Epsilon-independence between two processes

Tomasz Downarowicz¹ ; Paulina Grzegorek¹

¹ Institute of Mathematics and Computer Science Wroc/law University of Technology Wybrze/ze Wyspia/nskiego 27 50-370 Wroc/law, Poland

Studia Mathematica, Tome 188 (2008) no. 1, pp. 77-95

Résumé

We study the notion of $\varepsilon $-independence of a process on finitely (or countably) many states and that of $\varepsilon $-independence between two processes defined on the same measure preserving transformation. For that we use the language of entropy. First we demonstrate that if a process is $\varepsilon $-independent then its $\varepsilon $-independence from another process can be verified using a simplified condition. The main direction of our study is to find natural examples of $\varepsilon $-independence. In case of $\varepsilon $-independence of one process, we find an example among processes generated on the induced (first return time) transformation defined on a typical long cylinder set of any given process of positive entropy. To obtain examples of pairs of $\varepsilon $-independent processes we have to make an additional assumption on the master process. Then again, we find such pairs generated on the induced transformation as above. This is the most elaborate part of the paper. While the question whether our assumption is necessary remains open, we indicate a large class of processes where our assumption is satisfied.

Détail

DOI : 10.4064/sm188-1-5

Keywords: study notion varepsilon independence process finitely countably many states varepsilon independence between processes defined measure preserving transformation language entropy first demonstrate process varepsilon independent its varepsilon independence another process verified using simplified condition main direction study natural examples varepsilon independence varepsilon independence process example among processes generated induced first return time transformation defined typical long cylinder set given process positive entropy obtain examples pairs varepsilon independent processes have make additional assumption master process again pairs generated induced transformation above elaborate part paper while question whether assumption necessary remains indicate large class processes where assumption satisfied