It is known that each word of length $n$ contains at most $n+1$ distinct
palindromes. A finite rich word is a word with maximal number of palindromic
factors. The definition of palindromic richness can be naturally extended to
infinite words. Sturmian words and Rote complementary symmetric sequences form
two classes of binary rich words, while episturmian words and words coding
symmetric $d$-interval exchange transformations give us other examples on
larger alphabets. In this paper we look for morphisms of the free monoid, which
allow us to construct new rich words from already known rich words. We focus on
morphisms in Class $P_{ret}$. This class contains morphisms injective on the
alphabet and satisfying a particular palindromicity property: for every
morphism $\varphi$ in the class there exists a palindrome $w$ such that
$\varphi(a)w$ is a first complete return word to $w$ for each letter $a$. We
characterize $P_{ret}$ morphisms which preserve richness over a binary
alphabet. We also study marked $P_{ret}$ morphisms acting on alphabets with
more letters. In particular we show that every Arnoux-Rauzy morphism is
conjugated to a morphism in Class $P_{ret}$ and that it preserves richness.
Given a finite set of local constraints, we seek a cellular automaton (i.e.,
a local and uniform algorithm) that self-stabilises on the configurations that
satisfy these constraints. More precisely, starting from a finite perturbation
of a valid configuration, the cellular automaton must eventually fall back into
the space of valid configurations where it remains still. We allow the cellular
automaton to use extra symbols, but in that case, the extra symbols can also
appear in the initial finite perturbation. For several classes of local
constraints (e.g., $k$-colourings with $k\neq 3$, and North-East deterministic
constraints), we provide efficient self-stabilising cellular automata with or
without additional symbols that wash out finite perturbations in linear or
quadratic time, but also show that there are examples of local constraints for
which the self-stabilisation problem is inherently hard. We note that the
optimal self-stabilisation speed is the same for all local constraints that are
isomorphic to one another. We also consider probabilistic cellular automata
rules and show that in some cases, the use of randomness simplifies the
problem. In the deterministic case, we show that if finite perturbations are
corrected in linear time, then the cellular automaton self-stabilises even
starting from a random perturbation of a valid configuration, that is, when
errors in the initial configuration occur independently with a sufficiently low
density.
We study the biased $(2:b)$ Walker--Breaker games, played on the edge set of
the complete graph on $n$ vertices, $K_n$. These games are a variant of the
Maker--Breaker games with the restriction that Walker (playing the role of
Maker) has to choose her edges according to a walk. We look at the two standard
graph games -- the Connectivity game and the Hamilton Cycle game and show that
Walker can win both games even when playing against Breaker whose bias is of
the order of magnitude $n/ \ln n$.