In this paper we obtain exponential large-deviation bounds in the Shannon–McMillan–Breiman convergence formula for entropy in the case of weak Gibbs measures and topologically mixing subshifts of finite type. We also prove almost sure estimates for the error term in the convergence to entropy given by the Shannon–McMillan–Breiman formula for both uniformly and non-uniformly expanding shifts. Finally, we establish a topological characterization of large-deviation bounds for Gibbs measures and deduce some of their topological and geometrical aspects: the local entropy is zero and the topological pressure of positive measure sets is total. Some applications include large-deviation estimates for Lyapunov exponents, pointwise dimension and slow return times.

We study the relationship between pairs of topological dynamical systems $(X,T)$ and $(X^{\prime },T^{\prime })$, where $(X^{\prime },T^{\prime })$ is the quotient of $(X,T)$ under the action of a finite group $G$. We describe three phenomena concerning the behaviour of closed orbits in the quotient system, and the constraints given by these phenomena. We find upper and lower bounds for the extremal behaviour of closed orbits in the quotient system in terms of properties of $G$ and show that any growth rate in between these bounds can be achieved.

Irreducible interval exchange transformations are studied with regard to the whirly property, a condition for a non-trivial spatial factor. A uniformly whirly transformation is defined and is further studied. An equivalent condition is introduced for the whirly transformation. We will prove that almost all 3-interval exchange transformations are whirly, using a combinatorics approach with application of the Rauzy–Veech induction. It is still an open question whether the whirly property is a generic property for $m$-interval exchange transformations for $m\geq 4$.

We characterize Koopman one-parameter $C_{0}$-groups, in the class of all unitary one-parameter $C_{0}$-groups on $L_{2}(X)$, as those that preserve $L_{\infty }(X)$ and for which the infinitesimal generator is a derivation on the bounded functions in its domain.

It is well known that any $\mathbb{Z}$ subshift with the specification property has the property that every factor is intrinsically ergodic, i.e. every factor has a unique factor of maximal entropy. In recent work, other $\mathbb{Z}$ subshifts have been shown to possess this property as well, including $\unicode[STIX]{x1D6FD}$-shifts and a class of $S$-gap shifts. We give two results that show that the situation for $\mathbb{Z}^{d}$ subshifts with $d>1$ is quite different. First, for any $d>1$, we show that any $\mathbb{Z}^{d}$ subshift possessing a certain mixing property must have a factor with positive entropy which is not intrinsically ergodic. In particular, this shows that for $d>1$, $\mathbb{Z}^{d}$ subshifts with specification cannot have all factors intrinsically ergodic. We also give an example of a $\mathbb{Z}^{2}$ shift of finite type, introduced by Hochman, which is not even topologically mixing, but for which every positive-entropy subshift factor is intrinsically ergodic.

For a map $T:[0,1]\rightarrow [0,1]$ with an invariant measure $\unicode[STIX]{x1D707}$, we study, for a $\unicode[STIX]{x1D707}$-typical $x$, the set of points $y$ such that the inequality $|T^{n}x-y|<r_{n}$ is satisfied for infinitely many $n$. We give a formula for the Hausdorff dimension of this set, under the assumption that $T$ is piecewise expanding and $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D719}}$ is a Gibbs measure. In some cases we also show that the set has a large intersection property.

We generalize the notion of weakly mixing unitary representations to locally compact quantum groups, introducing suitable extensions of all standard characterizations of weak mixing to this setting. These results are used to complement the non-commutative Jacobs–de Leeuw–Glicksberg splitting theorem of Runde and the author [Ergodic theory for quantum semigroups. J. Lond. Math. Soc. (2) 89(3) (2014), 941–959]. Furthermore, a relation between mixing and weak mixing of state-preserving actions of discrete quantum groups and the properties of certain inclusions of von Neumann algebras, which is known for discrete groups, is demonstrated.

Copeland and Erdös [Note on normal numbers. Bull. Amer. Math. Soc. (N.S.)52 (1946), 857–860] showed that the concatenation of primes when written in base 10 yields a real number that is normal to base 10. We generalize this result to Pisot number bases in which all integers have finite expansion.