Interaction cohomology of forward or backward self-similar systems

TL;DR

This paper introduces a new interaction cohomology theory for self-similar systems, linking topological properties of invariant sets to inverse limits, with applications to polynomial semigroup dynamics.

Contribution

It develops a novel cohomology framework for self-similar systems, providing criteria for connectedness and infinite rank of cohomology groups, and applies it to complex polynomial dynamics.

Findings

Connected components of invariant sets relate to inverse limits of nerve realizations.

Criteria established for invariant set connectedness.

Conditions identified for infinite rank of first cohomology group.

Abstract

We investigate the dynamics of forward or backward self-similar systems (iterated function systems) and the topological structure of their invariant sets. We define a new cohomology theory (interaction cohomology) for forward or backward self-similar systems. We show that under certain conditions, the space of connected components of the invariant set is isomorphic to the inverse limit of the spaces of connected components of the realizations of the nerves of finite coverings${\cal U} $ of the invariant set, where each ${\cal U}$ consists of (backward)images of the invariant set under elements of finite word length. Inparticular, we give a criterion for the invariant set to be connected. Moreover, we give a sufficient condition for the first cohomology group to have infinite rank. As an application, we obtain many results on the dynamics of finitely generated semigroups of polynomials on the complex plane. Moreover, we define postunbranched systems and we investigate the interaction cohomology groups of such systems. Many examples are given.