Factors of non-stationary Bernoulli shifts
Ergodic Theory and Dynamical Systems Seminar
4th March 2021, 2:00 pm – 3:00 pm
The classical Bernoulli shift model is the shift map on a sequence of i.i.d. random variables. The Bernoulli systems are an important class of measure preserving dynamical systems and the Sinai factor theorem shows that there are many dynamical systems where they appear surprisingly as subsystems (factors). However the assumption of a stationary coin tossing process is in some ways too restrictive since due to several reasons such as noise, air pressure and etc., in most reasonable scenarios the coin tosses would be close to i.i.d. but not quite i.i.d.
A reasonable relaxation is to the setting of independent coin tosses where the shift map preserves the measure class. The ergodic theoretic study of these systems, which fall in the field of nonsingular ergodic theory is quite intricate and a lot of progress has been done only in recent years.
In this talk we will discuss a joint work with Terry Soo (UCL) where a surprising extension of the Sinai factor theorem is achieved. Namely we show that all nonsingular Bernoulli shifts on two symbols with the Doeblin condition have a factor that is equivalent to an independent and identically distributed system and prove that there are Bernoulli shifts of every possible ergodic index. The latter implies that the classification of nonsingular Bernoulli shifts according to metric isomorphism is much more subtle than its classical counterpart (Ornstein theory).