# On multiple Lebesgue functions

## TL;DR

This paper introduces the concept of k-fold Lebesgue functions in measure-preserving transformations, exploring their spectral properties and implications for classical ergodic theory problems.

## Contribution

It defines and studies a new class of functions in ergodic theory, extending the notion of Lebesgue functions to multiple-fold cases and analyzing their spectral invariants.

## Key findings

- Establishes the relationship between k-fold Lebesgue functions and classical ergodic invariants.
- Provides multiple analogs of Banach and Rokhlin problems for transformations with many such functions.
- Highlights the spectral characteristics associated with these new functions.

## Abstract

We introduce a notion being a $k$-fold Lebesgue function for measure preserving transformations, where any $2$-fold Lebesgue function is just ordinary Lebesgue. We discuss how this new metrical isomorphisms invariant of dynamical systems is related to others classical notions in ergodic theory, mostly focusing on its spectral aspects. In particular, for transformations with sufficiently many multiple Lebesgue functions we treat the corresponding multiple analogs of very well-known problems of Banach and Rokhlin.