Teoria ergodyczna - seminarium

1. In the 70's Gerard Rauzy proved that a number $y$ in $b$-ary expansion is deterministic if and only if, for any number $x$ normal in base $b$, $x+y$ is also normal in base $b.$ First of all, we will show that although normality of a number $x$ depends on the base, determinism of $y$ does not. Then we will prove a generalization of Rauzy's theorem: a number $y$ is deterministic if and only if, for any number $x$, $x+y$ has ``the same entropy'' as $x$ in any base (I will explain what that means).
2. It was shown by Donald D. Wall already in the 40's (in his dissertation) that if $x$ is normal in base $b$ then, for any rational $q\neq0$, $xq$ is normal in base $b.$ We will give examples (in base 2) that this fails if $q$ is replaced by a deterministic number $y$, where $y$ may have zero or positive density of $1$'s (these examples will take most of the time).

Entropy locking, or entropy plateau, is the phenomenon of (measure-theoretic or topological) entropy being constant on a subset of the parameter space for a family of dynamical systems. Maps related to $\mathrm{PSL}(2,\mathbb{Z})$ or to co-compact Fuchsian groups provide concrete examples, and a variety of techniques are used in the proofs of entropy locking for these families.

Consider the set of numbers that preserve normality under addition in base $b$. Directly from its definition, it is unclear if this set is even Borel. Rauzy obtained a characterization of this set as the set of deterministic (or "noise 0") numbers. One corollary of this characterization is that this set is indeed Borel, but it is unclear where it sits in the Borel hierarchy and what type of theorems we could expect to be true about it. Following the talks of Tomasz Downarowicz, who also proved some interesting facts about this set, we will determine the Borel complexity of this set and prove that it is $\Pi_0^3$-complete.

Continuing my previous talk, I will review the definition of the Borel hierarchy and describe several recent results.

In the article "Projectional Entropy in Higher Dimensional Shifts of Finite Type" (2007) A. Johnson, S. Kass and K. Madden provided an object called projectional entropy. In shortcut, it is the entropy of the shift $X_L$, which is obtained by restricting the points of a $\mathbb{Z}^d$ shift $X$ to a sublattice $L$. The authors gave some basic properties and a theorem stating that for a special class of $\mathbb{Z}^d$ shifts equality $h(X)=h(X_L)$ gives us some information about the structure of $X$. However, this theorem was proved only for a particular case of $\mathbb{Z}^2$ shifts and one dimensional sublattices $L$, and it was claimed that it can be generalized to shifts and sublattices of arbitrary dimensions.

In 2010 M. Schraudner improved the results in his paper ``Projectional Entropy And the Electrical Wire Shift''. He gave an example of a $\mathbb{Z}^3$ shift, for which claimed generalization doesn't hold. He proposed and proved a different version of the theorem. His work was an inspiration for me and B. Frej to generalize the term ``projectional entropy'' and its properties for actions of amenable groups.

I will briefly get through both mentioned papers and present our results.

The one-dimensional quantum harmonic oscillator is considered. The wave solution of the Schrodinger equation determines the model of motion of particles in $n$-quantum state as orbits of stationary random walk at the moment of jump from $n-1$ to $n$-quantum state. The numerical solutions are given for $n=2,3$.