Let $f:M\rightarrow M$ be a $C^{1}$ self-map of a smooth Riemannian manifold $M$ and $\unicode[STIX]{x1D707}$ be an $f$ -invariant ergodic Borel probability measure with a compact support $\unicode[STIX]{x1D6EC}$ . We prove that if $f$ is Hölder mild on the intersection of the singularity set and $\unicode[STIX]{x1D6EC}$ , then the pointwise dimension of $\unicode[STIX]{x1D707}$ can be controlled by the Lyapunov exponents of $\unicode[STIX]{x1D707}$ with respect to $f$ and the entropy of $f$ . Moreover, we establish the distinction of the Hausdorff dimension of the critical points sets of maps between the $C^{1,\unicode[STIX]{x1D6FC}}$ continuity and Hölder mildness conditions. Consequently, this shows that the Hölder mildness condition is much weaker than the $C^{1,\unicode[STIX]{x1D6FC}}$ continuity condition. As applications of our result, if we study the recurrence rate of $f$ instead of the pointwise dimension of $\unicode[STIX]{x1D707}$ , then we deduce that the analogous relation exists between recurrence rate, entropy and Lyapunov exponents.

We investigate the $K$ -theory of unital UCT Kirchberg algebras ${\mathcal{Q}}_{S}$ arising from families $S$ of relatively prime numbers. It is shown that $K_{\ast }({\mathcal{Q}}_{S})$ is the direct sum of a free abelian group and a torsion group, each of which is realized by another distinct $C^{\ast }$ -algebra naturally associated to $S$ . The $C^{\ast }$ -algebra representing the torsion part is identified with a natural subalgebra ${\mathcal{A}}_{S}$ of ${\mathcal{Q}}_{S}$ . For the $K$ -theory of ${\mathcal{Q}}_{S}$ , the cardinality of $S$ determines the free part and is also relevant for the torsion part, for which the greatest common divisor $g_{S}$ of $\{p-1:p\in S\}$ plays a central role as well. In the case where $|S|\leq 2$ or $g_{S}=1$ we obtain a complete classification for ${\mathcal{Q}}_{S}$ . Our results support the conjecture that ${\mathcal{A}}_{S}$ coincides with $\otimes _{p\in S}{\mathcal{O}}_{p}$ . This would lead to a complete classification of ${\mathcal{Q}}_{S}$ , and is related to a conjecture about $k$ -graphs.

We study the zero-temperature limit of the Gibbs measures of a class of long-range potentials on a full shift of two symbols $\{0,1\}$ . These potentials were introduced by Walters as a natural space for the transfer operator. In our case, they are constant on a countable infinity of cylinders and are Lipschitz continuous or, more generally, of summable variation. We assume that there exist exactly two ground states: the fixed points $0^{\infty }$ and $1^{\infty }$ . We fully characterize, in terms of the Peierls barrier between the two ground states, the zero-temperature phase diagram of such potentials, that is, the regions of convergence or divergence of the Gibbs measures as the temperature goes to zero.

It is known that volume hyperbolicity (partial hyperbolicity and uniform expansion or contraction of the volume in the extremal bundles) is a necessary condition for robust transitivity or robust chain recurrence and hence for tameness. In this paper, on any $3$ -manifold we build examples of quasi-attractors which are volume hyperbolic and wild at the same time. As a main corollary, we see that, for any closed $3$ -manifold $M$ , the space $\text{Diff}^{1}(M)$ admits a non-empty open set where every $C^{1}$ -generic diffeomorphism has no attractors or repellers. The main tool of our construction is the notion of flexible periodic points introduced in the authors’ previous paper. In order to eject the flexible points from the quasi-attractor, we control the topology of the quasi-attractor using the notion of partially hyperbolic filtrating Markov partitions, which we introduce in this paper.

Two notions of metric invariance entropy are constructed with respect to conditionally invariant measures for control systems in discrete time and it is shown that they are invariant under conjugacies.

We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the existence of a particular coding with two letters for the geodesics on these orbifolds.

For Hölder continuous cocycles over an invertible, Lipschitz base, we establish the Hölder continuity of Oseledets subspaces on compact sets of arbitrarily large measure. This extends a result of Araújo et al [On Hölder-continuity of Oseledets subspaces J. Lond. Math. Soc.93 (2016) 194–218] by considering possibly non-invertible cocycles, which, in addition, may take values in the space of compact operators on a Hilbert space. As a by-product of our work, we also show that a non-invertible cocycle with non-vanishing Lyapunov exponents exhibits non-uniformly hyperbolic behaviour (in the sense of Pesin) on a set of full measure.

We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and Mandelbrot percolation. In each setting we compute either the almost sure or the Baire typical Assouad dimension and consider some illustrative examples. Our results reveal a phenomenon common to each of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure-theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdorff or packing dimension.

We prove two generalizations of Furstenberg’s Diophantine result regarding the density of an orbit of an irrational point in the $1$ -torus under the action of multiplication by a non-lacunary multiplicative semigroup of $\mathbb{N}$ . We show that for any sequences $\{a_{n}\},\{b_{n}\}\subset \mathbb{Z}$ for which the quotients of successive elements tend to $1$ as $n$ goes to infinity, and any infinite sequence $\{c_{n}\}$ , the set $\{a_{n}b_{m}c_{k}x:n,m,k\in \mathbb{N}\}$ is dense modulo $1$ for every irrational $x$ . Moreover, by ergodic-theoretical methods, we prove that if $\{a_{n}\},\{b_{n}\}$ are a sequence having smooth $p$ -adic interpolation for some prime number $p$ , then for every irrational $x$ , the sequence $\{p^{n}a_{m}b_{k}x:n,m,k\in \mathbb{N}\}$ is dense modulo 1.

We consider commuting pairs of holomorphic endomorphisms of $\mathbb{P}^{2}$ with disjoint sequence of iterates. The case that has not been completely studied is when their degrees coincide after some number of iterations. We show in this case that they are either commuting Lattès maps or commuting homogeneous polynomial maps of $\mathbb{C}^{2}$ inducing a Lattès map on the line at infinity.

Model sets are projections of certain lattice subsets. It was realized by Moody [Uniform distribution in model sets. Canad. Math. Bull. 45(1) (2002), 123–130] that dynamical properties of such a set are induced from the torus associated with the lattice. We follow and extend this approach by studying dynamics on the graph of the map that associates lattice subsets to points of the torus and then we transfer the results to their projections. This not only leads to transparent proofs of known results on model sets, but we also obtain new results on so-called weak model sets. In particular, we prove pure point dynamical spectrum for the hull of a weak model set of maximal density together with the push forward of the torus Haar measure under the torus parametrization map, and we derive a formula for its pattern frequencies.

We study the topology and dynamics of subshifts and tiling spaces associated to non-primitive substitutions in one dimension. We identify a property of a substitution, which we call tameness, in the presence of which most of the possible pathological behaviours of non-minimal substitutions cannot occur. We find a characterization of tameness, and use this to prove a slightly stronger version of a result of Durand, which says that the subshift of a minimal substitution is topologically conjugate to the subshift of a primitive substitution. We then extend to the non-minimal setting a result obtained by Anderson and Putnam for primitive substitutions, which says that a substitution tiling space is homeomorphic to an inverse limit of a certain finite graph under a self-map induced by the substitution. We use this result to explore the structure of the lattice of closed invariant subspaces and quotients of a substitution tiling space, for which we compute cohomological invariants that are stronger than the Čech cohomology of the tiling space alone.

In this paper we show that any ergodic measure preserving transformation of a standard probability space which is $\text{AT}(n)$ for some positive integer $n$ has zero entropy. We show that for every positive integer $n$ any Bernoulli shift is not $\text{AT}(n)$ . We also give an example of a transformation which has zero entropy but does not have property $\text{AT}(n)$ for any integer $n\geq 1$ .

We establish self-norming central limit theorems for non-stationary time series arising as observations on sequential maps possessing an indifferent fixed point. These transformations are obtained by perturbing the slope in the Pomeau–Manneville map. We also obtain quenched central limit theorems for random compositions of these maps.

We prove that for the frame flow on a negatively curved, closed manifold of odd dimension other than 7, and a Hölder continuous potential that is constant on fibers, there is a unique equilibrium measure. Brin and Gromov’s theorem on the ergodicity of frame flows follows as a corollary. Our methods also give a corresponding result for automorphisms of the Heisenberg manifold fibering over the torus.

For a strongly dissipative Hénon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we effect a multifractal analysis, i.e. decompose the set of non-wandering points on the unstable manifold into level sets of an unstable Lyapunov exponent, and give a partial description of the Lyapunov spectrum which encodes this decomposition. We derive a formula for the Hausdorff dimension of the level sets in terms of the entropy and unstable Lyapunov exponent of invariant probability measures, and show the continuity of the Lyapunov spectrum. We also show that the set of points for which the unstable Lyapunov exponents do not exist carries the full Hausdorff dimension.