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Spectral Theory of Dynamical Systems

Part of the Texts and Readings in Mathematics Book Series (volume 15)

DOI: 10.1007/978-981-15-6225-9

ISBN: 978-981-15-6224-2

Publisher: Springer Singapore - Springer Journals

Aug 30, 2020

1 - 10 of 17 Chapters

LitStream

Chapter 1The Hahn-Hellinger Theorem

Nadkarni, Mahendra

Pages: 1 –17

[In this chapter we define spectral measure on a complex separable Hilbert space and state and prove the key theorem due to Hahn and Hellinger which describes such measures concretely and provides a complete invariant for unitary equivalence of two spectral measures.]
LitStream

Chapter 2The Spectral Theorem for Unitary Operators

Nadkarni, Mahendra

Pages: 19 –21

[In this chapter we briefly discuss the spectral theorem for unitary operators.]
LitStream

Chapter 3Symmetry and Denseness of the Spectrum

Nadkarni, Mahendra

Pages: 23 –28

[In this chapter we prove two properties of the spectrum of the unitary operator associated with an aperiodic non-singular automorphism, viz., that it is symmetric and fills the entire unit circle. We also discuss the spectrum of such an operator multiplied by a unitary function.]
LitStream

Chapter 4Multiplicity and Rank

Nadkarni, Mahendra

Pages: 29 –33

[The rank of a measure preserving automorphism σ is greater than or equal to the spectral multiplicity of Uσ. This is true for a class of non-singular automorphisms as well.]
LitStream

Chapter 5The Skew Product

Nadkarni, Mahendra

Pages: 35 –37

[In this chapter we will discuss the idea of skew product in its ‘simplest’ form. It is a very useful method, due to Anzai, of constructing new automorphisms from known ones.]
LitStream

Chapter 6A Theorem of Helson and Parry

Nadkarni, Mahendra

Pages: 39 –46

[Let τ be an ergodic measure preserving automorphism on a probability space. A theorem of H. Helson and W. Parry states that there is a measurable function ϕ taking values in {−1.1} such that the maximal spectral type of the unitary operator Vϕ: f → ϕf ○ τ is absolutely continuous with respect...
LitStream

Chapter 7Probability Measures on the Circle Group

Nadkarni, Mahendra

Pages: 47 –53

[In this chapter we will study certain aspects of finite measures on the circle group which can be applied to the spectral theory of non-singular automorphisms.]
LitStream

Chapter 8Baire Category Theorems of Ergodic Theory

Nadkarni, Mahendra

Pages: 55 –65

[In this chapter we will describe a topology on the class of non-singular automorphisms on a measure space and discuss the Baire category of some naturally occurring subclasses of it. We shall make a similar study of the class of measure preserving automorphisms.]
LitStream

Chapter 9Translations of Measures on the Circle

Nadkarni, Mahendra

Pages: 67 –77

[In this chapter we discuss the group of quasi-invariance of a finite measure on the circle group. We give a characterization of ergodic measures on the circle group supported on its group of quasi-invariance and apply it to prove a theorem of B. Host on marginal measures. These groups of...
LitStream

Chapter 10B. Host’s Theorem

Nadkarni, Mahendra

Pages: 79 –86

[A question of V. A. Rokhlin, whether a mixing automorphism is mixing of all orders, has an affirmative answer if the the spectrum of the automorphism is singular to Lebesgue measure. We prove this result of B. Host in this chapter. The general case remains open.]

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