Let $G$ be a topological group, let $\unicode[STIX]{x1D719}$ be a continuous endomorphism of $G$ and let $H$ be a closed $\unicode[STIX]{x1D719}$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is,

$$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$

$\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$

$\unicode[STIX]{x1D719}$

$G$

$H$

$\unicode[STIX]{x1D719}H=H$

$\ker (\unicode[STIX]{x1D719})\leq H$

$s(\unicode[STIX]{x1D719})$

$\log s(\unicode[STIX]{x1D719})$

$\unicode[STIX]{x1D719}$

$\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$

The critical dimension of an ergodic non-singular dynamical system is the asymptotic growth rate of sums of consecutive Radon–Nikodým derivatives. This has been shown to equal the average coordinate entropy for product odometers when the size of individual factors is bounded. We extend this result to $G$-measures with an asymptotic bound on the size of individual factors. Furthermore, unlike von Neumann–Krieger type, the critical dimension is an invariant property on the class of ergodic $G$-measures.

We study tangent sets of strictly self-affine sets in the plane. If a set in this class satisfies the strong separation condition and projects to a line segment for sufficiently many directions, then for each generic point there exists a rotation ${\mathcal{O}}$ such that all tangent sets at that point are either of the form ${\mathcal{O}}((\mathbb{R}\times C)\cap B(0,1))$, where $C$ is a closed porous set, or of the form ${\mathcal{O}}((\ell \times \{0\})\cap B(0,1))$, where $\ell$ is an interval.

We investigate a certain dual relationship between piecewise rotations of a circle and interval exchange maps. In 2005, Cruz and da Rocha [A generalization of the Gauss map and some classical theorems on continued fractions. Nonlinearity18 (2005), 505–525]  introduced a notion of ‘castles’ arising from piecewise rotations of a circle. We extend their idea and introduce a continuum version of castles, which we show to be equivalent to Veech’s zippered rectangles [Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. (2) 115 (1982), 201–242]. We show that a fairly natural map defined on castles represents the inverse of the natural extension of the Rauzy map.

We show that minimal shifts with zero topological entropy are topologically conjugate to interval exchange transformations, which are generally infinite. When these shifts have linear factor complexity (linear block growth), the conjugate interval exchanges are proved to satisfy strong finiteness properties.

Associated to any orthogonal representation of a countable discrete group, is a probability measure-preserving action called the Gaussian action. Using the Polish model formalism we developed before, we compute the entropy (in the sense of Bowen [J. Amer. Math. Soc.23 (2010) 217–245], Kerr and Li [Invent. Math.186 (2011) 501–558]) of Gaussian actions when the group is sofic. Computation of entropy for Gaussian actions has only been done when the acting group is abelian and thus our results are new, even in the amenable case. Fundamental to our approach are methods of non-commutative harmonic analysis and $C^{\ast }$-algebras which replace the Fourier analysis used in the abelian case.

We consider partitioned graphs, by which we mean finite directed graphs with a partitioned edge set ${\mathcal{E}}={\mathcal{E}}^{-}\cup {\mathcal{E}}^{+}$. Additionally given a relation ${\mathcal{R}}$ between the edges in ${\mathcal{E}}^{-}$ and the edges in ${\mathcal{E}}^{+}$, and under the appropriate assumptions on ${\mathcal{E}}^{-},{\mathcal{E}}^{+}$ and ${\mathcal{R}}$, denoting the vertex set of the graph by $\mathfrak{P}$, we speak of an ${\mathcal{R}}$-graph ${\mathcal{G}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$. From ${\mathcal{R}}$-graphs ${\mathcal{G}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$ we construct semigroups (with zero) ${\mathcal{S}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$ that we call ${\mathcal{R}}$-graph semigroups. We write a list of conditions on a topologically transitive subshift with property $(A)$ that together are sufficient for the subshift to have an ${\mathcal{R}}$-graph semigroup as its associated semigroup.

Generalizing previous constructions, we describe a method of presenting subshifts by means of suitably structured finite labeled directed graphs $({\mathcal{V}},~\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706}~)$ with vertex set ${\mathcal{V}}$, edge set $\unicode[STIX]{x1D6F4}$, and a label map that assigns to the edges in $\unicode[STIX]{x1D6F4}$ labels in an ${\mathcal{R}}$-graph semigroup ${\mathcal{S}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{-})$. We denote the presented subshift by $X({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$ and call $X({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$ an ${\mathcal{S}}_{{\mathcal{R}}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{-})$-presentation.

We introduce a property $(B)$ of subshifts that describes a relationship between contexts of admissible words of a subshift, and we introduce a property $(c)$ of subshifts that in addition describes a relationship between the past and future contexts and the context of admissible words of a subshift. Property $(B)$ and the simultaneous occurrence of properties $(B)$ and $(c)$ are invariants of topological conjugacy.

We consider subshifts in which every admissible word has a future context that is compatible with its entire past context. Such subshifts we call right instantaneous. We introduce a property $RI$ of subshifts, and we prove that this property is a necessary and sufficient condition for a subshift to have a right instantaneous presentation. We consider also subshifts in which every admissible word has a future context that is compatible with its entire past context, and also a past context that is compatible with its entire future context. Such subshifts we call bi-instantaneous. We introduce a property $BI$ of subshifts, and we prove that this property is a necessary and sufficient condition for a subshift to have a bi-instantaneous presentation.

We define a subshift as strongly bi-instantaneous if it has for every sufficiently long admissible word $a$ an admissible word $c$, that is contained in both the future context of $a$ and the past context of $a$, and that is such that the word $ca$ is a word in the future context of $a$ that is compatible with the entire past context of $a$, and the word $ac$ is a word in the past context of $a$, that is compatible with the entire future context of $a$. We show that a topologically transitive subshift with property $(A)$, and associated semigroup a graph inverse semigroup ${\mathcal{S}}$, has an ${\mathcal{S}}$-presentation, if and only if it has properties $(c)$ and $BI$, and a strongly bi-instantaneous presentation, if and only if it has properties $(c)$ and $BI$, and all of its bi-instantaneous presentations are strongly bi-instantaneous.

We construct a class of subshifts with property $(A)$, to which certain graph inverse semigroups ${\mathcal{S}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$ are associated, that do not have ${\mathcal{S}}(\mathfrak{P},{\mathcal{E}}^{-},{\mathcal{E}}^{+})$-presentations.

We associate to the labeled directed graphs $({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$ topological Markov chains and Markov codes, and we derive an expression for the zeta function of $X({\mathcal{V}},\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D706})$ in terms of the zeta functions of the topological Markov shifts and the generating functions of the Markov codes.

According to a conjecture of Lindenstrauss and Tsukamoto, a topological dynamical system $(X,T)$ is embeddable in the $d$-cubical shift $(([0,1]^{d})^{\mathbb{Z}},\text{shift})$ if both its mean dimension and periodic dimension are strictly bounded by $d/2$. We verify the conjecture for the class of systems admitting a finite-dimensional non-wandering set and a closed set of periodic points. This class of systems is closely related to systems arising in physics. In particular, we prove an embedding theorem for systems associated with the two-dimensional Navier–Stokes equations of fluid mechanics. The main tool in the proof of the embedding result is the new concept of local markers. Continuing the investigation of (global) markers initiated in previous work it is shown that the marker property is equivalent to a topological version of the Rokhlin lemma. Moreover, new classes of systems are found to have the marker property, in particular, extensions of aperiodic systems with a countable number of minimal subsystems. Extending work of Lindenstrauss we show that, for systems with the marker property, vanishing mean dimension is equivalent to the small boundary property.

In this article we consider Cherry flows on the torus which have two singularities, a source and a saddle, and no periodic orbits. We show that every Cherry flow admits a unique physical measure, whose basin has full volume. This proves a conjecture given by Saghin and Vargas [Invariant measures for Cherry flows. Comm. Math. Phys.317(1) (2013), 55–67]. We also show that the perturbation of Cherry flows depends on the divergence at the saddle: when the divergence is negative, this flow admits a neighborhood, such that any flow in this neighborhood belongs to one of the following three cases: it has a saddle connection; it is a Cherry flow; it is a Morse–Smale flow whose non-wandering set consists of two singularities and one periodic sink. In contrast, when the divergence is non-negative, this flow can be approximated by a non-hyperbolic flow with an arbitrarily large number of periodic sinks.

We provide an equivalent characterization for the existence of one-dimensional irrational rotation factors of conservative torus homeomorphisms that are not eventually annular. It states that an area-preserving non-annular torus homeomorphism $f$ is semiconjugate to an irrational rotation $R_{\unicode[STIX]{x1D6FC}}$ of the circle if and only if there exists a well-defined speed of rotation in some rational direction on the torus, and the deviations from the constant rotation in this direction are uniformly bounded. By means of a counterexample, we also demonstrate that a similar characterization does not hold for eventually annular torus homeomorphisms.

This paper considers the general question of when a topological action of a countable group can be factored into a direct product of non-trivial actions. In the early 1980s, D. Lind considered such questions for $\mathbb{Z}$-shifts of finite type. In particular, we study direct factorizations of subshifts of finite type over $\mathbb{Z}^{d}$ and other groups, and $\mathbb{Z}$-subshifts which are not of finite type. The main results concern direct factors of the multidimensional full $n$-shift, the multidimensional $3$-colored chessboard and the Dyck shift over a prime alphabet. A direct factorization of an expansive $\mathbb{G}$-action must be finite, but an example is provided of a non-expansive $\mathbb{Z}$-action for which there is no finite direct-prime factorization. The question about existence of direct-prime factorization of expansive actions remains open, even for $\mathbb{G}=\mathbb{Z}$.

We consider random dynamical systems generated by a special class of Volterra quadratic stochastic operators on the simplex $S^{m-1}$ . We prove that in contrast to the deterministic set-up the trajectories of the random dynamical system almost surely converge to one of the vertices of the simplex $S^{m-1}$ , implying the survival of only one species. We also show that the minimal random point attractor of the system equals the set of all vertices. The convergence proof relies on a martingale-type limit theorem, which we prove in the appendix.

On the torus $\mathbb{T}^{m}$ of dimension $m\geq 2$ we prove the existence of a real-analytic weak mixing diffeomorphism preserving a measurable Riemannian metric. The proof is based on a real-analytic version of the approximation by conjugation method with explicitly defined conjugation maps and partition elements.

The multicorns are the connectedness loci of unicritical antiholomorphic polynomials $\bar{z}^{d}+c$. We investigate the structure of boundaries of hyperbolic components: we prove that the structure of bifurcations from hyperbolic components of even period is as one would expect for maps that depend holomorphically on a complex parameter (for instance, as for the Mandelbrot set; in this setting, this is a non-obvious fact), while the bifurcation structure at hyperbolic components of odd period is very different. In particular, the boundaries of odd period hyperbolic components consist only of parabolic parameters, and there are bifurcations between hyperbolic components along entire arcs, but only of bifurcation ratio 2. We also count the number of hyperbolic components of any period of the multicorns. Since antiholomorphic polynomials depend only real-analytically on the parameters, most of the techniques used in this paper are quite different from the ones used to prove the corresponding results in a holomorphic setting.

The main result of the paper is classification of free multidimensional Borel flows up to Lebesgue orbit equivalence, by which we mean an orbit equivalence that preserves the Lebesgue measure on each orbit. Two non-smooth $\mathbb{R}^{d}$-flows are shown to be Lebesgue orbit equivalent if and only if they admit the same number of invariant ergodic probability measures.

We investigate the connectedness properties of the set $I^{\!+\!}(f)$ of points where the iterates of an entire function $f$ are unbounded. In particular, we show that $I^{\!+\!}(f)$ is connected whenever iterates of the minimum modulus of $f$ tend to $\infty$. For a general transcendental entire function $f$, we show that $I^{\!+\!}(f)\cup \{\infty \}$ is always connected and that, if $I^{\!+\!}(f)$ is disconnected, then it has uncountably many components, infinitely many of which are unbounded.

In the paper we study relations of rigidity, equicontinuity and pointwise recurrence between an invertible topological dynamical system (t.d.s.) $(X,T)$ and the t.d.s. $(K(X),T_{K})$ induced on the hyperspace $K(X)$ of all compact subsets of $X$, and provide some characterizations. Among other examples, we construct a minimal, non-equicontinuous, distal and uniformly rigid t.d.s. and a weakly mixing t.d.s. which induces dense periodic points on the hyperspace $K(X)$ but itself does not have dense distal points, solving in that way a few open questions from earlier articles by Dong, and Li, Yan and Ye.

Recently, Einsiedler and the authors provided a bound in terms of escape of mass for the amount by which upper-semicontinuity for metric entropy fails for diagonal flows on homogeneous spaces $\unicode[STIX]{x1D6E4}\setminus G$, where $G$ is any connected semisimple Lie group of real rank one with finite center, and $\unicode[STIX]{x1D6E4}$ is any non-uniform lattice in $G$. We show that this bound is sharp, and apply the methods used to establish bounds for the Hausdorff dimension of the set of points that diverge on average.

We determine limit distributions for return- and hitting-time functions of certain asymptotically rare events for conservative ergodic infinite measure preserving transformations with regularly varying asymptotic type. Our abstract result applies, in particular, to shrinking cylinders around typical points of null-recurrent renewal shifts and infinite measure preserving interval maps with neutral fixed points.