Symbolic Dynamics of Odd Discontinuous Bimodal Maps

TL;DR

This paper explores the symbolic dynamics of symmetric discontinuous bimodal maps, applying kneading theory and matrix spectral analysis to understand their complex behavior.

Contribution

It introduces a symbolic dynamics framework for odd discontinuous bimodal maps and connects kneading theory with homological analysis of these systems.

Findings

Spectral radius relates to kneading determinant roots.

Kneading theory effectively characterizes the dynamics.

Homology provides insights into map behavior.

Abstract

Iterations of odd piecewise continuous maps with two discontinuities, i.e., symmetric discontinuous bimodal maps, are studied. Symbolic dynamics is introduced. The tools of kneading theory are used to study the homology of the discrete dynamical systems generated by the iterations of that type of maps. When there is a Markov matrix, the spectral radius of this matrix is the inverse of the least root of the kneading determinant.