Spectral Properties and Combinatorial Constructions in Ergodic Theory

Katok, Anatole; Thouvenot, J

doi:10.1016/s1874-575x(06)80036-6

Cited by 72 publications

(95 citation statements)

References 127 publications

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“…Thus, Theorem 1.1 settles the open question (which appears, for example, in Forni [For02] and in the survey [KT06,§6.3.2] by Katok and Thouvenot) of whether a typical minimal multi-valued Hamiltonian flow with only simple saddles is mixing. Even if nonmixing, such flows are nevertheless typically weakly mixing 2 ([Ulc09]; see also §1.3).…”

Section: Definitions and Main Resultsmentioning

confidence: 73%

Absence of mixing in area-preserving flows on surfaces

Ulcigrai

2011

Ann. Math.

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We prove that minimal area-preserving flows locally given by a smooth Hamiltonian on a closed surface of genus g ≥ 2 are typically (in the measure-theoretical sense) not mixing. The result is obtained by considering special flows over interval exchange transformations under roof functions with symmetric logarithmic singularities and proving absence of mixing for a full measure set of interval exchange transformations. Definitions and main results1.1. Flows given by multi-valued Hamiltonians. Let us consider the following natural construction of area-preserving flows on surfaces. On a closed connected orientable surface S of genus g ≥ 1 with a fixed smooth area form ω, consider a smooth closed real-valued differential 1-form η. Let X be the vector field determined by η = i X ω = ω(η, ·) and consider the flow {ϕ t } t∈R on S associated to X. Since η is closed, the transformations ϕ t , t ∈ R, are areapreserving. The flow {ϕ t } t∈R is known as the multi-valued Hamiltonian flow associated to η. Indeed, the flow {ϕ t } t∈R is locally Hamiltonian; i.e., locally one can find coordinates (x, y) on S in which it is given by the solution to the equationsẋ = ∂H/∂y,ẏ = −∂H/∂x for some smooth real-valued Hamiltonian function H. A global Hamiltonian H cannot be in general be defined (see [NZ99, §1.3.4]), but one can think of {ϕ t } t∈R as globally given by a multi-valued Hamiltonian function.The study of flows given by multi-valued Hamiltonians was initiated by S. P. Novikov [Nov82] in connection with problems arising in solid-state physics i.e., the motion of an electron in a metal under the action of a magnetic field. The orbits of such flows arise also in pseudo-periodic topology, as hyperplane sections of periodic surfaces in T n (see e.g. Zorich [Zor99]).From the point of view of topological dynamics, a decomposition into minimal components (i.e., subsurfaces on which the flow is minimal) and periodic components on which all orbits are periodic (elliptic islands around a center and cylinders filled by periodic orbits) was proved independently by Maier

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Section: Definitions and Main Resultsmentioning

confidence: 73%

Absence of mixing in area-preserving flows on surfaces

Ulcigrai

2011

Ann. Math.

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“…For necessary notions and facts from ergodic theory and spectral theory of dynamical systems we refer the reader to e.g. [7], [14]. For the theory of substitutions, also from the dynamical system point of view, see [22].…”

Section: Consider Now X = O(x) ⊂ {0 1}mentioning

confidence: 99%

0-1 sequences of the Thue-Morse type and Sarnak’s conjecture

Abdalaoui

Kasjan

Lemańczyk

2015

Proc. Amer. Math. Soc.

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We show that the images via z → z m of the continuous part of the spectral measures of the dynamical systems generated by the 0-1 sequences of the Thue-Morse type are pairwise mutually singular for different odd numbers m ∈ N. Sarnak's conjecture on orthogonality with the Möbius function is shown to hold for such dynamical systems. The same conjecture is shown to hold for all systems induced by regular Toeplitz sequences. A non-regular Toeplitz sequence for which Sarnak's conjecture fails is constructed.

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“…It has already been shown that all subsets containing 1 are realizable [19] (reproved by a different argument in [3]). For more information on the subject see also earlier articles by Robinson [29], [30], and the surveys [11,18]. Less is known about Koopman realization of sets which do not contain 1.…”

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confidence: 99%

“…and called the multiplicity function (see e.g. the Appendix in [28] or [18] for more information on the spectral theory of unitary operators). Each member of the essential range of M V is called an essential value of the multiplicity function.…”

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confidence: 99%