In this paper, we address the problem of computing the topological entropy of a map $\psi : G \to G$, where G is a Lie group, given by some power $\psi (g) = g^k$, with k a positive integer. When G is abelian, $\psi $ is an endomorphism and its topological entropy is given by $h(\psi ) = \dim (T(G)) \log (k)$, where $T(G)$ is the maximal torus of G, as shown by Patrão [The topological entropy of endomorphisms of Lie groups. Israel J. Math. 234 (2019), 55–80]. However, when G is not abelian, $\psi $ is no longer an endomorphism and these previous results cannot be used. Still, $\psi $ has some interesting symmetries, for example, it commutes with the conjugations of G. In this paper, the structure theory of Lie groups is used to show that $h(\psi ) = \dim (T)\log (k)$, where T is a maximal torus of G, generalizing the formula in the abelian case. In particular, the topological entropy of powers on compact Lie groups with discrete center is always positive, in contrast to what happens to endomorphisms of such groups, which always have null entropy.

We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, or (ii) is a non-locally-finite product of two infinite groups, then the actions have the same sofic topological entropy. This fact is then used to show that if two free uniquely ergodic and entropy regular probability-measure-preserving actions of such groups are boundedly orbit equivalent then the actions have the same sofic measure entropy. Our arguments are based on a relativization of property SC to sofic approximations and yield more general entropy inequalities.

We consider graphs consisting of finitely many internal rays for degenerating Newton maps and state a convergence result. As an application, we prove that a hyperbolic component in the moduli space of quartic Newton maps is bounded if and only if every element has degree $2$ on the immediate basin of each root. This provides the first complete description of bounded hyperbolic components in a complex two-dimensional moduli space.

We generalize Berg’s notion of quasi-disjointness to actions of countable groups and prove that every measurably distal system is quasi-disjoint from every measure-preserving system. As a corollary, we obtain easy to check necessary and sufficient conditions for two systems to be disjoint, provided one of them is measurably distal. We also obtain a Wiener–Wintner-type theorem for countable amenable groups with distal weights and applications to weighted multiple ergodic averages and multiple recurrence.

Consider a component ${\cal Q}$ of a stratum in the moduli space of area-one abelian differentials on a surface of genus g. Call a property ${\cal P}$ for periodic orbits of the Teichmüller flow on ${\cal Q}$ typical if the growth rate of orbits with property ${\cal P}$ is maximal. We show that the following property is typical. Given a continuous integrable cocycle over the Teichmüller flow with values in a vector bundle $V\to {\cal Q}$ , the logarithms of the eigenvalues of the matrix defined by the cocycle and the orbit are arbitrarily close to the Lyapunov exponents of the cocycle for the Masur–Veech measure.

We prove that for any transitive subshift X with word complexity function $c_n(X)$ , if $\liminf ({\log (c_n(X)/n)}/({\log \log \log n})) = 0$ , then the quotient group ${{\mathrm {Aut}(X,\sigma )}/{\langle \sigma \rangle }}$ of the automorphism group of X by the subgroup generated by the shift $\sigma $ is locally finite. We prove that significantly weaker upper bounds on $c_n(X)$ imply the same conclusion if the gap conjecture from geometric group theory is true. Our proofs rely on a general upper bound for the number of automorphisms of X of range n in terms of word complexity, which may be of independent interest. As an application, we are also able to prove that for any subshift X, if ${c_n(X)}/{n^2 (\log n)^{-1}} \rightarrow 0$ , then $\mathrm {Aut}(X,\sigma )$ is amenable, improving a result of Cyr and Kra. In the opposite direction, we show that for any countable infinite locally finite group G and any unbounded increasing $f: \mathbb {N} \rightarrow \mathbb {N}$ , there exists a minimal subshift X with ${{\mathrm {Aut}(X,\sigma )}/{\langle \sigma \rangle }}$ isomorphic to G and ${c_n(X)}/{nf(n)} \rightarrow 0$ .

This is the first paper in a two-part series containing some results on dimension estimates for $C^1$ iterated function systems and repellers. In this part, we prove that the upper box-counting dimension of the attractor of any $C^1$ iterated function system (IFS) on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of any ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Similar results are obtained for the repellers for $C^1$ expanding maps on Riemannian manifolds.

Let $\mathcal {P}$ be an (unbounded) countable multiset of primes (that is, every prime may appear multiple times) and let $G=\bigoplus _{p\in \mathcal {P}}\mathbb {F}_p$ . We develop a Host–Kra structure theory for the universal characteristic factors of an ergodic G-system. More specifically, we generalize the main results of Bergelson, Tao and Ziegler [An inverse theorem for the uniformity seminorms associated with the action of $\mathbb {F}_p^\infty $ . Geom. Funct. Anal. 19(6) (2010), 1539–1596], who studied these factors in the special case $\mathcal {P}=\{p,p,p,\ldots \}$ for some fixed prime p. As an application we deduce a Khintchine-type recurrence theorem in the flavor of Bergelson, Tao and Ziegler [Multiple recurrence and convergence results associated to $F_p^\omega $ -actions. J. Anal. Math. 127 (2015), 329–378] and Bergelson, Host and Kra [Multiple recurrence and nilsequences. Invent. Math. 160(2) (2005), 261–303, with an appendix by I. Ruzsa].

We consider the McMullen maps $f_{\unicode{x3bb} }(z)=z^{n}+\unicode{x3bb} z^{-n}$ with $\unicode{x3bb} \in \mathbb {C}^{*}$ and $n \geq 3$. We prove that the closures of escape hyperbolic components are pairwise disjoint and the boundaries of all bounded escape components (the McMullen domain and Sierpiński holes) are quasi-circles with Hausdorff dimension strictly between $1$ and $2$.

We prove an explicit characterization of the points in Thurston’s Master Teapot, which can be implemented algorithmically to test whether a point in $\mathbb {C}\times \mathbb {R}$ belongs to the complement of the Master Teapot. As an application, we show that the intersection of the Master Teapot with the unit cylinder is not symmetrical under reflection through the plane that is the product of the imaginary axis of $\mathbb {C}$ and $\mathbb {R}$.

We study approximation schemes for shift spaces over a finite alphabet using (pseudo)metrics connected to Ornstein’s ${\bar d}$ metric. This leads to a class of shift spaces we call ${\bar d}$ -approachable. A shift space is ${\bar d}$ -approachable when its canonical sequence of Markov approximations converges to it also in the ${\bar d}$ sense. We give a topological characterization of chain-mixing ${\bar d}$ -approachable shift spaces. As an application we provide a new criterion for entropy density of ergodic measures. Entropy density of a shift space means that every invariant measure $\mu $ of such a shift space is the weak $^*$ limit of a sequence $\mu _n$ of ergodic measures with the corresponding sequence of entropies $h(\mu _n)$ converging to $h(\mu )$ . We prove ergodic measures are entropy-dense for every shift space that can be approximated in the ${\bar d}$ pseudometric by a sequence of transitive sofic shifts. This criterion can be applied to many examples that were beyond the reach of previously known techniques including hereditary $\mathscr {B}$ -free shifts and some minimal or proximal systems. The class of symbolic dynamical systems covered by our results includes also shift spaces where entropy density was established previously using the (almost) specification property.

We prove the first orbit equivalence superrigidity results for actions of type III$_\unicode{x3bb} $ when $\unicode{x3bb} \neq 1$. These actions arise as skew products of actions of dense subgroups of $\operatorname {SL}(n,\mathbb {R})$ on the sphere $S^{n-1}$ and they can have any prescribed associated flow.

For a pseudo-Anosov flow $\varphi $ without perfect fits on a closed $3$-manifold, Agol–Guéritaud produce a veering triangulation $\tau $ on the manifold M obtained by deleting the singular orbits of $\varphi $. We show that $\tau $ can be realized in M so that its 2-skeleton is positively transverse to $\varphi $, and that the combinatorially defined flow graph $\Phi $ embedded in M uniformly codes the orbits of $\varphi $ in a precise sense. Together with these facts, we use a modified version of the veering polynomial, previously introduced by the authors, to compute the growth rates of the closed orbits of $\varphi $ after cutting M along certain transverse surfaces, thereby generalizing the work of McMullen in the fibered setting. These results are new even in the case where the transverse surface represents a class in the boundary of a fibered cone of M. Our work can be used to study the flow $\varphi $ on the original closed manifold. Applications include counting growth rates of closed orbits after cutting along closed transverse surfaces, defining a continuous, convex entropy function on the ‘positive’ cone in $H^1$ of the cut-open manifold, and answering a question of Leininger about the closure of the set of all stretch factors arising as monodromies within a single fibered cone of a $3$-manifold. This last application connects to the study of endperiodic automorphisms of infinite-type surfaces and the growth rates of their periodic points.

We present several applications of the weak specification property and certain topological Markov properties, recently introduced by Barbieri, García-Ramos, and Li [Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group. Adv. Math. 397 (2022), 52], and implied by the pseudo-orbit tracing property, for general expansive group actions on compact spaces. First we show that any expansive action of a countable amenable group on a compact metrizable space satisfying the weak specification and strong topological Markov properties satisfies the Moore property, that is, every surjective endomorphism of such dynamical system is pre-injective. This together with an earlier result of Li (where the strong topological Markov property is not needed) of the Myhill property [Garden of Eden and specification. Ergod. Th. & Dynam. Sys. 39 (2019), 3075–3088], which we also re-prove here, establishes the Garden of Eden theorem for all expansive actions of countable amenable groups on compact metrizable spaces satisfying the weak specification and strong topological Markov properties. We hint how to easily generalize this result even for uncountable amenable groups and general compact, not necessarily metrizable, spaces. Second, we generalize the recent result of Cohen [The large scale geometry of strongly aperiodic subshifts of finite type. Adv. Math. 308 (2017), 599–626] that any subshift of finite type of a finitely generated group having at least two ends has weakly periodic points. We show that every expansive action of such a group having a certain Markov topological property, again implied by the pseudo-orbit tracing property, has a weakly periodic point. If it has additionally the weak specification property, the set of such points is dense.

In this paper, directional sequence entropy and directional Kronecker algebra for $\mathbb {Z}^q$ -systems are introduced. The relation between sequence entropy and directional sequence entropy are established. Meanwhile, directional discrete spectrum systems and directional null systems are defined. It is shown that a $\mathbb {Z}^q$ -system has directional discrete spectrum if and only if it is directional null. Moreover, it turns out that a $\mathbb {Z}^q$ -system has directional discrete spectrum along q linearly independent directions if and only if it has discrete spectrum.

We consider skew-product maps over circle rotations $x\mapsto x+\alpha \;(\mod 1)$ with factors that take values in ${\textrm {SL}}(2,{\mathbb {R}})$ . In numerical experiments, with $\alpha $ the inverse golden mean, Fibonacci iterates of maps from the almost Mathieu family exhibit asymptotic scaling behavior that is reminiscent of critical phase transitions. In a restricted setup that is characterized by a symmetry, we prove that critical behavior indeed occurs and is universal in an open neighborhood of the almost Mathieu family. This behavior is governed by a periodic orbit of a renormalization transformation. An extension of this transformation is shown to have a second periodic orbit as well, and we present some evidence that this orbit attracts supercritical almost Mathieu maps.

In this paper, we study the ergodicity of the geodesic flows on surfaces with no focal points. Let M be a smooth connected and closed surface equipped with a $C^{\infty }$ Riemannian metric g, whose genus $\mathfrak {g} \geq 2$. Suppose that $(M,g)$ has no focal points. We prove that the geodesic flow on the unit tangent bundle of M is ergodic with respect to the Liouville measure, under the assumption that the set of points on M with negative curvature has at most finitely many connected components.

We prove an extension of the Moore–Schmidt theorem on the triviality of the first cohomology class of cocycles for the action of an arbitrary discrete group on an arbitrary measure space and for cocycles with values in an arbitrary compact Hausdorff abelian group. The proof relies on a ‘conditional’ Pontryagin duality for spaces of abstract measurable maps.

For any $r\in [0,1]$ we give an example of a rigid operator whose spectrum is the annulus $\{\lambda\in \mathbb{C} : r \le |\lambda| \le 1 \} $ . In particular, when $r=0$ this operator is rigid and non-invertible, and when $r\in {\kern1pt}] 0,1 [ $ this operator is invertible but its inverse is not rigid. This answers two questions of Costakis, Manoussos and Parissis [Recurrent linear operators. Complex Anal. Oper. Theory 8 (2014), 1601–1643].