Absolutely Continuous Invariant Measures for Piecewise Convex Maps - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T00:36:18Z http://mathoverflow.net/feeds/question/114059 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114059/absolutely-continuous-invariant-measures-for-piecewise-convex-maps Absolutely Continuous Invariant Measures for Piecewise Convex Maps Tom Kempton 2012-11-21T14:06:00Z 2012-12-03T21:29:34Z <p>Hi all,</p> <p>I'm interested in a class of 'generalised tent maps' $f:[0,1]\to[0,1]$ for which</p> <p>1) $f$ is strictly increasing on $[0, \frac{1}{2}]$, $f(0)=0$ and $f(\frac{1}{2})=1$</p> <p>2) $f$ is symmetric about $\frac{1}{2}$, i.e. $f(x)=f(1-x)$.</p> <p>3) $f$ is differentiable at 0 with $f'(0)>1$</p> <p>4) $f$ is piecewise convex, but not strictly convex, on pieces $[0,1/2]$ and $[\frac{1}{2},1]$</p> <p>5) $f$ is continuous.</p> <p>Is it known that such functions f preserve absolutely continuous invariant probability measures? </p> <p>I've seen a few papers proving the existence of acips for certain classes of piecewise convex functions, such as Lasota and Yorke (Trans AMS, 1982) and Bose et al (Studia Math 2003), but they always require that the function f is increasing on each of the pieces, which doesn't hold for the tent like constructions I'm interested in.</p> <p>Thanks,</p> <p>Tom</p> http://mathoverflow.net/questions/114059/absolutely-continuous-invariant-measures-for-piecewise-convex-maps/115337#115337 Answer by Banach for Absolutely Continuous Invariant Measures for Piecewise Convex Maps Banach 2012-12-03T21:22:43Z 2012-12-03T21:29:34Z <p>If $f$ is piecewise $C^2$, since your map is piecewise expanding it has an ACIM. See Lasota and York (Trans AMS, 1973). </p> <p>See also the <em><a href="http://books.google.it/books?id=2xsb6iveF9QC&amp;lpg=PA167&amp;ots=c7HDvka-L-&amp;dq=chaos%2520fractals%2520and%2520noise%2520asymptotically%2520stable%25206.2.2&amp;pg=PA148#v=onepage&amp;q=theorem%25206.2.2&amp;f=false" rel="nofollow">Chaos, Fractals, and Noise</a></em> by A. Lasota and M. Mackey. One of the theorems in Ch. 6 might work for your maps.</p>