  In this paper we study the transition to chaos for the restriction to the
real and imaginary axes of the tangent family $\{ T_t(z)=i t\tan z\}_{0< t\leq
\pi}$. Because tangent maps have no critical points but have an essential
singularity at infinity and two symmetric asymptotic values, there are new
phenomena: as $t$ increases we find single instances of "period quadrupling",
"period splitting" and standard "period doubling"; there follows a general
pattern of "period merging" where two attracting cycles of period $2^n$ "merge"
into one attracting cycle of period $2^{n+1}$, and "cycle doubling" where an
attracting cycle of period $2^{n+1}$ "becomes" two attracting cycles of the
same period.
  We use renormalization to prove the existence of these bifurcation
parameters. The uniqueness of the cycle doubling and cycle merging parameters
is quite subtle and requires a new approach. To prove the cycle doubling and
merging parameters are, indeed, unique, we apply the concept of "holomorphic
motions" to our context.
  In addition, we prove that there is an "infinitely renormalizable" tangent
map $T_{t_\infty}$. It has no attracting or parabolic cycles. Instead, it has a
strange attractor contained in the real and imaginary axes which is forward
invariant and minimal under $T^2_{t_\infty}$. The intersection of this strange
attractor with the real line consists of two binary Cantor sets and the
intersection with the imaginary line is totally disconnected, perfect and
unbounded.
