Let be {\ displaystyle X} Is a topological vector space (for example, a Banach space ). Linear Continuous Operator called hypercyclic if an element exists such that many tight in . This item called a hypercyclic vector for the operator .
The concept of hypercyclicity is a special case of the broader concept of topological transitivity .
Examples
The first example of a hypercyclic operator was received by Birkhoff in 1929.
In 1969, Rolevich proved that the backward shift operator in space is hypercyclic multiplied by a constant translating sequence in sequence .
In 1988, Charles Reid came up with an example of an operator in a Banach space. such that all its nonzero vectors are hypercyclic. This is a counterexample to the well-known problem of the existence of an invariant subspace for Banach spaces. For Hilbert spaces, the problem remains open.
Links
- K.-G. Grosse-Erdmann. Universal families and hypercyclic operators.
- Read, CJ (1988), " The invariant subspace problem for a class of Banach spaces, 2: hypercyclic operators ", Israel Journal of Mathematics T. 63 (1): 1–40, MR : 0959046 , ISSN 0021-2172 , DOI 10.1007 / BF02765019