We show that the set of points on $C^{1}$ curves which are badly approximable by rationals in a number field form a winning set in the sense of Schmidt. As a consequence, we obtain a number field version of Schmidt’s conjecture in Diophantine approximation.

For a general group $G$ we consider various weak mixing properties of non-singular actions. In the case where the action is actually measure preserving all these properties coincide, and our purpose here is to check which implications persist in the non-singular case.

We prove that the Karoubi envelope of a shift—defined as the Karoubi envelope of the syntactic semigroup of the language of blocks of the shift—is, up to natural equivalence of categories, an invariant of flow equivalence. More precisely, we show that the action of the Karoubi envelope on the Krieger cover of the shift is a flow invariant. An analogous result concerning the Fischer cover of a synchronizing shift is also obtained. From these main results, several flow equivalence invariants—some new and some old—are obtained. We also show that the Karoubi envelope is, in a natural sense, the best possible syntactic invariant of flow equivalence of sofic shifts. Another application concerns the classification of Markov–Dyck and Markov–Motzkin shifts: it is shown that, under mild conditions, two graphs define flow equivalent shifts if and only if they are isomorphic. Shifts with property ($\mathscr{A}$) and their associated semigroups, introduced by Wolfgang Krieger, are interpreted in terms of the Karoubi envelope, yielding a proof of the flow invariance of the associated semigroups in the cases usually considered (a result recently announced by Krieger), and also a proof that property ($\mathscr{A}$) is decidable for sofic shifts.

A classical result in ergodic theory says that there always exists a topological model for any factor map ${\it\pi}:(X,{\mathcal{X}},{\it\mu},T)\rightarrow (Y,{\mathcal{Y}},{\it\nu},S)$ of ergodic systems. That is, there are some topological factor map $\hat{{\it\pi}}:(\hat{X},\hat{T})\rightarrow ({\hat{Y}},{\hat{S}})$ and invariant measures $\hat{{\it\mu}}$, $\hat{{\it\nu}}$ such that the diagram

$$\begin{eqnarray}\displaystyle & & \displaystyle \nonumber\end{eqnarray}$$

${\it\phi}$

${\it\psi}$

$\hat{{\it\pi}}$

We calculate the almost sure dimension for a general class of random affine code tree fractals in $\mathbb{R}^{d}$. The result is based on a probabilistic version of the Falconer–Sloan condition $C(s)$ introduced in Falconer and Sloan [Continuity of subadditive pressure for self-affine sets. Real Anal. Exchange 34 (2009), 413–427]. We verify that, in general, systems having a small number of maps do not satisfy condition $C(s)$. However, there exists a natural number $n$ such that for typical systems the family of all iterates up to level $n$ satisfies condition $C(s)$.

This article generalizes our previous results [Le Maître. The number of topological generators for full groups of ergodic equivalence relations. Invent. Math. 198 (2014), 261–268] to the non-ergodic case by giving a formula relating the topological rank of the full group of an aperiodic probability-measure-preserving (pmp) equivalence relation to the cost of its ergodic components. Furthermore, we obtain examples of full groups that have a dense free subgroup whose rank is equal to the topological rank of the full group, using a Baire category argument. We then study the automatic continuity property for full groups of aperiodic equivalence relations, and find a connected metric for which they have the automatic continuity property. This allows us to provide an algebraic characterization of aperiodicity for pmp equivalence relations, namely the non-existence of homomorphisms from their full groups into totally disconnected separable groups. A simple proof of the extreme amenability of full groups of hyperfinite pmp equivalence relations is also given, generalizing a result of Giordano and Pestov to the non-ergodic case [Giordano and Pestov. Some extremely amenable groups related to operator algebras and ergodic theory. J. Inst. Math. Jussieu6(2) (2007), 279–315, Theorem 5.7].

We consider the general non-vanishing, divergence-free vector fields defined on a domain in $3$-space and tangent to its boundary. Based on the theory of finite-type invariants, we define a family of invariants for such fields, in the style of Arnold’s asymptotic linking number. Our approach is based on the configuration space integrals due to Bott and Taubes.

For a continuous semicascade on a metrizable compact set ${\rm\Omega}$, we consider the weak$^{\ast }$ convergence of generalized operator ergodic means in $\text{End}\,C^{\ast }({\rm\Omega})$. We discuss conditions under which: every ergodic net contains a convergent sequence; all ergodic nets converge; all ergodic sequences converge. We study the relationships between the convergence of ergodic means and the properties of transitivity of the proximality relation on ${\rm\Omega}$, minimality of supports of ergodic measures, and uniqueness of minimal sets in the closure of trajectories of a semicascade. These problems are solved in terms of three associated algebraic-topological objects: the Ellis semigroup $E$, the Köhler operator semigroup ${\rm\Gamma}\subset \text{End}\,C^{\ast }({\rm\Omega})$, and the semigroup $G=\overline{\text{co}}\,{\rm\Gamma}$. The main results are stated for semicascades with metrizable $E$ and for tame semicascades.

The study of quadratic polynomials is a foundational part of modern complex dynamics. In this article, we study quasiregular counterparts to these in the plane. More specifically, let $h:\mathbb{C}\rightarrow \mathbb{C}$ be an $\mathbb{R}$-linear map and consider the quasiregular mapping $H=g\circ h$, where $g$ is a quadratic polynomial. By studying $H$ and via the Böttcher-type coordinate constructed in A. Fletcher and R. Fryer [On Böttcher coordinates and quasiregular maps. Contemp. Math.575 (2012), 53–76], we are able to obtain results on the dynamics of any degree-two mapping of the plane with constant complex dilatation. We show that any such mapping has either one, two or three fixed external rays, that all cases can occur and exhibit how the dynamics changes in each case. We use results from complex dynamics to prove that these mappings are nowhere uniformly quasiregular in a neighbourhood of infinity. We also show that in most cases, two such mappings are not quasiconformally conjugate on a neighbourhood of infinity.

Let ${\mathcal{T}}=(T_{t}^{f})_{t\in \mathbb{R}}$ be a special flow built over an IET $T:\mathbb{T}\rightarrow \mathbb{T}$ of bounded type, under a roof function $f$ with symmetric logarithmic singularities at a subset of discontinuities of $T$ . We show that ${\mathcal{T}}$ satisfies the so-called switchable Ratner’s property which was introduced in Fayad and Kanigowski [On multiple mixing for a class of conservative surface flows. Invent. Math. to appear]. A consequence of this fact is that such flows are mildly mixing (before, they were only known to be weakly mixing [Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations.J. Mod. Dynam.3 (2009), 35–49] and not mixing [Ulcigrai. Absence of mixing in area-preserving flows on surfaces. Ann. of Math. (2) 173 (2011), 1743–1778]). Thus, on each compact, connected, orientable surface of genus greater than one there exist flows that are mildly mixing and not mixing.

We consider a class of maps from the unit square to itself preserving a contracting foliation and inducing a one-dimensional map having an absolutely continuous invariant measure. We show how the physical measure of those systems can be rigorously approximated with an explicitly given bound on the error with respect to the Wasserstein distance. We present a rigorous implementation of our algorithm using interval arithmetics, and the result of the computation on a non-trivial example of a Lorenz-like two-dimensional map and its attractor, obtaining a statement on its local dimension.

We study a problem that arises from the study of Lorentz surfaces and Anosov flows. For a non-decreasing map of degree one $h:\mathbb{S}^{1}\rightarrow \mathbb{S}^{1}$, we are interested in groups of circle diffeomorphisms that act on the complement of the graph of $h$ in $\mathbb{S}^{1}\times \mathbb{S}^{1}$ by preserving a volume form. We show that such groups are semiconjugate to subgroups of $\text{PSL}(2,\mathbb{R})$ and that, when $h\in \text{Homeo}(\mathbb{S}^{1})$, we have a topological conjugacy. We also construct examples where $h$ is not continuous, for which there is no such conjugacy.

We study double averages along orbits for measure-preserving actions of $\mathbb{A}^{{\it\omega}}$, the direct sum of countably many copies of a finite abelian group $\mathbb{A}$. We show an $\text{L}^{p}$ norm-variation estimate for these averages, which in particular re-proves their convergence in $\text{L}^{p}$ for any finite $p$ and for any choice of two $\text{L}^{\infty }$ functions. The result is motivated by recent questions on quantifying convergence of multiple ergodic averages.

To a Toeplitz flow $(X,T)$ we associate an ordered $K^{0}$-group, denoted $K^{0}(X,T)$, which is order isomorphic to the $K^{0}$-group of the associated (non-commutative) $C^{\ast }$-crossed product $C(X)\rtimes _{T}\mathbb{Z}$. However, $K^{0}(X,T)$ can be defined in purely dynamical terms, and it turns out to be a complete invariant for (strong) orbit equivalence. We characterize the $K^{0}$-groups that arise from Toeplitz flows $(X,T)$ as exactly those simple dimension groups $(G,G^{+})$ that contain a non-cyclic subgroup $H$ of rank one that intersects $G^{+}$ non-trivially. Furthermore, the Bratteli diagram realization of $(G,G^{+})$ can be chosen to have the ERS property, i.e. the incidence matrices of the Bratteli diagram have equal row sums. We also prove that for any Choquet simplex $K$ there exists an uncountable family of pairwise non-orbit equivalent Toeplitz flows $(X,T)$ such that the set of $T$-invariant probability measures $M(X,T)$ is affinely homeomorphic to $K$, where the entropy $h(T)$ may be prescribed beforehand. Furthermore, the analogous result is true if we substitute strong orbit equivalence for orbit equivalence, but in that case we can actually prescribe both the entropy and the maximal equicontinuous factor of $(X,T)$. Finally, we present some interesting concrete examples of dimension groups associated to Toeplitz flows.

We study full groups of minimal actions of countable groups by homeomorphisms on a Cantor space $X$, showing that these groups do not admit a compatible Polish group topology and, in the case of $\mathbb{Z}$-actions, are coanalytic non-Borel inside $\text{Homeo}(X)$. We point out that the full group of a minimal homeomorphism is topologically simple. We also study some properties of the closure of the full group of a minimal homeomorphism inside $\text{Homeo}(X)$.

The dynamical degrees of a rational map $f:X{\dashrightarrow}X$ are fundamental invariants describing the rate of growth of the action of iterates of $f$ on the cohomology of $X$ . When $f$ has non-empty indeterminacy set, these quantities can be very difficult to determine. We study rational maps $f:X^{N}{\dashrightarrow}X^{N}$ , where $X^{N}$ is isomorphic to the Deligne–Mumford compactification $\overline{{\mathcal{M}}}_{0,N+3}$ . We exploit the stratified structure of $X^{N}$ to provide new examples of rational maps, in arbitrary dimension, for which the action on cohomology behaves functorially under iteration. From this, all dynamical degrees can be readily computed (given enough book-keeping and computing time). In this paper, we explicitly compute all of the dynamical degrees for all such maps $f:X^{N}{\dashrightarrow}X^{N}$ , where $\text{dim}(X^{N})\leq 3$ and the first dynamical degrees for the mappings where $\text{dim}(X^{N})\leq 5$ . These examples naturally arise in the setting of Thurston’s topological characterization of rational maps.

For strongly dissipative Hénon maps at the first bifurcation parameter where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set, we establish a thermodynamic formalism, i.e. we prove the existence and uniqueness of an invariant probability measure that minimizes the free energy associated with a non-continuous geometric potential $-t\log J^{u}$, where $t\in \mathbb{R}$ is in a certain large interval and $J^{u}$ denotes the Jacobian in the unstable direction. We obtain geometric and statistical properties of these measures.

Let $\{h_{n}\}_{n\in \mathbb{N}}$ be a sequence of self maps on a metric space $X$. We say that $Q\subset X$ is a mixing set on $\{h_{n}\}_{n\in \mathbb{N}}$ if for every $V\subset Q$ such that $\text{int}_{Q}(V)\not =\emptyset$ and every $\unicode[STIX]{x1D716}>0$ there exists $N=N(V,\unicode[STIX]{x1D716})$ such that $\text{d}_{H}(Q,h_{n}(V))<\unicode[STIX]{x1D716}$ for all $n\geq N$, where $\text{d}_{H}$ is the Hausdorff metric. It is shown that if $Q$ is a non-degenerate mixing set for a sequence of homeomorphisms on a continuum, then the continuum must be non-Suslinean. This is generalized to the notion of a $\unicode[STIX]{x1D719}$-mixing set. As a corollary, it is shown that a continuum must be non-Suslinean in order to admit a positive entropy homeomorphism.

We present an algorithm to compute the exact value of the packing measure of self-similar sets satisfying the so called Strong Separation Condition (SSC) and prove its convergence to the value of the packing measure. We also test the algorithm with examples that show both the accuracy of the algorithm for the most regular cases and the possibility of using the additional information provided by it to obtain formulas for the packing measure of certain self-similar sets. For example, we are able to obtain a formula for the packing measure of any Sierpinski gasket with contraction factor in the interval $(0,\frac{1}{3}]$.

Let $1,-1,-1,1,-1,1,1,-1,-1,1,1,\ldots$ be the $\{-1,1\}$-valued Thue–Morse sequence. Its correlation dimension is $D_{2}$, satisfying

$$\begin{eqnarray}\mathop{\sum }_{k=0}^{K-1}|{\it\gamma}(k)|^{2}\asymp K^{1-D_{2}}\end{eqnarray}$$

$\infty$

$K\rightarrow \infty$

${\it\gamma}$

$$\begin{eqnarray}D_{2}=1-\log \frac{1+\sqrt{17}}{4}\bigg/\log 2=0.64298\ldots .\end{eqnarray}$$

${\it\mu}$

$[0,1)$

$T$

$Tx=2x$

$\text{mod}~1$

$D_{2}\log 2$

$T^{-1}$

$$\begin{eqnarray}P_{1}((1/2)x+(1/2)j\mid x)=(1/2)(1-\cos ({\it\pi}(x+j)))\quad (j=0,1).\end{eqnarray}$$

${\it\mu}$

$$\begin{eqnarray}\lim _{N\rightarrow \infty }\int P_{N}((x-{\it\varepsilon},x+{\it\varepsilon})|x)\,d{\it\mu}(x)\asymp {\it\varepsilon}^{D_{2}}\quad (\text{as}~{\it\varepsilon}\rightarrow 0),\end{eqnarray}$$

$P_{N}(\cdot \mid \cdot )$

$N$