Stability of smooth extensions of Bernoulli shifts

Zbigniew S. Kowalski¹

¹ Faculty of Pure and Applied Mathematics Wrocław University of Science and Technology 50-370 Wrocław, Poland

Applicationes Mathematicae, Tome 44 (2017) no. 1, pp. 85-104

Résumé

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.

Détail

DOI : 10.4064/am2298-2-2017

Keywords: homeomorphisms epsilon epsilon reals epsilon epsilon here homeomorphism varomega mathcal sigma one sided bernoulli shift where varomega mathbb measure space varomega times define skew product omega sigma omega omega class distribution functions probability measures epsilon epsilon epsilon epsilon epsilon sufficient conditions existence exactly pair homeomorphisms above times s invariant here measure determined example consequence above x every sqrt possesses ergodic invariant measure times which kind sinai ruelle bowen measure apply above results quantum harmonic oscillator binomial model asset prices