\relax 
\ifx\hyper@anchor\@undefined
\global \let \oldcontentsline\contentsline
\gdef \contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global \let \oldnewlabel\newlabel
\gdef \newlabel#1#2{\newlabelxx{#1}#2}
\gdef \newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\let \contentsline\oldcontentsline
\let \newlabel\oldnewlabel}
\else
\global \let \hyper@last\relax 
\fi

\citation{Hof79}
\citation{G86}
\citation{Mal91}
\citation{Cnly89}
\citation{T92}
\citation{DGNW}
\citation{JR05}
\citation{CCT}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}}
\citation{CCT}
\citation{Cnly89}
\citation{Mal91}
\citation{T92}
\citation{Miles}
\newlabel{defn: vrmFs}{{1.1}{2}{Introduction\relax }{thm.1.1}{}}
\citation{E01}
\citation{E05}
\citation{BH92}
\citation{Cnly89}
\@writefile{toc}{\contentsline {section}{\numberline {2}Examples}{4}{section.2}}
\newlabel{sect: Examples}{{2}{4}{Examples\relax }{section.2}{}}
\newlabel{eg: b(n) max growth}{{2.1}{4}{Examples\relax }{thm.2.1}{}}
\newlabel{eg: short growth = 3}{{2.2}{5}{Examples\relax }{thm.2.2}{}}
\newlabel{eg: r(n) = 2,3... }{{2.4}{5}{Examples\relax }{thm.2.4}{}}
\citation{Hof79}
\citation{Hof79}
\citation{G86}
\newlabel{eg: exp growth}{{2.6}{6}{Examples\relax }{thm.2.6}{}}
\newlabel{eg: lin growth}{{2.7}{6}{Examples\relax }{thm.2.7}{}}
\newlabel{eg: log growth}{{2.8}{6}{Examples\relax }{thm.2.8}{}}
\citation{T92}
\citation{T92}
\citation{Miles}
\citation{CCT}
\citation{T92}
\citation{HT93b}
\citation{JR05}
\citation{CCT}
\citation{DGNW}
\citation{DGNW}
\@writefile{toc}{\contentsline {section}{\numberline {3}Asymptotic Growth}{8}{section.3}}
\newlabel{sect: Asymptotics}{{3}{8}{Asymptotic Growth\relax }{section.3}{}}
\newlabel{lem: Delta r(n) = 1 -> growth =2}{{3.3}{9}{Asymptotic Growth\relax }{thm.3.3}{}}
\newlabel{prop: max growth 2 n-1}{{3.4}{9}{Asymptotic Growth\relax }{thm.3.4}{}}
\citation{Miles}
\newlabel{prop: exp growth}{{3.5}{10}{Asymptotic Growth\relax }{thm.3.5}{}}
\newlabel{lem: b(n) growth}{{3.7}{10}{Asymptotic Growth\relax }{thm.3.7}{}}
\newlabel{thm: growth, case r(n) = 1}{{a}{10}{Asymptotic Growth\relax }{Item.1}{}}
\newlabel{thm: growth, case Delta < 1}{{b}{10}{Asymptotic Growth\relax }{Item.2}{}}
\newlabel{thm: growth, case Delta = 1}{{c}{10}{Asymptotic Growth\relax }{Item.3}{}}
\newlabel{thm: growth, case Delta > 1}{{d}{10}{Asymptotic Growth\relax }{Item.4}{}}
\newlabel{cor: lim sup growth < 2 -> r(n) const}{{3.8}{10}{Asymptotic Growth\relax }{thm.3.8}{}}
\newlabel{lem: lim inf growth <= 2}{{3.9}{11}{Asymptotic Growth\relax }{thm.3.9}{}}
\newlabel{thm: sigma approx}{{1}{11}{Asymptotic Growth\relax }{bigthm.1}{}}
\newlabel{cor: sigma models}{{3.10}{12}{Asymptotic Growth\relax }{thm.3.10}{}}
\newlabel{lem: r(n) econst => r-bon}{{3.12}{12}{Asymptotic Growth\relax }{thm.3.12}{}}
\newlabel{lem: b(n)/b(n-1) = 2}{{3.13}{12}{Asymptotic Growth\relax }{thm.3.13}{}}
\citation{S74}
\newlabel{thm: growth restrictions}{{2}{13}{Asymptotic Growth\relax }{bigthm.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {4}Estimates on Growth}{13}{section.4}}
\newlabel{sect: Estimates}{{4}{13}{Estimates on Growth\relax }{section.4}{}}
\citation{Du89}
\newlabel{eq: b(n)/b(n-1)}{{1}{14}{Estimates on Growth\relax }{equation.1}{}}
\newlabel{lem: swap bounds}{{4.1}{14}{Estimates on Growth\relax }{thm.4.1}{}}
\citation{Du89}
\newlabel{lem: b(n) <= r(n)}{{4.2}{15}{Estimates on Growth\relax }{thm.4.2}{}}
\newlabel{lem: b(n)/b(n-1) geq 1 + 1/r(n-1)}{{4.3}{15}{Estimates on Growth\relax }{thm.4.3}{}}
\newlabel{lem: b(n)/b(n-1) > 1 +1/r(n-1)}{{4.3}{15}{Estimates on Growth\relax }{thm.4.3}{}}
\citation{CN70}
\citation{K83}
\citation{CN70}
\newlabel{thm: upper/lower bounds}{{4.6}{16}{Estimates on Growth\relax }{thm.4.6}{}}
\newlabel{eq: NZC recursion}{{3}{16}{Estimates on Growth\relax }{equation.3}{}}
\newlabel{eq: NZC est}{{4}{16}{Estimates on Growth\relax }{equation.4}{}}
\newlabel{lem: lower bound for NZC ratio}{{4.8}{17}{Estimates on Growth\relax }{thm.4.8}{}}
\citation{E01}
\citation{E05}
\@writefile{toc}{\contentsline {section}{\numberline {5}A Generalization}{18}{section.5}}
\newlabel{sect: Generalization}{{5}{18}{A Generalization\relax }{section.5}{}}
\newlabel{defn: ex vrmFs}{{5.1}{18}{A Generalization\relax }{thm.5.1}{}}
\newlabel{eg: dyn seq}{{5.2}{18}{A Generalization\relax }{thm.5.2}{}}
\bibcite{BH92}{1}
\newlabel{prp: ex vrmFs , r(n)=R n < 1}{{5.4}{19}{A Generalization\relax }{thm.5.4}{}}
\newlabel{eq: beta(n) for n < 0}{{5}{19}{A Generalization\relax }{equation.5}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6}Acknowledgements}{19}{section.6}}
\bibcite{CCT}{2}
\bibcite{CN70}{3}
\bibcite{Cnly89}{4}
\bibcite{DGNW}{5}
\bibcite{Du89}{6}
\bibcite{E01}{7}
\bibcite{E05}{8}
\bibcite{G86}{9}
\bibcite{Hof79}{10}
\bibcite{HT93b}{11}
\bibcite{JR05}{12}
\bibcite{K83}{13}
\bibcite{Mal91}{14}
\bibcite{Miles}{15}
\bibcite{S74}{16}
\bibcite{T92}{17}