In this paper we prove that permutations of even length with maximal entropy are not unique. There are in fact two distinct permutations, $\theta_n$ and $\widetilde{\theta_n}$, with maximal entropy, both of which are self-dual and $\widetilde{\theta_n}$ is the reverse of $\theta_n$.

We consider a class of dynamical systems generated by finite sets of Möbius transformations acting on the unit disc. Compositions of such Möbius transformations give rise to sequences of transformations that are used in the theory of continued fractions. In that theory, the distinction between sequences of limit-point type and sequences of limit-disc type is of central importance. We prove that sequences of limit-disc type only arise in exceptional circumstances, and we give necessary and sufficient conditions for a sequence to be of limit-disc type. We also calculate the Hausdorff dimension of the set of sequences of limit-disc type in some significant cases. Finally, we obtain strong and complete results on the convergence of these dynamical systems.

Let $(X,\mathfrak{B},\unicode[STIX]{x1D707})$ be a Borel probability space. Let $T_{n}:X\rightarrow X$ be a sequence of continuous transformations on $X$. Let $\unicode[STIX]{x1D708}$ be a probability measure on $X$ such that $(1/N)\sum _{n=1}^{N}(T_{n})_{\ast }\unicode[STIX]{x1D708}\rightarrow \unicode[STIX]{x1D707}$ in the weak-$\ast$ topology. Under general conditions, we show that for $\unicode[STIX]{x1D708}$ almost every $x\in X$, the measures $(1/N)\sum _{n=1}^{N}\unicode[STIX]{x1D6FF}_{T_{n}x}$ become equidistributed towards $\unicode[STIX]{x1D707}$ if $N$ is restricted to a set of full upper density. We present applications of these results to translates of closed orbits of Lie groups on homogeneous spaces. As a corollary, we prove equidistribution of exponentially sparse orbits of the horocycle flow on quotients of $\text{SL}(2,\mathbb{R})$, starting from every point in almost every direction.

Let $f$ and $g$ be two class $P$ -homeomorphisms of the circle $S^{1}$ with break point singularities, which are differentiable maps except at some singular points where the derivative has a jump. Assume that $f$ and $g$ have irrational rotation numbers and the derivatives $\text{Df}$ and $\text{Dg}$ are absolutely continuous on every continuity interval of $\text{Df}$ and $\text{Dg}$ , respectively. We prove that if the product of the $f$ -jumps along all break points of $f$ is distinct from that of $g$ then the homeomorphism $h$ conjugating $f$ and $g$ is a singular function, i.e. it is continuous on $S^{1}$ , but $\text{Dh}(x)=0$  almost everywhere with respect to the Lebesgue measure. This result generalizes previous results for one and two break points obtained by Dzhalilov, Akin and Temir, and Akhadkulov, Dzhalilov and Mayer. As a consequence, we get in particular Dzhalilov–Mayer–Safarov’s theorem: if the product of the $f$ -jumps along all break points of $f$ is distinct from $1$ , then the invariant measure $\unicode[STIX]{x1D707}_{f}$ is singular with respect to the Lebesgue measure.

Berend [Multi-invariant sets on tori. Trans. Amer. Math. Soc.280(2) (1983), 509–532] obtained necessary and sufficient conditions on a ℤr-action α on a torus 𝕋d by toral automorphisms in order for every orbit to be either finite or dense. One of these conditions is that for every common eigendirection of the ℤr-action there is an element n∈ℤr such that αn expands this direction. In this paper, we investigate what happens when this condition is removed; more generally, we consider a partial orbit {αn⋅x:n∈Ω} where Ω is a set of elements which acts in an approximately isometric way on a given set of eigendirections. This analysis is used in an essential way in the work of the author with E. Lindenstrauss [Topological self-joinings of Cartan actions by toral automorphisms. Preprint, 2010] classifying topological self-joinings of maximal ℤr-actions on tori for r≥3 .

In this paper, we present results for the forcing relation on the set of braid types of periodic orbits of a rotational horseshoe on the annulus. Precisely, we are concerned with a family of periodic orbits, called the Boyland family, and we prove that for each pair $(r,s)$ of rational numbers with $r<s$ in $(0,1)$, there exists a non-monotone orbit $B_{r,s}$ in this family which has pseudo-Anosov type and rotation interval $[r,s]$. Furthermore, the forcing relation among these orbits is given by the inclusion order on their rotation sets. It is also proved that the Markov partition associated to each Boyland orbit comes from a pruning map which projects to a bimodal circle map. This family also contains the Holmes orbits $H_{p/q}$, which are the largest for the forcing order among all the $(p,q)$-orbits of the rotational horseshoe.

In this paper, we point out a close relationship between two standard classical problems in mechanics which have coexisted in textbooks for many decades: (1) the pendulum whose suspension point executes fast periodic motion along a given curve; and (2) the skate (known also as the Prytz planimeter, or the ‘bicycle’). More generally, we deal with dynamical systems subjected to rapidly oscillating forcing. Examples include: charged particles in rapidly oscillating electromagnetic fields, in particular the Paul trap; particles in an acoustic wave; a bead sliding on a rapidly vibrating hoop. It turns out that the averaged systems of such kind are approximated by a non-holonomic system. The holonomy turns out to have a transparent geometrical or physical interpretation. For the example of a particle in an acoustic wave the holonomy is directly proportional to the speed of the vibration-induced drift.

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.

This paper shows that any symplectic diffeomorphism $\mathbf{F} :(\Sigma^{2n},\eta) \to (\Sigma^{2n},\eta)$ can be embedded as a subsystem of a Liouville-integrable Hamiltonian flow on some symplectic manifold. If F is real-analytic, then the flow can be chosen to be real-analytic, but it is Liouville integrable with smooth first integrals. Examples are constructed of integrable, volume-preserving Hamiltonian flows on Poisson manifolds whose metric entropy with respect to the volume form is positive. Completely integrable Hamiltonian flows on a symplectic manifold are constructed which have positive metric entropy with respect to an invariant probability measure that is absolutely continuous with respect to the canonical volume form.

In this paper we study topological cocycles for minimal homeomorphisms on a compact metric space. We introduce a notion of an essential range for topological cocycles with values in a locally compact group, and we show that this notion coincides with the well-known topological essential range if the group is abelian. We then define a regularity condition for cocycles and prove several results on the essential ranges and the orbit closures of the skew product of regular cocycles. Furthermore, we show that recurrent cocycles for a minimal rotation on a locally connected compact group are always regular, assuming that their ranges are in a nilpotent group, and then their essential ranges are almost connected.

Consider an exact action of a discrete group $G$ on a separable C*-algebra $A$. It is shown that the reduced crossed product $A\rtimes _{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D706}}G$ is strongly purely infinite—provided that the action of $G$ on any quotient $A/I$ by a $G$-invariant closed ideal $I\neq A$ is element-wise properly outer and that the action of $G$ on $A$ is $G$-separating (cf. Definition 5.1). This is the first non-trivial sufficient general criterion for strong pure infiniteness of reduced crossed products of C*-algebras $A$ that are not $G$-simple. In the case $A=\text{C}_{0}(X)$, the notion of a $G$-separating action corresponds to the property that two compact sets $C_{1}$ and $C_{2}$, that are contained in open subsets $C_{j}\subseteq U_{j}\subseteq X$, can be mapped by elements $g_{1},g_{2}$ of $G$ onto disjoint sets $\unicode[STIX]{x1D70E}_{g_{j}}(C_{j})\subseteq U_{j}$, but satisfy not necessarily the contraction property $\unicode[STIX]{x1D70E}_{g_{j}}(U_{j})\subseteq \overline{U_{j}}$. A generalization of strong boundary actions on compact spaces to non-unital and non-commutative C*-algebras $A$ (cf. Definition 7.1) is also introduced. It is stronger than the notion of $G$-separating actions by Proposition 7.6, because $G$-separation does not imply $G$-simplicity and there are examples of $G$-separating actions with reduced crossed products that are stably projection-less and non-simple.

In the first part of this paper, we consider several natural problems about locally homogeneous rigid geometric structures. In particular, we formulate a notion of topological completeness which is adapted to the study of global rigidity of chaotic dynamical systems. In the second part of the paper, we prove the following result: let φ be a C∞ expanding map of a closed manifold. If φ preserves a topologically complete C∞ rigid geometric structure, then φ is C∞ conjugate to an expanding infra-nilendomorphism.

We first consider the dynamics of a class of meromorphic skew products having superattracting fixed points or fixed indeterminacy points at the origin. Our theorem asserts that, if a map has a suitable weight, then it is conjugate to the associated monomial map on an invariant open set whose closure contains the origin. We next extend this result to a wider class of meromorphic maps such that the eigenvalues of the associated matrices are real and greater than $1$.

We show that any non-trivial self-similar subset of the real line that is invariant under a lattice iterated function system (IFS) satisfying the open set condition (OSC) is not Minkowski measurable. So far, this has only been known for special classes of such sets. Thus, we provide the last puzzle-piece in proving that under the OSC a non-trivial self-similar subset of the real line is Minkowski measurable if and only if it is invariant under a non-lattice IFS, a 25-year-old conjecture.

Let $h:D\rightarrow {\mathbb R}^2$ be a homeomorphism of a disk into the plane. Let $C\subset {\rm int}\,D$ be a Cantor set such that $(C,h|_C)$ is an adding machine. We give a condition under which there are points outside of C which have their entire orbits contained in D.

Consider a three-dimensional partially hyperbolic diffeomorphism. It is proved that under some rigid hypothesis on the tangent bundle dynamics, the map is (modulo finite covers and iterates) an Anosov diffeomorphism, a (generalized) skew-product or the time-one map of an Anosov flow, thus recovering a well-known classification conjecture of the second author to this restricted setting.

Let Rw(z): W × C∞ → C∞ be an analytic family of rational functions, J(w) the Julia set of Rw and J*(w) the upper semicontinuous regularization of J(w). We shall discuss the relationship between J(w) and J*(w) as well as some related problems.

Given a factor code $\unicode[STIX]{x1D70B}$ from a shift of finite type $X$ onto a sofic shift $Y$, an ergodic measure $\unicode[STIX]{x1D708}$ on $Y$, and a function $V$ on $X$ with sufficient regularity, we prove an invariant upper bound on the number of ergodic measures on $X$ which project to $\unicode[STIX]{x1D708}$ and maximize the measure pressure $h(\unicode[STIX]{x1D707})+\int V\,d\unicode[STIX]{x1D707}$ among all measures in the fiber $\unicode[STIX]{x1D70B}^{-1}(\unicode[STIX]{x1D708})$. If $\unicode[STIX]{x1D708}$ is fully supported, this bound is the class degree of $\unicode[STIX]{x1D70B}$. This generalizes a previous result for the special case of $V=0$ and thus settles a conjecture raised by Allahbakhshi and Quas.

We study ${\Bbb Z}^2$ actions arising as base point actions of tiling representations of ${\Bbb R}^2$ flows. We cast an equivalence relation between such actions in terms of a simple arithmetic condition on an orbit equivalence. Stated as such, our equivalence class is easily seen to be a restricted even Kakutani equivalence, as we well as a higher-dimensional generalization of $\alpha$-equivalence, defined by Fieldsteel, del Junco and Rudolph for ${\Bbb Z}$ actions.

We consider here a certain class of groupoids obtained via an equivalence relation (the so-called subgroupoids of pair groupoids). We generalize to Haar systems in these groupoids some results related to entropy and pressure which are well known in thermodynamic formalism. We introduce a transfer operator, where the equivalence relation (which defines the groupoid) plays the role of the dynamics and the corresponding transverse function plays the role of the a priori probability. We also introduce the concept of invariant transverse probability and of entropy for an invariant transverse probability, as well as of pressure for transverse functions. Moreover, we explore the relation between quasi-invariant probabilities and transverse measures. Some of the general results presented here are not for continuous modular functions but for the more general class of measurable modular functions.