What is the "category of bifurcations"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T03:14:12Z http://mathoverflow.net/feeds/question/12272 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12272/what-is-the-category-of-bifurcations What is the "category of bifurcations"? Darsh Ranjan 2010-01-19T03:30:55Z 2010-01-19T03:30:55Z <p>While reading the introduction to <a href="http://www.math.harvard.edu/~ctm/papers/home/text/papers/muniv/muniv.pdf" rel="nofollow">this paper by Curtis McMullen</a>, I came to the following (bold added): </p> <blockquote> <p>In this paper we show that <i>every</i> bifurcation set contains a copy of the boundary of the Mandelbrot set or its degree $d$ generalization. The Mandelbrot sets $M_d$ are thus <i>universal</i>; they are <b>initial objects in the category of bifurcations</b>, providing a lower bound on the complexity of $B(f)$ for all families $f_t$. </p> </blockquote> <p>(Here $f_t$ is a family of rational functions mapping $P^1$ to itself and depending holomorphically on a parameter $t$ ranging over some complex manifold, and the bifurcation set (or locus) $B(f)$ is the set of $t$ at which the dynamics of $f_t$ undergo a discontinuous change. For a positive integer $d\geq 2$, $M_d$ is the set of $t$ for which the function $z\mapsto z^d+t$ has a connected Julia set, and $\partial M_d$, the boundary of $M_d$, is the bifurcation locus for the family $\{z\mapsto z^d+t\}_t$.)</p> <p>Even without knowing exactly what the "category of bifurcations" is, I can see an analogy between the claim about Mandelbrot sets and the concept of initial objects, but presumably something more definite is intended. My question is thus how to interpret the second sentence quoted from McMullen's paper, or more simply: what is the "category of bifurcations"? </p>