Emergence and non-typicality of the finiteness of the attractors in many topologies

TL;DR

This paper introduces the concept of Emergence in dynamical systems, conjectures its typicality, and provides conditions under which families of systems exhibit infinitely many sinks across parameters, with new results for specific topologies.

Contribution

It defines Emergence for dynamical systems, conjectures its local typicality, and establishes conditions for families to have infinitely many sinks, including new results for certain topologies.

Findings

Conditions for open sets of families to contain generic sets with infinitely many sinks

Introduction of the notion of Emergence in dynamical systems

New results for the case d=r=1 in topologies on families

Abstract

We will introduce the notion of Emergence for a dynamical system, and we will conjecture the local typicality of super-polynomial ones. Then, as part of this program, we will provide sufficient conditions for an open set of Cd-families of Cr-dynamics to contain a Baire generic set formed by families displaying infinitely many sinks at every parameter, for all 1 \le d \le r with d finite (and r possibly infinite) for two different topologies on families. In particular the case d=r=1 is new.