Fractional powers of Dehn twists about nonseparating curves

Let $S_g$ be a closed orientable surface of genus $g\ge 2$ and $C$ a simple closed nonseparating curve in $F$. Let $t_C$ denote a left handed Dehn twist about $C$. A $\mathit{\text{fractional}}\mathit{\text{power}}$ of $t_C$ of $\mathit{\text{exponent}}$ $\text{fraction}\mathrm{\ell }n$ is an $h\in \text{Mod}(S_g)$ such that $h^n=t_C^{\mathrm{\ell }}$. Unlike a root of a $t_C$, a fractional power $h$ can exchange the sides of $C$. We derive necessary and sufficient conditions for the existence of both side-exchanging and side-preserving fractional powers. We show in the side-preserving case that if $\gcd (\mathrm{\ell },n)=1$, then $h$ will be isotopic to the ${\mathrm{\ell }}^{th}$ power of an $n^{th}$ root of $t_C$ and that $n\le 2g+1$. In general, we show that $n\le 4g$, and that side-preserving fractional powers of exponents $\text{fraction}2g2g+2$ and $\text{fraction}2g4g$ always exist. For a side-exchanging fractional power of exponent $\text{fraction}\mathrm{\ell }2n$, we show that $2n\ge 2g+2$, and that side-exchanging fractional powers of exponent $\text{fraction}2g+24g+2$ and $\text{fraction}4g+14g+2$ always exist. We give a complete listing of certain side-preserving and side-exchanging fractional powers on $S_5$.