Finding f such that f(f(x))=g(x) given g - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T18:45:30Z http://mathoverflow.net/feeds/question/66538 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66538/finding-f-such-that-ffxgx-given-g Finding f such that f(f(x))=g(x) given g David Corwin 2011-05-31T12:13:25Z 2011-05-31T12:35:42Z <p>Suppose \$g(x)\$ is a smooth increasing function defined for \$x \ge 0\$ such that \$g(x) \ge x\$ for all \$x\$. Does there exist a function \$f\$ with similar properties such that \$f(f(x))=g(x)\$ for all \$x \ge 0\$? (You can interpret "similar" as widely as you'd like - smoothness would be great, but even continuity would be nice)</p> <p>I asked the question given these conditions on \$g\$ since it seems reasonable that they would produce a positive answer. However, I'm just as interested in the same question for more general classes of \$g\$. For example, suppose we only assume \$g\$ is continuous, or even measurable - can we find an \$f\$ with the same properties? And let's suppose we relax the requirement \$g(x) \ge x\$, etc (I included that because it helps ensure the existence of a set-theoretic \$f\$).</p> <p>Under the given conditions, how many such \$f\$ exist?</p> <p>I'm not entirely what the tag should be, so please feel free to edit it.</p>