### Jarkko Kari

**Title: ** Decidability in Group Cellular Automata

**Abstract:** Many undecidable questions concerning cellular automata are known to be decidable when the cellular automaton has a suitable algebraic structure. Typical situations include linear cellular automata where the states come from a finite field or a finite commutative ring, and so called additive cellular automata in the case the states come from a finite commutative group and the cellular automaton is a group homomorphism. In this talk we generalize the setup and consider so-called group cellular automata whose state set is any (possibly non-commutative) finite group and the cellular automaton is a group homomorphism. The configuration space may be any group shift – a subshift that is a subgroup of the full shift – and still many properties are decidable in any dimension of the cellular space. Our decidability proofs are based on algorithms to manipulate group shifts – using results by B. Kitchens and K. Schmidt – and on viewing the set of space-time diagrams of group cellular automata as multidimensional group shifts. The trace shift and the limit set of the cellular automaton are lower dimensional projections of the space-time diagrams and they can be effectively constructed. This view provides algorithms to decide injectivity, surjectivity, equicontinuity, sensitivity and nilpotency of the cellular automaton. Non-transitivity is semi-decidable. We also easily establish that injectivity always implies surjectivity, which in turn implies pre-injectivity, that transitivity implies mixingness, that non-sensitivity implies equicontinuity, and that jointly periodic points are dense in the limit set. The talk is based on a joint work with Pierre Béaur.

### Marta Maggioni

**Title: ** Random matching for random interval maps

**Abstract:** In this talk we extend the notion of matching for deterministic transformations to random matching for random interval maps. For a large class of piecewise affine random systems of the interval, we prove that this property implies that any invariant density of a stationary measure is piecewise constant. We provide examples of random matching for a variety of families of random dynamical systems, including generalised beta-transformations, continued fraction maps and a family of random maps producing signed binary expansions. We finally apply the property of random matching and its consequences to this family to study minimal weight expansions. Based on a joint work with Karma Dajani and Charlene Kalle.

### Claire Merriman

**Title: ** \(\alpha\)-odd continued fractions

**Abstract:** Nakada’s \(\alpha\)-expansions move from the regular continued fractions (\(\alpha = 1\)), Hurwitz singular continued fractions (obtained at \(\alpha =\) little golden ratio), and nearest integer continued fractions (\(\alpha =1/2\)). This talk will look at similar continued fraction expansions where all of the denominators are odd. I will describe how restricting the parity of the partial quotients changes the Gauss map and natural extension domain. This is join work with Florin Boca.

### Pierre Guillon

**Title: ** Soficity of the one-marked-component shift

**Abstract:** The one-marked-component shift is the set of colorings of the grid by alphabet \(\{0,1,2\}\) such that every finite connected component (maximal subgraph) colored by \(\{1,2\}\) contains exactly one occurrence of \(2\). We build a shift of finite type that factors onto it. The construction involves combinatorial local constraints which force acyclicity, chords in undirected cycles, trees spanning finite bunches. This is joint work with Julien Cassaigne.

### Anh Le

**Title: ** Additive averages of multiplicative recurrence sequences and applications

**Abstract:**

Motivated by Frantzikinakis and Host’s work on partition regularity of quadratic equations, we study sets of recurrence for actions of \((N, x)\) and \((Q, x)\). In particular, we show that multiple recurrence sequences arising from multiplicative systems have positive additive averages. We also give criteria for when sets of the forms \(\{(an+b)^k/(cn + d)^k: n \in N\}\) are sets of multiplicative recurrence. As a result, we recover two recent results in number theory regarding completely multiplicative functions and Omega functions. This is based on a joint work with Sebastián Donoso, Joel Moreira and Wenbo Sun.

### Tullio Ceccherini-Silberstein

**Title: ** Linear shifts and their endomorphisms

**Abstract:** Let \(G\) be a group and let \(A\) be a vector space over a field \(K\). We study linear subshifts (over the alphabet \(A\) and universe \(G\)) and their endomorphims (= linear cellular automata). In particular, we focus on linear subshifts of finite type and limit sets and nilpotency for linear cellular automata.

Joint work with Michel Coornaert (Strasbourg) and Xuan Kien Phung (Montreal).

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