# When a smooth self-map of a semi-simple Lie group can realize the least number of periodic points

## TL;DR

This paper investigates conditions under which the minimal number of periodic points of a smooth self-map on a semi-simple Lie group matches algebraic lower bounds, linking eigenvalues of induced homomorphisms to periodic point counts.

## Contribution

It establishes a precise criterion involving eigenvalues for when the algebraic lower bounds of periodic points coincide for self-maps of semi-simple Lie groups.

## Key findings

- NF_n(f) equals NJD_n(f) iff all eigenvalues have modulus ≤ 1
- The equality holds for self-maps inducing the identity on fundamental groups
- Eigenvalue conditions determine the minimal number of periodic points

## Abstract

There are two algebraic lower bounds of the number of n-periodic points of a self-map f:M\to M of a compact smooth manifold of dimension at least 3 : NF_n(f)=min {#Fix(g^n) ;g\sim f; g continuous} and NJD_n(f)=min {#Fix}(g^n) ;g\sim f; g smooth}}. In general NJD_n(f) may be much greater than NF_n(f). We show that for a self-map of a semi-simple Lie group, inducing the identity fundamental group homomorphism, the equality NF_n(f)=NJD_n(f) holds for all n iff all eigenvalues of a quotient cohomology homomorphism induced by f have moduli \le 1.