Category Archives: Classic results

Distal systems and Expansive systems

A topological dynamical system is a pair where is a compact metric space and is a continuous map. This gives rise to an action of the semigroup on , where acts via the iterated map (and as usual denotes the … Continue reading

Posted in Classic results, Topological Dynamics | | 2 Comments

A viewpoint on Katai’s orthogonality criterion

The Liouville function, defined as the completely multiplicative function which sends every prime to , encodes several important properties of the primes. For instance, the statement that is equivalent to the prime number theorem, while the improved (and essentially best … Continue reading

Entropy of measure preserving systems

A measure preserving system is a quadruple where is a set, is a -algebra, is a probability measure and is a measurable map satisfying for every . The notion of isomorphism in the category of measure preserving systems (defined, for … Continue reading

Posted in Classic results, Ergodic Theory, Tool | Tagged | 1 Comment

Piecewise syndetic sets, topological dynamics and ultrafilters

In this post I explore the notion of piecewise syndeticity and its relation to topological dynamical systems and the Stone-Čech compactification. I restrict attention to the additive semigroup but most results presented are true in much bigger generality (and I … Continue reading

Gaussian systems

Examples of measure preserving systems with varied behaviours are vital in ergodic theory, to understand the general properties and to have counter examples to false statements. One classical method to craft examples with specific properties is the so-called Gaussian construction. … Continue reading

The horocycle flow is mixing of all orders

— 1. Introduction — The main purpose of this post is to present a proof, due to Brian Marcus, that the horocycle flow is mixing of all orders. The precise definition of mixing of all orders for -actions is given … Continue reading

Primes of the form x^2+2y^2

In this post I will present a quite nice proof of the following fact from elementary number theory: Theorem 1 Let be a prime number. There are such that if and only if is a quadratic residue . Recall that … Continue reading

Banach density with respect to a single Folner sequence

In this short post I show that in any countable amenable group the (left) upper Banach density of a set can be obtained by looking only at translations of a given Følner sequence. Definitions and the precise statement are given … Continue reading