Random Continuous Functions

Paul Brodhead, Douglas Cenzer, Jeffrey B. Remmel

Abstract: We investigate notions of algorithmic randomness in the space C(2N)<math><mi mathvariant="script" is="true">C</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mn is="true">2</mn><mi mathvariant="double-struck" is="true">N</mi></msup><mo stretchy="false" is="true">)</mo></math> of continuous functions on 2N<math><msup is="true"><mn is="true">2</mn><mi mathvariant="double-struck" is="true">N</mi></msup></math>. A probability measure is given and a version of the Martin-Löf test for randomness is defined which allows us to define a class of (Martin-Löf) random continuous functions. We show that random Δ20<math><msubsup is="true"><mi mathvariant="normal" is="true">Δ</mi><mn is="true">2</mn><mn is="true">0</mn></msubsup></math> continuous functions exist, but no computable function can be random. We show that a random function maps any computable real to a random real and that the image of a random continuous function is always a perfect set and hence uncountable. We show that for any y∈2N<math><mi is="true">y</mi><mo is="true">∈</mo><msup is="true"><mn is="true">2</mn><mi mathvariant="double-struck" is="true">N</mi></msup></math>, there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set.