# On the ellipticity of operators associated with Morse-Smale diffeomorphisms

## TL;DR

This paper investigates how the Fredholm property of operators, generated by pseudodifferential operators and Morse-Smale diffeomorphisms on surfaces, depends on Sobolev space smoothness levels.

## Contribution

It provides a detailed analysis of the ellipticity and Fredholm properties of these operators within Sobolev scales, linking dynamical systems with operator theory.

## Key findings

- Fredholm property varies with Sobolev smoothness exponent s
- Ellipticity conditions depend on the Morse-Smale diffeomorphism dynamics
- Characterization of operator ellipticity in the algebra generated by pseudodifferential and shift operators

## Abstract

We consider the operator algebra generated by pseudodifferential operators on a closed smooth surface and shift operator induced by a Morse--Smale diffeomorphism of this surface. Elements in this algebra are considered as operators in the scale of Sobolev spaces and the aim of this paper is to describe how Fredholm property of a given operator depends on the Sobolev smoothness exponent $s$.