# Topics in Dynamics: I: Flows

EDWARD NELSON
Pages: 122
https://www.jstor.org/stable/j.ctt13x187h

1. Front Matter
(pp. [i]-[ii])
(pp. [iii]-[iv])
3. 1. Differential calculus
(pp. 1-13)

In recent years there has been an upsurge of interest in infinite dimensional manifolds. The theory has had important applications to Morse theory, transversality theory, and in other areas. It might be thought that an infinite dimensional manifold with a smooth vector field on it is a suitable framework for discussing classical dynamical systems with infinitely many degrees of freedom. However, classical dynamical systems of infinitely many degrees of freedom are usually described in terms of partial differential operators, and partial differential operators cannot be formulated as everywhere-defined operators on a Banach space. We will be concerned only with finite...

4. 2. Picard's method
(pp. 13-27)

We shall be studying non-linear time-independent differential equations. In doing so, however, it will be useful to have some information about linear time-dependent differential equations.

Recall that the differential equation$\frac{\text {df}(\text {t})}{\text {dt}}={\text {g}}(\text {t}),$with g a continuous function of t, is solved by integration:$\text{f}(\text{t})=\text{f}({{\text{t}}_{0}})+\int_{{{\text{t}}_{\text{O}}}}^{\text{t}}{\text{g}(\text{s})\text{ds}},$and we have sketched how the integral may be defined. In very close analogy, the linear time-dependent equation may be solved by the product integral. This is an ancient device going back at least to Volterra at the turn of the century, but it keeps being rediscovered.

If E is a Banach space and A is in...

5. 3. The local structure of vector fields
(pp. 28-56)

We have been discussing mappings X: U → E. Loosely speaking, these may be termed vector fields on U. If R is a diffeomorphism of U onto an open set V in the Banach space F then we define the transformed vector field Y = R*X on V by$\text {Y}(\text {y})=\text{DR}({{\text{R}}^{-1}}\text{y})\centerdot \text{X}({{\text{R}}^{-1}}\text{y}).$The reason for this definition is as follows. If X is locally Lipschitz it generates a local flow φ(t,x) on U with$\frac{\text{d}}{\text{dt}}\varphi (\text{t},\text{x})\left| _{\text{t}=\text{O}} \right.=\text{X}(\text{x}).$

This flow may be transported to V by setting

(R*φ(t,y) = R(φ(t,R-1y)).

The vector field generating this local flow is the above defined Y = R*X by...

6. 4. Sums and Lie products of vector fields
(pp. 56-59)

If X is a locally Lipschitz vector field in an open subset of a Banach space, we denote by UX(t) the local flow it generates.

Theorem 1. Let X and Y be locally Lipschitz vector fields defined in an open subset U of the Banach space E. For all x0in U there is a neighborhood V of x0with V contained in U and an ɛ > 0 such that${{\text{U}}_{\text{X}+\text{Y}}}(\text{t})\text{x}=\underset{\text{n}\to \infty }{\mathop{\lim }}\,\ {{({{\text{U}}_{\text{X}}}(\frac{\text{t}}{\text{n}}){{\text{U}}_{\text{Y}}}(\frac{\text{t}}{\text{n}}))}^{\text{n}}}\text{x}$uniformly for x in V and |t| I ≤ ɛ.

Proof. We have

UX= 1 + hX + o(h),

and for each x0this holds uniformly for x...

7. 5. Self-adjoint operators on Hilbert space
(pp. 60-77)

We shall develop briefly those aspects of Hilbert space theory which are of greatest relevance to dynamics. A knowledge of integration theory is assumed.

Let ℋ be a complex vector space. A sesquilinear form on ℋ is a mapping ℋ × ℋ → C which takes each ordered pair (u,v) into a complex number , such that is conjugate linear in u and linear in v (we follow the physicists' convention). For a sesquilinear form, computation shows that the polarization identity holds:

4 = - - i + i.

This means that...

8. 6. Commutative multiplicity theory
(pp. 77-97)

An unsatisfactory aspect of the spectral theorem as we have presented it is the lack of uniqueness in the choice of the measure space (Μ,μ) and the unitary transformation Ā. In this section we will study the problem more thoroughly and obtain a complete classification of self-adjoint operators up to unitary equivalence. On a finite dimensional Hilbert space this is easy: two self-adjoint operators are unitarily equivalent if and only if they have the same eigenvalues with the same multiplicities.

Multiplicity theory for unbounded self-adjoint operators is essentially the same as for bounded self-adjoint operators, and without any genuine increase...

9. 7. Extensions of Hermitean operators
(pp. 97-102)

A Hermitean operator A on a Hilbert space ℋ is called essentially self-adjoint in case Ā is self-adjoint. A complex number λ is in the resolvent set of an operator A in case λ-A is injective and (λ-A)-lis in L(ℋ).

Theorem 1. Let A be a Hermitean operator on a Hilbert space. Then the following are equivalent:

(i) A is essentially self-adjoint, Ā*= Ā,

(ii) Ā = A*,

(iii) A*= A**,

(iv) A*⊂ A**,

(v) ℛ(i-A) and ℛ(-i-A) are dense,

(vii) i and -i are not eigenvalues of A*,

(viii) (i-Ā)-1is in L(ℋ) and is...

10. 8. Sums and Lie products of self-adjoint operators
(pp. 103-111)

A theorem of Paul Chernoff [23] gives us the result needed in order to discuss the one-parameter unitary group generated by the sum or Lie product of two self-adjoint operators. The natural context for the discussion is given by the notion of a contraction semigroup on a Banach space.

Let 𝔛 be a Banach space. A contraction semigroup on 𝔛 is a family of operators ptin L(𝔛), for O ≤ t ∞, such that ∥Pt∥ ≤ = 1, PO= 1, PtPs= Pt+s, and$\caption {(1)} \underset{\text{t}\to 0 }{\mathop{\lim }}\,\ {{\text{P}}^{\text{t}}}\text{u}=\text{u},\quad \quad \quad \text{u}\in {\mathfrak {X}}.$

This is usually called a "contraction semigroup of class (CO)", the last phrase...

11. Notes and references
(pp. 112-118)