Ergodicity of certain cylinder flows
Let \(T\) be an ergodic transformation on the torus \(\mathbb{T}\) that preserves the normalized Haar measure \(\lambda\). Let \(f:\mathbb{T}\to\mathbb{R}\) be a Borel measurable function. Suppose further that the closed subgroup of \(\mathbb{R}\) generated by the values of \(f\) equals \(\mathbb{R}\). Then it makes sense to define the skew product extension \(T_ f\) of \(T\) by \(f\) on the product space \(\mathbb{T}\times\mathbb{R}\): \[ T_ f:\mathbb{T}\times\mathbb{R}\to\mathbb{T}\times\mathbb{R},\quad T_ f(x,y):=(Tx,y+f(x)). \] Denote the measure theoretic product of \(\lambda\) with Lebesgue measure \(\lambda_ 1\) on \(\mathbb{R}\) by \(\mu\). Ergodicity of the transformation \(T_ f\) with respect to \(\mu\) obviously depends on the interplay between \(f\) and \(T\). A necessary condition for ergodicity is that \(f\) has integral zero. Define \(f_ n(x):=\sum^{n-1}_{k=0}f(T^ kx)\) \((x\in\mathbb{T},n\in\mathbb{N})\). Then the \(n\)-th iterate of \(T_ f\) has the form \(T^ n_ f(x,y)=(T^ nx,y+f_ n(x))\). Boundedness of the sequence \(f_ n\) prohibits \(T_ f\) from being ergodic. In this case, \(f\) is a coboundary. For this notion and a survey of related results see \textit{P. Liardet} [Compos. Math. 61, 267-293 (1987; Zbl 0619.10053)]. Let \(T\) be the irrational rotation of \(\mathbb{T}\) defined by \(\alpha\), \(Tx:=x+\alpha\;\bmod 1\). The author studies the set \[ {\mathcal E}(f):=\{\alpha\text{ irrational : }T_ f\text{ is ergodic}\} \] for certain classes of differentiable functions \(f\). If \(f\) is in \(C^ 1[0,1]\), with \(f(0)\neq f(1)\), then, by \textit{P. Hellekalek} and \textit{G. Larcher} [Isr. J. Math. 54, 301-306 (1986; Zbl 0609.28007)], \({\mathcal E}(f)\) is equal to the set of all irrational numbers. This result was generalized by the author [Isr. J. Math. 69, No. 1, 65-74 (1990; Zbl 0703.28009)] to piecewise absolutely continuous functions \(f\) with Riemann-integrable derivative \(f'\) with nonzero integral. In the case \(f\in C^ 1[0,1]\), \(f(0)=f(1)\), \(f'\in Lip^ 1[0,1]\), it is known that \(\lambda_ 1({\mathcal E}(f))=0\) [see \textit{P. Hellekalek} and \textit{G. Larcher}, Theor. Comput. Sci. 65, No. 2, 189-196 (1989; Zbl 0674.28008)]. In this paper, the author proves that, for \(r\)-times differentiable functions \(f\), \(r>1\), where \(f^{(r-2)}\) is continuous with zero integral, \(f^{(r-1)}\) is piecewise continuous with zero integral and \(f^{(r)}\) is Riemann integrable with nonzero integral, we have \[ {\mathcal E}(f)\supseteq\{\alpha:\limsup{a_{i+1}\over q^ r_ i}>0\}. \] Here \(a_ i\) denotes the \(i\)-th partial quotient and \(q_ i\) the denominator of the \(i\)-th convergent to \(\alpha\). For von Neumann-Kakutani adding machine transformations \(T\) on \(\mathbb{T}\), the author shows two results: (i) If \(f\) is piecewise continuous and if \(f'\) is Riemann integrable with nonzero integral, then \(T_ f\) is ergodic; (ii) If \(f'\) is of bounded variation, then \(\sup_ n| f_ n(x)|<\infty\) for all \(x\in\mathbb{T}\), for a certain class of adding machines. Hence \(f\) defines an \(L^ \infty\)-coboundary in this case. The method of proof is an exemplary application of K. Schmidt's concept of essential values of cocycles [see \textit{K. Schmidt}: ``Cocycles of ergodic transformation groups'' (1977; Zbl 0421.28017)]. The author shows that the set of essential values of the cocycle defined by \(f\) equals \(\mathbb{R}\). This implies the ergodicity of the skew product transformation \(T_ f\).
von Neumann-Kakutani adding machine transformations
scientific article; zbMATH DE number 78198
scientific article