# Open maps: the boundary of chaos for piecewise monotonic functions with one discontinuity

• Clement Hege

Student thesis: Phd

### Abstract

The study of open dynamical systems is the study of systems with holes. This thesis focuses on the sets of orbits that avoid these holes, called the survivor sets, and on the families of open strictly monotonic functions with one discontinuity, all restricted to their survivor set. Such functions have applications in electricity (switching systems) and mechanics (oscillators with multiple impacts). The aim of this thesis is to find and describe the transition to chaos from zero to positive topological entropy for these families. The literature uses two different approaches for open maps: combinatorics on words and piecewise monotone maps without holes. Combinatorics on words have led to a full description, on the parameter space, of the transition to chaos for one family: the doubling map with a hole. Studies on piecewise monotone maps have proved more detailed results on entropy for maps without holes and described the main tool of our method: renormalisation. This thesis combines these two approaches to describe the transition to chaos for the tent map and two other similar functions with a hole. It also describes the route to chaos, i.e. the evolution of the functionsâ dynamics as parameters continuously move across the boundary of chaos. Our method uses a set of sub-regions of the parameter space that contains the boundary of chaos. Each of these sub-regions is called a renormalisation box and is associated with one specific renormalisation. The results of this thesis are stated in four theorems. The first theorem describes the possible dynamics of a function for its parameters in one renormalisation box. This allows to create a simple network of renormalisations to define sub-regions within sub-regions, and create an induction process. The second theorem proves that at each step of this process the set of sub-regions contains the boundary of chaos. The third theorem proves that the limit of our process is the full transition to chaos, which is a connected curve. Finally, the last theorem computes the speed of convergence of our method for a new kind of route to chaos which does not exist for the doubling map with a hole. These four theorems describe the transition and routes to chaos for the tent map and two other similar functions with holes. Our work is a first full description of the transition to chaos for these families of open strictly monotonic functions with one discontinuity.
Date of Award 1 Aug 2023 English The University of Manchester Tuomas Sahlsten (Supervisor), Nikita Sidorov (Supervisor) & Paul Glendinning (Supervisor)

### Keywords

• Transition to chaos
• Topological entropy
• Tent map
• Survivor set
• Route to chaos
• Renormalisation
• Plateau functions
• Open maps
• Piecewise monotonic maps
• Hausdorff dimension
• Boundary of chaos
• Lorenz maps
• Combinatorics on words
• Circle maps
• Dynamical systems
• Full family
• Doubling map

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