Abstract
"We study the size of products of matrices taken from a finite set of matrices. In particular we describe
the maximal growth rate of the norms of products of matrices from our set. For a set of matrices
we define an associated “FMIRF” to be the function mapping a natural number n to the maximum
of the norm of all possible n length products from our set. Switching systems are models defined
by a finite set of matrices, by understanding the associated FMIRF, we gain insight into the stability
of the switching system. Switching systems have attracted a lot of interest from control theorists,
as they pose a complex challenge and as they have a wide range of applications. Switching systems
are needed to describe systems in nature and engineering, because such systems do not fit a single
model and exhibit switching between different models depending on various environments. Zhendong
and Shuzi have uncovered many tools for analysing nonlinear behaviour in switching systems. We
focus on switching systems with FMIRFs that grow slower than exponentially, such FMIRFs are called
marginally unstable. We are interested in the different possible rates of growth a marginally unstable
system can achieve, significant research by Jungers, Protasov and Blondel in 2008 showed that in
particular cases marginally unstable systems grow like a polynomial. This is not the case though for
all systems and in 2015 Jungers and Protasov released an example of a FMIRF with sublinear growth.
We prove that their example is bounded almost everywhere and adapt it to achieve other sublinear
grow rates. We show that if we have a set of two dimensional matrices then the corresponding FMIRF
has bounded growth or linear growth. We prove that if we have a set containing a single matrix the
the corresponding FMIRF grows like a polynomial. This begs the question, does the cardinality of our
set always affect the possible growth of the corresponding FMIRF? We prove that, given a FMIRF
associated to finite sets of matrices it is possible to construct a set of two matrices with a FMIRF
that has the same growth. We also explore the effect of taking direct and Kronecker products on sets
of matrices has on their associated FMIRFs. Finally instead of starting with a set of matrices we start
with a function that maps from a dynamical system onto a matrix of θ-H¨older continuous functions.
In this more general setting we show how the generalised FMIRF’s growth depends on the θ-H¨older
continuous functions. An example of a generalised FMIRF with unbounded but sublinear growth is
given."