Dynamics of Chebyshev endomorphisms on some affine algebraic varieties

TL;DR

This paper investigates the chaotic dynamics of Chebyshev endomorphisms on complex affine varieties, focusing on cases where the dimension is 2 or 3, and explores their behavior via invariant theory and quotient spaces.

Contribution

It introduces a new approach to analyze the chaotic properties of Chebyshev endomorphisms on affine varieties using invariant theory and quotient space embeddings.

Findings

Chaotic behavior of Chebyshev endomorphisms on specific affine varieties.

Embedding quotient spaces as affine subvarieties for analysis.

Detailed study of dynamics when dimension n=2 and 3.

Abstract

Chebyshev polynomials in one variable are typical chaotic maps on the complex 1-space. Chebyshev endomorphisms f on the complex n-space A are also chaotic. The endomorphisms f induce mappings on the quotient space A/G, where G is the dihedral group of order 2(n+1). Using invariant theory we embed A/G as an affine subvariety X in the complex m-space. Then we have morphisms g on X. We study the chaotic properties of g when n = 2 and 3.