Compactification and Linearisation of Abstract Dynamical Systems
by
Ludvík Janoš

By an abstract dynamical system (ADS) we understand a pair (X, T) where X is aset of cardinality not exceeding that of the continuum, and T : X --> X is a selfmap of X. We adopt the axiom CH. By a compact dynamical system we understand a triple (Y, S, \tau) where (Y, S) is an ADS, \tau is a compact metric topology on Y and S is continuous. Similarly by a linear dynamical system (LDS) we mean an ADS (H, L) where H is a separable Hilbert space or an Euclidean space and L is a continuous linear operator. We prove the following statements:

Theorem 1 Every ADS (X, T) can be equivariantly embedded in a CDS (Y, S, \tau).

Theorem 2 Every ADS (X, T) can be equivariantly embedded in a LDS (H, L) with the norm |L| <= 1.

Theorems 1 and 2 read that every system (X, T) can be pre-compactified and linearised. If a system (X, T) is such that the embedding of X into Y mentioned in Theorem 1 can be chosen bijective then we say that the system (X, T) is compactifiable. We show that there are systems which are not compactifiable.

We introduce numerical invariants Dim(X, T) and Lin(X, T) with values in N \cup {\infty} as follows:

Dim(X, T) = min{dim(Y, T) : (X, T) can be pre-compactified in (Y, S, \tau) }

Lin(X, T) is the smallest dimension of linear spaces in which the system (X, T) can be linearised.

Theorem 3 If the map T:X --> X is surjective and not injective then Lin(X, T) = \infty.

Remark. We also characterise those ADS's (X, T) for which the system (Y, S, \tau) involved in Theorem 1 can be chosen equicontinuous, i.e., such that the family of iterations of S, { Sⁿ : n in N } is equicontinuous.

Date received: June 24, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caah-44.