Kifer, Peres, and Weiss proved in [A dimension gap for continued fractions with independent digits. Israel J. Math.124 (2001), 61–76] that there exists $c_{0}>0$, such that $\dim \unicode[STIX]{x1D707}\leq 1-c_{0}$ for any probability measure $\unicode[STIX]{x1D707}$, which makes the digits of the continued fraction expansion independent and identically distributed random variables. In this paper we prove that amongst this class of measures, there exists one whose dimension is maximal. Our results also apply in the more general setting of countable branched systems.

We study the asymptotic dynamics of piecewise-contracting maps defined on a compact interval. For maps that are not necessarily injective, but have a finite number of local extrema and discontinuity points, we prove the existence of a decomposition of the support of the asymptotic dynamics into a finite number of minimal components. Each component is either a periodic orbit or a minimal Cantor set and such that the $\unicode[STIX]{x1D714}$-limit set of (almost) every point in the interval is exactly one of these components. Moreover, we show that each component is the $\unicode[STIX]{x1D714}$-limit set, or the closure of the orbit, of a one-sided limit of the map at a discontinuity point or at a local extremum.

Let $\unicode[STIX]{x1D6E4}$ denote the mapping class group of the plane minus a Cantor set. We show that every action of $\unicode[STIX]{x1D6E4}$ on the circle is either trivial or semiconjugate to a unique minimal action on the so-called simple circle.

Let $\operatorname{Homeo}_{+}(D_{n}^{2})$ be the group of orientation-preserving homeomorphisms of $D^{2}$ fixing the boundary pointwise and $n$ marked points as a set. The Nielsen realization problem for the braid group asks whether the natural projection $p_{n}:\operatorname{Homeo}_{+}(D_{n}^{2})\rightarrow B_{n}:=\unicode[STIX]{x1D70B}_{0}(\operatorname{Homeo}_{+}(D_{n}^{2}))$ has a section over subgroups of $B_{n}$. All of the previous methods use either torsion or Thurston stability, which do not apply to the pure braid group $PB_{n}$, the subgroup of $B_{n}$ that fixes $n$ marked points pointwise. In this paper, we show that the pure braid group has no realization inside the area-preserving homeomorphisms using rotation numbers.

We focus on various dynamical invariants associated to monomial correspondences on toric varieties, using algebraic and arithmetic geometry. We find a formula for their dynamical degrees, relate the exponential growth of the degree sequences to a strict log-concavity condition on the dynamical degrees and compute the asymptotic rate of the growth of heights of points of such correspondences.

Our main result states that, under an exponential map whose Julia set is the whole complex plane, on each piecewise smooth Jordan curve there is a point whose orbit is dense. This has consequences for the boundaries of nice sets, used in induction methods to study ergodic and geometric properties of the dynamics.

We define a new class of shift spaces which contains a number of classes of interest, like Sturmian shifts used in discrete geometry. We show that this class is closed under two natural transformations. The first one is called conjugacy and is obtained by sliding block coding. The second one is called the complete bifix decoding, and typically includes codings by non-overlapping blocks of fixed length.

Given a ${\mathcal{C}}^{\infty }$ expanding map $T$ of the circle, we construct a Hilbert space ${\mathcal{H}}$ of smooth functions on which the transfer operator ${\mathcal{L}}$ associated to $T$ acts as a compact operator. This result is made quantitative (in terms of singular values of the operator ${\mathcal{L}}$ acting on ${\mathcal{H}}$) using the language of Denjoy–Carleman classes. Moreover, the nuclear power decomposition of Baladi and Tsujii can be performed on the space ${\mathcal{H}}$, providing a bound on the growth of the dynamical determinant associated to ${\mathcal{L}}$.

We provide a detailed study of actions of the integers on compact quantum metric spaces, which includes general criteria ensuring that the associated crossed product algebra is again a compact quantum metric space in a natural way. Moreover, we provide a flexible set of assumptions ensuring that a continuous family of $\ast$-automorphisms of a compact quantum metric space yields a field of crossed product algebras which varies continuously in Rieffel’s quantum Gromov–Hausdorff distance. Finally, we show how our results apply to continuous families of Lip-isometric actions on compact quantum metric spaces and to families of diffeomorphisms of compact Riemannian manifolds which vary continuously in the Whitney $C^{1}$-topology.

Let $\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ from the unit interval $\mathbf{I}= [\!0,1\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if

$$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$

$[T_{\unicode[STIX]{x1D703}},\text{Id}]$

$\unicode[STIX]{x1D703}$

$[T_{\unicode[STIX]{x1D703}},\text{Id}]$

$$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$

$$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}|<+\infty ,\end{eqnarray}$$

$[0;a_{1},a_{2},\ldots ]$

$(p_{n}/q_{n})_{n\geq 0}$

$\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$

$\unicode[STIX]{x1D703}$

We study the independence density for finite families of finite tuples of sets for continuous actions of discrete groups on compact metrizable spaces. We use it to show that actions with positive naive entropy are Li–Yorke chaotic and untame. In particular, distal actions have zero naive entropy. This answers a question of Lewis Bowen.

In this paper, we construct and study a semigroup associated to an action of a countable discrete group on a compact Hausdorff space that can be regarded as a higher dimensional generalization of the type semigroup. We study when this semigroup is almost unperforated. This leads to a new characterization of dynamical comparison and thus answers a question of Kerr and Schafhauser. In addition, this paper suggests a definition of comparison for dynamical systems in which neither the acting group is necessarily amenable nor the action is minimal.

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a 3-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan’s homological characterization of geodesible flows in the volume-preserving case. As an application, we show that steady Euler flows cannot be constructed using plugs (as in Wilson’s or Kuperberg’s constructions). Analogous results in higher dimensions are also proved.

We study the thermodynamic formalism of a $C^{\infty }$ non-uniformly hyperbolic diffeomorphism on the 2-torus, known as the Katok map. We prove for a Hölder continuous potential with one additional condition, or geometric $t$-potential $\unicode[STIX]{x1D711}_{t}$ with $t<1$, the equilibrium state exists and is unique. We derive the level-2 large deviation principle for the equilibrium state of $\unicode[STIX]{x1D711}_{t}$. We study the multifractal spectra of the Katok map for the entropy and dimension of level sets of Lyapunov exponents.

For $G$ a Polish group, we consider $G$-flows which either contain a comeager orbit or have all orbits meager. We single out a class of flows, the maximally highly proximal (MHP) flows, for which this analysis is particularly nice. In the former case, we provide a complete structure theorem for flows containing comeager orbits, generalizing theorems of Melleray, Nguyen Van Thé, and Tsankov and of Ben Yaacov, Melleray, and Tsankov. In the latter, we show that any minimal MHP flow with all orbits meager has a metrizable factor with all orbits meager, thus ‘reflecting’ complicated dynamical behavior to metrizable flows. We then apply this to obtain a structure theorem for Polish groups whose universal minimal flow is distal.