Flexibility of statistical properties for smooth systems satisfying the central limit theorem

Abstract

In this paper we exhibit new classes of smooth systems which satisfy the Central Limit Theorem (CLT) and have (at least) one of the following properties:• zero entropy;• weak but not strong mixing;• (polynomially) mixing but not K;• K but not Bernoulli;• non Bernoulli and mixing at arbitrary fast polynomial rate. We also give an example of a system satisfying the CLT where the normalizing sequence is regularly varying with index 1.

Exponential mixing implies Bernoulli

“…Except for the Rokhlin problem it is known that all the above inclusions are strict (also in the smooth setting see e.g. the discussion in [20]). All the above mentioned properties do not require a smooth structure and can be defined for an arbitrary measure preserving system.…”

“…All the three properties imply ergodicity, but central limit theorem and large deviations do not imply weak mixing and hence also do not imply stronger ergodic properties, see e.g. [20]. In this paper we focus on consequences of exponential mixing.…”

“…Thus a natural next step is to understand which additional features of the system are responsible for Bernoullicity. The recent works [28,22,20] seem to indicate that an important role is played by the competition between the rate of mixing and the complexity of the system restricted to the subspace with zero exponents. In particular, in the present paper we show that the exponential mixing implies the Bernoulli property as the growth in the zero exponents directions is always sub-exponential.…”

“…We note that [20] constructs non Bernoulli systems with arbitrary fast polynomial mixing rate. However, in order to get the mixing rate of n −α [20] considers manifolds of dimension growing quadratically with α.…”

“…We note that [20] constructs non Bernoulli systems with arbitrary fast polynomial mixing rate. However, in order to get the mixing rate of n −α [20] considers manifolds of dimension growing quadratically with α. In fact, lowering the dimension of the phase space makes it more difficult to construct non Bernoulli systems.…”