In this article we prove estimates for the topological pressure of the set of points whose Birkhoff time averages are far from the space averages corresponding to the unique equilibrium state that has a weak Gibbs property. In particular, if $f$ has an expanding repeller and $\unicode[STIX]{x1D719}$ is a Hölder continuous potential, we prove that the topological pressure of the set of points whose accumulation values of Birkhoff averages belong to some interval $I\subset \mathbb{R}$ can be expressed in terms of the topological pressure of the whole system and the large deviations rate function. As a byproduct we deduce that most irregular sets for maps with the specification property have topological pressure strictly smaller than the whole system. Some extensions to a non-uniformly hyperbolic setting, level-2 irregular sets and hyperbolic flows are also given.

We give a criterion for the rigidity of the action of a group of affine transformations of a homogeneous space of a real Lie group. Let $G$ be a real Lie group, $\unicode[STIX]{x1D6EC}$ a lattice in $G$ , and $\unicode[STIX]{x1D6E4}$ a subgroup of the affine group $\text{Aff}(G)$ stabilizing $\unicode[STIX]{x1D6EC}$ . Then the action of $\unicode[STIX]{x1D6E4}$ on $G/\unicode[STIX]{x1D6EC}$ has the rigidity property in the sense of Popa [On a class of type $\text{II}_{1}$ factors with Betti numbers invariants. Ann. of Math. (2) 163(3) (2006), 809–899] if and only if the induced action of $\unicode[STIX]{x1D6E4}$ on $\mathbb{P}(\mathfrak{g})$ admits no $\unicode[STIX]{x1D6E4}$ -invariant probability measure, where $\mathfrak{g}$ is the Lie algebra of $G$ . This generalizes results of Burger [Kazhdan constants for $\text{SL}(3,\mathbf{Z})$ . J. Reine Angew. Math. 413 (1991), 36–67] and Ioana and Shalom [Rigidity for equivalence relations on homogeneous spaces. Groups Geom. Dyn. 7(2) (2013), 403–417]. As an application, we establish rigidity for the action of a class of groups acting by automorphisms on nilmanifolds associated to two-step nilpotent Lie groups.

We study ergodic finite and infinite measures defined on the path space $X_{B}$ of a Bratteli diagram $B$ which are invariant with respect to the tail equivalence relation on $X_{B}$. Our interest is focused on measures supported by vertex and edge subdiagrams of $B$. We give several criteria when a finite invariant measure defined on the path space of a subdiagram of $B$ extends to a finite invariant measure on $B$. Given a finite ergodic measure on a Bratteli diagram $B$ and a subdiagram $B^{\prime }$ of $B$, we find the necessary and sufficient conditions under which the measure of the path space $X_{B^{\prime }}$ of $B^{\prime }$ is positive. For a class of Bratteli diagrams of finite rank, we determine when they have maximal possible number of ergodic invariant measures. The case of diagrams of rank two is completely studied. We also include an example which explicitly illustrates the proven results.

We prove a generalization of the fixed point theorem of Cartwright and Littlewood. Namely, suppose that $h:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ is an orientation preserving planar homeomorphism, and let $C$ be a continuum such that $h^{-1}(C)\cup C$ is acyclic. If there is a $c\in C$ such that $\{h^{-i}(c):i\in \mathbb{N}\}\subseteq C$, or $\{h^{i}(c):i\in \mathbb{N}\}\subseteq C$, then $C$ also contains a fixed point of $h$. Our approach is based on Brown’s short proof of the result of Cartwright and Littlewood. In addition, making use of a linked periodic orbits theorem of Bonino, we also prove a counterpart of the aforementioned result for orientation reversing homeomorphisms, that guarantees a $2$-periodic orbit in $C$ if it contains a $k$-periodic orbit ($k>1$).

Let $G$ be a semisimple Lie group of rank one and $\unicode[STIX]{x1D6E4}$ be a torsion-free discrete subgroup of $G$ . We show that in $G/\unicode[STIX]{x1D6E4}$ , given $\unicode[STIX]{x1D716}>0$ , any trajectory of a unipotent flow remains in the set of points with injectivity radius larger than $\unicode[STIX]{x1D6FF}$ for a $1-\unicode[STIX]{x1D716}$ proportion of the time, for some $\unicode[STIX]{x1D6FF}>0$ . The result also holds for any finitely generated discrete subgroup $\unicode[STIX]{x1D6E4}$ and this generalizes Dani’s quantitative non-divergence theorem [On orbits of unipotent flows on homogeneous spaces. Ergod. Th. & Dynam. Sys.4(1) (1984), 25–34] for lattices of rank-one semisimple groups. Furthermore, for a fixed $\unicode[STIX]{x1D716}>0$ , there exists an injectivity radius $\unicode[STIX]{x1D6FF}$ such that, for any unipotent trajectory $\{u_{t}g\unicode[STIX]{x1D6E4}\}_{t\in [0,T]}$ , either it spends at least a $1-\unicode[STIX]{x1D716}$ proportion of the time in the set with injectivity radius larger than $\unicode[STIX]{x1D6FF}$ , for all large $T>0$ , or there exists a $\{u_{t}\}_{t\in \mathbb{R}}$ -normalized abelian subgroup $L$ of $G$ which intersects $g\unicode[STIX]{x1D6E4}g^{-1}$ in a small covolume lattice. We also extend these results to when $G$ is the product of rank-one semisimple groups and $\unicode[STIX]{x1D6E4}$ a discrete subgroup of $G$ whose projection onto each non-trivial factor is torsion free.

We consider the family of quadratic Hénon diffeomorphisms of the plane $\mathbb{R}^{2}$. A map will be said to be a ‘horseshoe’ if its restriction to the non-wandering set is hyperbolic and conjugate to the full 2-shift. We give a criterion for being a horseshoe based on an auxiliary coding which describes positions of points relative to the stable manifold of one of the fixed points. In addition we describe the topological conjugacy type of maps on the boundary of the horseshoe locus. We use complex techniques and we work with maps in a parameter region which is a two-dimensional analog of the familiar ‘$1/2$-wake’ for the quadratic family $p_{c}(z)=z^{2}$.

Fix $d\geq 2$. Given a finite undirected graph ${\mathcal{H}}$ without self-loops and multiple edges, consider the corresponding ‘vertex’ shift, $\text{Hom}(\mathbb{Z}^{d},{\mathcal{H}})$, denoted by $X_{{\mathcal{H}}}$. In this paper, we focus on ${\mathcal{H}}$ which is ‘four-cycle free’. There are two main results of this paper. Firstly, that $X_{{\mathcal{H}}}$ has the pivot property, meaning that, for all distinct configurations $x,y\in X_{{\mathcal{H}}}$, which differ only at a finite number of sites, there is a sequence of configurations $x=x^{1},x^{2},\ldots ,x^{n}=y\in X_{{\mathcal{H}}}$ for which the successive configurations $x^{i},x^{i+1}$ differ exactly at a single site. Secondly, if ${\mathcal{H}}$ is connected ,then $X_{{\mathcal{H}}}$ is entropy minimal, meaning that every shift space strictly contained in $X_{{\mathcal{H}}}$ has strictly smaller entropy. The proofs of these seemingly disparate statements are related by the use of the ‘lifts’ of the configurations in $X_{{\mathcal{H}}}$ to the universal cover of ${\mathcal{H}}$ and the introduction of ‘height functions’ in this context.

Thurston parameterized quadratic invariant laminations with a non-invariant lamination, the quotient of which yields a combinatorial model for the Mandelbrot set. As a step toward generalizing this construction to cubic polynomials, we consider slices of the family of cubic invariant laminations defined by a fixed critical leaf with non-periodic endpoints. We parameterize each slice by a lamination just as in the quadratic case, relying on the techniques of smart criticality previously developed by the authors.

We give explicit formulae for fringe lengths of the Calegari–Walker ziggurats—i.e., graphs of extremal rotation numbers associated with positive words in free groups. These formulae reveal (partial) integral projective self-similarity in ziggurat fringes, which are low-dimensional projections of characteristic polyhedra on the bounded cohomology of free groups. This explains phenomena observed experimentally by Gordenko, Calegari and Walker.

Hoehn and Mouron [Hierarchies of chaotic maps on continua. Ergod. Th. & Dynam. Sys.34 (2014), 1897–1913] constructed a map on the universal dendrite that is topologically weakly mixing but not mixing. We modify the Hoehn–Mouron example to show that there exists a transitive (even weakly mixing) dendrite map with zero topological entropy. This answers the question of Baldwin [Entropy estimates for transitive maps on trees. Topology40(3) (2001), 551–569].

We consider for each $t$ the set $K(t)$ of points of the circle whose forward orbit for the doubling map does not intersect $(0,t)$, and look at the dimension function $\unicode[STIX]{x1D702}(t):=\text{H.dim}\,K(t)$. We prove that at every bifurcation parameter $t$, the local Hölder exponent of the dimension function equals the value of the function $\unicode[STIX]{x1D702}(t)$ itself. A similar statement holds for general expanding maps of the circle: namely, we consider the topological entropy of the map restricted to the survival set, and obtain bounds on its local Hölder exponent in terms of the value of the function.

We generalize Petridis’s new proof of Plünnecke’s graph inequality to graphs whose vertex set is a measure space. Consequently, by a recent work of Björklund and Fish, this gives new Plünnecke inequalities for measure-preserving actions which enable us to deduce, via a Furstenberg correspondence principle, Banach density estimates in countable abelian groups that extend those given by Jin.

The evolution of cellular automata and of finitely anticipative transformations is studied by using right sets. These are the sets of symbols that are compatible with a past of a position and the respective coordinate of the transformation. Our main result shows, under some suitable conditions, that if the entropy converges to zero then the right sets increase towards the whole alphabet. We discuss these concepts with Wolfram automata.

In 1985, Barnsley and Harrington defined a ‘Mandelbrot Set’ ${\mathcal{M}}$ for pairs of similarities: this is the set of complex numbers $z$ with $0<|z|<1$ for which the limit set of the semigroup generated by the similarities

$$\begin{eqnarray}x\mapsto zx\quad \text{and}\quad x\mapsto z(x-1)+1\end{eqnarray}$$

${\mathcal{M}}$

$\{-1,0,1\}$

${\mathcal{M}}$

${\mathcal{M}}$

${\mathcal{M}}$

We prove that any ergodic non-atomic probability-preserving action of an irreducible lattice in a semisimple group, with at least one factor being connected and of higher-rank, is essentially free. This generalizes the result of Stuck and Zimmer [Stabilizers for ergodic actions of higher rank semisimple groups. Ann. of Math. (2)139(3) (1994), 723–747], who found that the same statement holds when the ambient group is a semisimple real Lie group and every simple factor is of higher-rank. We also prove a generalization of a result of Bader and Shalom [Factor and normal subgroup theorems for lattices in products of groups. Invent. Math.163(2) (2006), 415–454] by showing that any probability-preserving action of a product of simple groups, with at least one having property $(T)$, which is ergodic for each simple subgroup, is either essentially free or essentially transitive. Our method involves the study of relatively contractive maps and the Howe–Moore property, rather than relying on algebraic properties of semisimple groups and Poisson boundaries, and introduces a generalization of the ergodic decomposition to invariant random subgroups, which is of independent interest.

We consider a non-uniquely ergodic dynamical system given by a $\mathbb{Z}^{l}$-action (or $(\mathbb{N}\cup \{0\})^{l}$-action) $\unicode[STIX]{x1D70F}$ on a non-empty compact metrisable space $\unicode[STIX]{x1D6FA}$, for some $l\in \mathbb{N}$. Let (D) denote the following property: the graph of the restriction of the entropy map $h^{\unicode[STIX]{x1D70F}}$ to the set of ergodic states is dense in the graph of $h^{\unicode[STIX]{x1D70F}}$. We assume that $h^{\unicode[STIX]{x1D70F}}$ is finite and upper semi-continuous. We give several criteria in order that (D) holds, each of which is stated in terms of a basic notion: Gateaux differentiability of the pressure map $P^{\unicode[STIX]{x1D70F}}$ on some sets dense in the space $C(\unicode[STIX]{x1D6FA})$ of real-valued continuous functions on $\unicode[STIX]{x1D6FA}$, level-two large deviation principle, level-one large deviation principle, convexity properties of some maps on $\mathbb{R}^{n}$ for all $n\in \mathbb{N}$. The one involving the Gateaux differentiability of $P^{\unicode[STIX]{x1D70F}}$ is of particular relevance in the context of large deviations since it establishes a clear comparison with another well-known sufficient condition: we show that for each non-empty $\unicode[STIX]{x1D70E}$-compact subset $\unicode[STIX]{x1D6F4}$ of $C(\unicode[STIX]{x1D6FA})$, (D) is equivalent to the existence of an infinite dimensional vector space $V$ dense in $C(\unicode[STIX]{x1D6FA})$ such that $f+g$ has a unique equilibrium state for all $(f,g)\in \unicode[STIX]{x1D6F4}\times V\setminus \{0\}$; any Schauder basis $(f_{n})$ of $C(\unicode[STIX]{x1D6FA})$ whose linear span contains $\unicode[STIX]{x1D6F4}$ admits an arbitrary small perturbation $(h_{n})$ so that one can take $V=\text{span}(\{f_{n}+h_{n}:n\in \mathbb{N}\})$. Taking $\unicode[STIX]{x1D6F4}=\{0\}$, the existence of an infinite dimensional vector space dense in $C(\unicode[STIX]{x1D6FA})$ constituted by functions admitting a unique equilibrium state is equivalent to (D) together with the uniqueness of the measure of maximum entropy.

Suppose that $X$ is a Polish space, $E$ is a countable Borel equivalence relation on $X$, and $\unicode[STIX]{x1D707}$ is an $E$-invariant Borel probability measure on $X$. We consider the circumstances under which for every countable non-abelian free group $\unicode[STIX]{x1D6E4}$, there is a Borel sequence $(\cdot _{r})_{r\in \mathbb{R}}$ of free actions of $\unicode[STIX]{x1D6E4}$ on $X$, generating subequivalence relations $E_{r}$ of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic, with the further property that $(E_{r})_{r\in \mathbb{R}}$ is an increasing sequence of relations which are pairwise incomparable under $\unicode[STIX]{x1D707}$-reducibility. In particular, we show that if $E$ satisfies a natural separability condition, then this is the case as long as there exists a free Borel action of a countable non-abelian free group on $X$, generating a subequivalence relation of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic.

For each $n=1,2,\ldots ,$ let $\text{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^{n}$ be the affine group over the integers. For every point $x=(x_{1},\ldots ,x_{n})\in \mathbb{R}^{n}$ let $\text{orb}(x)=\{\unicode[STIX]{x1D6FE}(x)\in \mathbb{R}^{n}\mid \unicode[STIX]{x1D6FE}\in \text{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^{n}\}.$ Let $G_{x}$ be the subgroup of the additive group $\mathbb{R}$ generated by $x_{1},\ldots ,x_{n},1$. If $\text{rank}(G_{x})\neq n$ then $\text{orb}(x)=\{y\in \mathbb{R}^{n}\mid G_{y}=G_{x}\}$. Thus, $G_{x}$ is a complete classifier of $\text{orb}(x)$. By contrast, if $\text{rank}(G_{x})=n$, knowledge of $G_{x}$ alone is not sufficient in general to uniquely recover $\text{orb}(x)$; as a matter of fact, $G_{x}$ determines precisely $\max (1,\unicode[STIX]{x1D719}(d)/2)$ different orbits, where $d$ is the denominator of the smallest positive non-zero rational in $G_{x}$ and $\unicode[STIX]{x1D719}$ is the Euler function. To get a complete classification, rational polyhedral geometry provides an integer $1\leq c_{x}\leq \max (1,d/2)$ such that $\text{orb}(y)=\text{orb}(x)$ if and only if $(G_{x},c_{x})=(G_{y},c_{y})$. Applications are given to lattice-ordered abelian groups with strong unit and to AF $C^{\ast }$-algebras.

Let $\unicode[STIX]{x1D6E4}$ be a lattice in a simply connected nilpotent Lie group $G$. Given an infinite measure-preserving action $T$ of $\unicode[STIX]{x1D6E4}$ and a ‘direction’ in $G$ (i.e. an element $\unicode[STIX]{x1D703}$ of the projective space $P(\mathfrak{g})$ of the Lie algebra $\mathfrak{g}$ of $G$), some notions of recurrence and rigidity for $T$ along $\unicode[STIX]{x1D703}$ are introduced. It is shown that the set of recurrent directions ${\mathcal{R}}(T)$ and the set of rigid directions for $T$ are both $G_{\unicode[STIX]{x1D6FF}}$. In the case where $G=\mathbb{R}^{d}$ and $\unicode[STIX]{x1D6E4}=\mathbb{Z}^{d}$, we prove that (a) for each $G_{\unicode[STIX]{x1D6FF}}$-subset $\unicode[STIX]{x1D6E5}$ of $P(\mathfrak{g})$ and a countable subset $D\subset \unicode[STIX]{x1D6E5}$, there is a rank-one action $T$ such that $D\subset {\mathcal{R}}(T)\subset \unicode[STIX]{x1D6E5}$ and (b) ${\mathcal{R}}(T)=P(\mathfrak{g})$ for a generic infinite measure-preserving action $T$ of $\unicode[STIX]{x1D6E4}$. This partly answers a question from a recent paper by Johnson and Şahin. Some applications to the directional entropy of Poisson actions are discussed. In the case where $G$ is the Heisenberg group $H_{3}(\mathbb{R})$ and $\unicode[STIX]{x1D6E4}=H_{3}(\mathbb{Z})$, a rank-one $\unicode[STIX]{x1D6E4}$-action $T$ is constructed for which ${\mathcal{R}}(T)$ is not invariant under the natural ‘adjoint’ $G$-action.

The Lyapunov exponents of locally constant $\text{GL}(2,\mathbb{C})$-cocycles over Bernoulli shifts vary continuously with the cocycle and the invariant probability measure.