Spectral properties of skew-product diffeomorphisms of tori
A. Iwanik (1997)
Colloquium Mathematicum
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A. Iwanik (1997)
Colloquium Mathematicum
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Isaac Kornfeld, Michael Lin (2000)
Studia Mathematica
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It is well known that a weakly almost periodic operator T in a Banach space is mean ergodic, and in the complex case, also λT is mean ergodic for every |λ|=1. We prove that a positive contraction on is weakly almost periodic if (and only if) it is mean ergodic. An example shows that without positivity the result is false. In order to construct a contraction T on a complex such that λT is mean ergodic whenever |λ|=1, but T is not weakly almost periodic, we prove the following: Let...
Geoffrey Goodson (2000)
Colloquium Mathematicae
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We study certain symmetries that arise when automorphisms S and T defined on a Lebesgue probability space (X, ℱ, μ) satisfy the equation . In an earlier paper [6] it was shown that this puts certain constraints on the spectrum of T. Here we show that it also forces constraints on the spectrum of . In particular, has to have a multiplicity function which only takes even values on the orthogonal complement of the subspace . For S and T ergodic satisfying this equation further constraints...
Mahesh Nerurkar (2000)
Colloquium Mathematicae
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We construct continuous G-valued cocycles that are not cohomologous to any compact constant via a measurable transfer function, provided the underlying dynamical system is rigid and the range group G satisfies a certain general condition. For more general ergodic aperiodic systems, we also show that the set of continuous ergodic cocycles is residual in the class of all continuous cocycles provided the range group G is a compact connected Lie group. The first construction is based on...
I. Filipowicz, J. Kwiatkowski, M. Lemanczyk (1988)
Publicacions Matemàtiques
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Let Gbar = G{nt, nt | nt+1, t ≥ 0} be a subgroup of all roots of unity generated by exp(2πi/nt}, t ≥ 0, and let τ: (X, β, μ) O be an ergodic transformation with pure point spectrum Gbar. Given a cocycle φ, φ: X → Z2, admitting an approximation with speed 0(1/n1+ε, ε>0) there exists a Morse cocycle φ such that the corresponding transformations τφ...
Zbigniew Kowalski (1993)
Studia Mathematica
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We consider skew products preserving a measure which is absolutely continuous with respect to the product measure. Here f is a 1-sided Markov shift with a finite set of states or a Lasota-Yorke type transformation and , i = 1,..., max e, are nonsingular transformations of some probability space. We obtain the description of the set of eigenfunctions of the Frobenius-Perron operator for T and consequently we get the conditions ensuring the ergodicity, weak mixing and exactness of T....
Ryotaro Sato (1995)
Studia Mathematica
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Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average converges almost everywhere to a function f* in , where (pq) and are assumed to be in the set . Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized...
M. Lemańczyk (1987)
Compositio Mathematica
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Alan Forrest (2000)
Colloquium Mathematicae
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E. Muehlegger, A. Raich, C. Silva, M. Touloumtzis, B. Narasimhan, W. Zhao (1999)
Colloquium Mathematicae
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We construct infinite measure preserving and nonsingular rank one -actions. The first example is ergodic infinite measure preserving but with nonergodic, infinite conservative index, basis transformations; in this case we exhibit sets of increasing finite and infinite measure which are properly exhaustive and weakly wandering. The next examples are staircase rank one infinite measure preserving -actions; for these we show that the individual basis transformations have conservative...