 b(1989.)515 1851 y([17])40 b(A.-S.)33 b(Sznitman.)53 b(Lifsc)n(hitz)33 b(tail)g(on)f(h)n(yp)r(erb)r(olic)h (space:)46 b(Neumann)33 b(conditions.)685 1951 y Ff(Comm.)e(Pur)l(e)f (Appl.)h(Math.)p Fq(,)e(43\(1\):1{30,)c(1990.)515 2117 y([18])40 b(M.)27 b(E.)f(T)-7 b(a)n(ylor.)33 b Ff(Partial)d(di\013er)l (ential)g(e)l(quations.)f(I)p Fq(.)35 b(Springer-V)-7 b(erlag,)24 b(New)j(Y)-7 b(ork,)685 2217 y(1996.)35 b(Basic)27 b(theory)-7 b(.)515 2383 y([19])40 b(A.)35 b(T)-7 b(emp)r(elman.)57 b(Ergo)r(dic)33 b(theorems)h(for)g(general)e(dynamical)i(sytems.)57 b Ff(T)-6 b(r)l(ans.)685 2482 y(Mosc)l(ow)31 b(Math.)h(So)l(c.)p Fq(,)c(26:95{132,)c(1972.)35 b(class.)26 b(28A65.)515 2648 y([20])40 b(A.)h(T)-7 b(emp)r(elman.)74 b Ff(Er)l(go)l(dic)43 b(the)l(or)l(ems)e(for)i(gr)l(oup)e(actions.)i(Informational)f(and)685 2748 y(thermo)l(dynamic)l(al)35 b(asp)l(e)l(cts)p Fq(,)d(v)n(olume)e (78)g(of)g Ff(Mathematics)35 b(and)e(its)f(Applic)l(ations)p Fq(.)685 2848 y(Klu)n(w)n(er)27 b(Academic)g(Publishers,)g(1992.)515 3014 y([21])40 b(J.)45 b(W)-7 b(eidmann.)88 b Ff(Line)l(ar)l(e)46 b(Op)l(er)l(ator)l(en)f(in)g(Hilb)l(ertr\177)-42 b(aumen)p Fq(.)88 b(B.)44 b(G.)h(T)-7 b(eubner,)685 3113 y(Stuttgart,)28 b(1976.)35 b(Mathematisc)n(he)27 b(Leitf\177)-42 b(aden.)515 3279 y([22])40 b(J.)c(W)-7 b(eidmann.)62 b Ff(Line)l(ar)38 b(op)l(er)l(ators)g(in)g(Hilb)l(ert)f(sp)l(ac)l(es)p Fq(.)62 b(Springer-V)-7 b(erlag,)36 b(New)685 3379 y(Y)-7 b(ork,)27 b(1980.)35 b(T)-7 b(ranslated)27 b(from)g(the)h(German)f(b)n (y)g(Joseph)g(Sz)r(\177)-44 b(ucs.)1905 5255 y(11)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0009010838161--