We examine the class of increasing sequences of natural numbers which are IP-rigidity sequences for some weakly mixing probability-preserving transformation. This property is closely related to the uncountability of the eigenvalue group of a corresponding non-singular transformation. We give examples, including a super-lacunary sequence which is not IP-rigid.

The squiral inflation rule is equivalent to a bijective block substitution rule and leads to an interesting lattice dynamical system under the action of ${ \mathbb{Z} }^{2} $. In particular, its balanced version has purely singular continuous diffraction. The dynamical spectrum is of mixed type, with pure point and singular continuous components. We present a constructive proof that admits a generalization to bijective block substitutions of trivial height on ${ \mathbb{Z} }^{d} $.

We prove that if ${\Sigma }_{\mathbf{A} } ( \mathbb{N} )$ is an irreducible Markov shift space over $ \mathbb{N} $ and $f: {\Sigma }_{\mathbf{A} } ( \mathbb{N} )\rightarrow \mathbb{R} $ is coercive with bounded variation then there exists a maximizing probability measure for $f$, whose support lies on a Markov subshift over a finite alphabet. Furthermore, the support of any maximizing measure is contained in this same compact subshift. To the best of our knowledge, this is the first proof beyond the finitely primitive case in the general irreducible non-compact setting. It is also noteworthy that our technique works for the full shift over positive real sequences.

For a positive measure set of non-uniformly expanding quadratic maps on the interval we effect a multifractal formalism, i.e., decompose the phase space into level sets of time averages of a given continuous function and consider the associated Birkhoff spectrum which encodes this decomposition. We derive a formula which relates the Hausdorff dimension of level sets to entropies and Lyapunov exponents of invariant probability measures, and then use this formula to show that the spectrum is continuous. In order to estimate the Hausdorff dimension from above, one has to ‘see’ sufficiently many points. To this end, we construct a family of towers. Using these towers we establish a large deviation principle of empirical distributions, with Lebesgue as a reference measure.

Mixing (of all orders) rank-one actions $T$ of the Heisenberg group ${H}_{3} ( \mathbb{R} )$ are constructed. The restriction of $T$ to the center of ${H}_{3} ( \mathbb{R} )$ is simple and commutes only with $T$. Mixing Poisson and mixing Gaussian actions of ${H}_{3} ( \mathbb{R} )$ are also constructed. A rigid weakly mixing rank-one action $T$ is constructed such that the restriction of $T$ to the center of ${H}_{3} ( \mathbb{R} )$ is not isomorphic to its inverse.

We give a sharp polynomial bound on the number of Pollicott–Ruelle resonances. These resonances, which are complex numbers in the lower half-plane, appear in expansions of correlations for Anosov contact flows. The bounds follow the tradition of upper bounds on the number of scattering resonances and improve a recent bound of Faure and Sjöstrand. The complex scaling method used in scattering theory is replaced by an approach using exponentially weighted spaces introduced by Helffer and Sjöstrand in scattering theory and by Faure and Sjöstrand in the theory of Anosov flows.

We provide a strengthened version of the famous Jakobson's theorem. Consider an interval map $f$ satisfying a summability condition. For a generic one-parameter family ${f}_{t} $ of maps with ${f}_{0} = f$, we prove that $t= 0$ is a Lebesgue density point of the set of parameters for which ${f}_{t} $ satisfies both the Collet–Eckmann condition and a strong polynomial recurrence condition.

We extend the classical mean ergodic theorem to the setting of norming dual pairs. It turns out that, in general, not all equivalences from the Banach space setting remain valid in our situation. However, for Markovian semigroups on the norming dual pair $({C}_{b} (E), \mathscr{M} (E))$ all classical equivalences hold true under an additional assumption which is slightly weaker than the e-property.

Semi-invertible multiplicative ergodic theorems establish the existence of an Oseledets splitting for cocycles of non-invertible linear operators (such as transfer operators) over an invertible base. Using a constructive approach, we establish a semi-invertible multiplicative ergodic theorem that for the first time can be applied to the study of transfer operators associated to the composition of piecewise expanding interval maps randomly chosen from a set of cardinality of the continuum. We also give an application of the theorem to random compositions of perturbations of an expanding map in higher dimensions.

We consider low-dimensional systems with the shadowing property and we study the problem of existence of periodic orbits. In dimension two, we show that the shadowing property for a homeomorphism implies the existence of periodic orbits in every $\epsilon $-transitive class, and in contrast we provide an example of a ${C}^{\infty } $ Kupka–Smale diffeomorphism with the shadowing property exhibiting an aperiodic transitive class. Finally, we consider the case of transitive endomorphisms of the circle, and we prove that the $\alpha $-Hölder shadowing property with $\alpha \gt 1/ 2$ implies that the system is conjugate to an expanding map.

Let $C$ be a locally convex closed subset of a negatively curved Riemannian manifold $M$. We define the skinning measure ${\sigma }_{C} $ on the outer unit normal bundle to $C$ in $M$ by pulling back the Patterson–Sullivan measures at infinity, and give a finiteness result for ${\sigma }_{C} $, generalizing the work of Oh and Shah, with different methods. We prove that the skinning measures, when finite, of the equidistant hypersurfaces to $C$ equidistribute to the Bowen–Margulis measure ${m}_{\mathrm{BM} } $ on ${T}^{1} M$, assuming only that ${m}_{\mathrm{BM} } $ is finite and mixing for the geodesic flow. Under additional assumptions on the rate of mixing, we give a control on the rate of equidistribution.

Nathanial Brown [Topological dynamical systems associated to ${\mathit{II}}_{1} $-factors. Adv. Math. 227(4), 1665–1699] introduced a convex-like structure on the set of unitary equivalence classes of unital *-homomorphisms of a separable type ${\mathit{II}}_{1} $ factor into ${R}^{\omega } $ (ultrapower of the hyperfinite factor). The goal of this paper is to introduce such a structure on the set of sofic representations of groups. We prove that if the commutant of a representation acts ergodically on the Loeb measure space then that representation is an extreme point.

Let $G$ be a countable abelian group and let ${T}^{(1)} , {T}^{(2)} , \ldots , {T}^{(s)} $ be measure preserving $G$-actions on a probability space. We prove that joint ergodicity of ${T}^{(1)} , {T}^{(2)} , \ldots , {T}^{(s)} $ implies total joint ergodicity if each ${T}^{(i)} $ is totally ergodic. We also show that if $G= { \mathbb{Z} }^{d} $, $s\geq d+ 1$ and the actions ${T}^{(1)} , {T}^{(2)} , \ldots , {T}^{(s)} $ commute, then total joint ergodicity of ${T}^{(1)} , {T}^{(2)} , \ldots , {T}^{(s)} $ follows from joint ergodicity. This can be seen as a generalization of Berend’s result for commuting $ \mathbb{Z} $-actions.

To any automorphism, $\alpha $, of a totally disconnected, locally compact group, $G$, there is associated a compact, $\alpha $-stable subgroup of $G$, here called the nub of $\alpha $, on which the action of $\alpha $ is ergodic. Ergodic actions of automorphisms of compact groups have been studied extensively in topological dynamics and results obtained transfer, via the nub, to the study of automorphisms of general locally compact groups. A new proof that the contraction group of $\alpha $ is dense in the nub is given, but it is seen that the two-sided contraction group need not be dense. It is also shown that each pair $(G, \alpha )$, with $G$ compact and $\alpha $ ergodic, is an inverse limit of pairs that have ‘finite depth’ and that analogues of the Schreier refinement and Jordan–Hölder theorems hold for pairs with finite depth.

For analytic quasiperiodic Schrödinger cocycles, Goldshtein and Schlag [Hölder continuity of the integrated density of states for quasi-periodic Schrödinger equations and averages of shifts of subharmonic functions. Ann. of Math. (2) 154 (2001), 155–203] proved that the Lyapunov exponent is Hölder continuous provided that the base frequency $\omega $ satisfies a strong Diophantine condition. In this paper, we give a refined large deviation theorem, which implies the Hölder continuity of the Lyapunov exponent for all Diophantine frequencies $\omega $, even for weak Liouville $\omega $, which improves the result of Goldshtein and Schlag.