Stability of smooth extensions of Bernoulli shifts
Volume 44 / 2017
Abstract
Let $ S_i,\,i=0,1$, be homeomorphisms of $I=[0,1]$ such that $S_i^{-1}(x)=(1-\epsilon _i)x+\epsilon _ig(x)$, $i=0,1$, for some reals $\epsilon _0 \lt 0$ and $\epsilon _1 \gt 0.$ Here $g$ is a $C^1(0,1)$ homeomorphism and $g(x) \lt x$ for $x\in (0,1).$ Let $(\varOmega ,\mathcal B,\mu _p,\sigma )$ be the one-sided Bernoulli shift where $\varOmega =\{0,1\}^{\mathbb {N}}$ and $\mu _p$ is the $(p,q)$ measure for some $p\in I.$ In the space $\varOmega \times I$ we define the skew product $S(\omega ,x)=(\sigma (\omega ),S_{\omega (0)}(x)) .$ For some class of distribution functions $F \in C^2(0,1)$ of probability measures and all $\epsilon _0 \lt 0$, $\epsilon _1 \gt 0 ,$ and $p\in ({\epsilon _1/(\epsilon _1-\epsilon _0)},1)$, we give sufficient conditions for existence of exactly one pair of homeomorphisms as above such that $\mu _p\times \mu _F$ is $S$-invariant. Here $\mu _F$ is the measure determined by $F.$ For example, as a consequence of the above, we show that if $S_0^{-1}(x)=1.307x-0.307x^2$ and $S_1^{-1}(x)=0.26x+0.74x^2 ,$ then for every $p\in [0.706781,{\sqrt {2}/2})$, $S$ possesses ergodic invariant measure $\mu _p\times \mu _{G_p}$ which is a kind of Sinai–Ruelle–Bowen measure. We apply the above results to the quantum harmonic oscillator and a binomial model for asset prices.
