We construct strictly ergodic 0-1 Toeplitz flows with pure point spectrum and irrational eigenvalues. It is also shown that the property of being regular is not a measure-theoretic invariant for strictly ergodic Toeplitz flows.
2
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
If a rotation α of 𝕋 has unbounded partial quotients then "most" of its skew-product diffeomorphic extensions to the 2-torus 𝕋 × 𝕋 defined by $C^1$ cocycles of topological degree zero enjoy nontrivial ergodic properties. In fact they admit a cyclic approximation with speed o(1/n) and have nondiscrete (simple) spectrum. Similar results are obtained for $C^r$ cocycles if α admits a sufficiently good approximation by rationals. For a.e. α and generic $C^1$ cocycles the speed can be improved to o(1/(nlogn)). For generic α and generic $C^r$ cocycles (r = 1,...,∞) the spectral measure of the skew product has a continuous component and Hausdorff dimension zero.
5
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
Let α be an ergodic rotation of the d-torus $\mathbb{T}^d = ℝ^d/ℤ^d$. For any piecewise smooth function $f: \mathbb{T}^d → ℝ$ with sufficiently regular pieces the unitary operator Vh(x) = exp(2π if(x))h(x + α) acting on $L^2(\mathbb{T}^d)$ is shown to have a continuous non-Dirichlet spectrum if the gradient of f has nonzero integral. In particular, the resulting skew product $S_f: \mathbb{T}^{d+1} → \mathbb{T}^{d+1}$ must be ergodic. If in addition α is sufficiently well approximated by rational vectors and f is represented by a linear function with noninteger coefficients then the spectrum of V is singular. In the case d = 1 our technique allows us to extend Pask's result on ergodicity of cylinder flows on T×ℝ to arbitrary piecewise absolutely continuous real-valued cocycles f satisfying ʃf = 0 and ʃf' ≠ 0.
6
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
Let T be a stochastic operator on a σ-finite standard measure space with an equivalent σ-finite infinite subinvariant measure λ. Then T possesses a natural "conservative deterministic factor" Φ which is the Frobenius-Perron operator of an invertible measure preserving transformation φ. Moreover, T is mixing ("sweeping") iff φ is a mixing transformation. Some stronger versions of mixing are also discussed. In particular, a notion of *L¹-s.o.t. mixing is introduced and characterized in terms of weak compactness. Finally, it is shown that most stochastic operators are completely mixing and that the same holds for convolution stochastic operators on l.c.a. groups.
7
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
We give a necessary and sufficient condition for a Toeplitz flow to be strictly ergodic. Next we show that the regularity of a Toeplitz flow is not a topological invariant and define the "eventual regularity" as a sequence; its behavior at infinity is topologically invariant. A relation between regularity and topological entropy is given. Finally, we construct strictly ergodic Toeplitz flows with "good" cyclic approximation and non-discrete spectrum.
8
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
We study the factors of Gaussian dynamical systems which are generated by functions depending only on a finite number of coordinates. As an application, we show that for Gaussian automorphisms with simple spectrum, the partition ${(X_0 ≤ 0), (X_0 > 0)}$ is generating.
11
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
Topological and combinatorial properties of dynamical systems called odometers and arising from number systems are investigated. First, a topological classification is obtained. Then a rooted tree describing the carries in the addition of 1 is introduced and extensively studied. It yields a description of points of discontinuity and a notion of low scale, which is helpful in producing examples of what the dynamics of an odometer can look like. Density of the orbits is also discussed.