Absolutely Continuous Invariant Measures for Piecewise Convex Maps

2012-11-21T14:06:00Z

Hi all,

I'm interested in a class of 'generalised tent maps' $f:[0,1]\to[0,1]$ for which

1) $f$ is strictly increasing on $[0, \frac{1}{2}]$, $f(0)=0$ and $f(\frac{1}{2})=1$

2) $f$ is symmetric about $\frac{1}{2}$, i.e. $f(x)=f(1-x)$.

3) $f$ is differentiable at 0 with $f'(0)>1$

4) $f$ is piecewise convex, but not strictly convex, on pieces $[0,1/2]$ and $[\frac{1}{2},1]$

5) $f$ is continuous.

Is it known that such functions f preserve absolutely continuous invariant probability measures?

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.

Thanks,

Tom

Answer by Banach for Absolutely Continuous Invariant Measures for Piecewise Convex Maps

2012-12-03T21:22:43Z

If $f$ is piecewise $C^2$, since your map is piecewise expanding it has an ACIM. See Lasota and York (Trans AMS, 1973).

See also the Chaos, Fractals, and Noise by A. Lasota and M. Mackey. One of the theorems in Ch. 6 might work for your maps.

Absolutely Continuous Invariant Measures for Piecewise Convex Maps - MathOverflow

Absolutely Continuous Invariant Measures for Piecewise Convex Maps

Answer by Banach for Absolutely Continuous Invariant Measures for Piecewise Convex Maps