Skew products arising from ergodic transformations

Abstract

In this thesis, we are concerned with the skew products of ergodic transformations. Investigating the ergodic properties of skew products is equivalent to investigating the existence of solution of the corresponding coboundary equation. The main purpose of this thesis is to investigate the solvability of coboundary equations and to generalize the Borel``s normal number theorem. In Chapter 3, on the unit interval [0,1) it is proved that a real-valued function $f(x)=exp(πi 1_I(x))$ is not of the form $f(x)=\overline{q(2x)}q(x)$, $|q(x)|=1$ a.e. if the interval I has dyadic endpoints. Its relation with the uniform distribution mod 2 is also shown. In Chapter 4, for the transformation T: x → kx (mod 1) for k ≥ 2, it is proved that a real-valued function f(x) of modulus 1 is not a multiplicative coboundary if the discontinuities $0 ＜ x_1 ＜ … < x_n ≤ 1$ of f(x) are k-adic points and $x_1 ≥ \frac {1}{k}$. It is also proved that the induced skew product is weakly mixing. In Chapter 5, let ρ: G → U(H) be an irreducible unitary representation of a compact group G. For Bernoulli shifts, the solvability of ρ(φ(x))g(Tx)=g(x) is investigated if φ(x) is a step function. In Chapter 6, $Y= ∏_{- ∞}^{∞}{0, 1, … , k-1}$ where k ≤ ∞ and σ be a shift map on Y. The solvability of φ(x)g(Tx)=g(x) is investigated if $φ(x)=∑_{j=0}^{n}a_j 1_{B_j}(x)$ with complex values $a_j$ and cylinder sets $B_j$. Its relation with the uniform distribution mod M is also shown and we give a different proof of S. Siboni``s results.