  A three-stranded DNA with short range base pairings only is known to exhibit
a classical analog of the quantum Efimov effect, viz., a three chain bound
state at the two chain melting point where no two are bound. By using a
non-perturbative renormalization group method for a rigid duplex DNA and a
flexible third strand, with base pairings and strand exchange, we show that the
Efimov-DNA is associated with a limit cycle type behavior of the flow of an
effective three chain interaction. The analysis also shows that thermally
generated bubbles play an essential role in producing the effect. A toy model
for the flow equations shows the limit cycle in an extended three dimensional
parameter space of the two-chain coupling and a complex three chain
interaction.
