In this talk, based on joint work with Tao Chen and Yunping Jiang, we study the transition to chaos for the restriction to the real and imaginary axes of the tangent family $z \mapsto it\tan z$. 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, in addition to standard "period doubling", we find "period merging" where two attracting cycles of period $2n$ "merge" into one attracting cycle of period $2n+1$, and "cycle doubling" where an attracting cycle of period $2n+1$ "becomes" two attracting cycles of the same period. Describing these new phenomena involves adapting the concepts of "renormalization" and "holomorphic motions" to our context. The parameters where these bifurcations occur limit at an "infinitely renormalizable" tangent map with a "strange attractor" that has a Cantor set structure. Speaker(s): Linda Keen (CUNY)