The paper by Froyland, González-Tokman and Quas [Stability and approximation of random invariant densities for Lasota–Yorke map cocycles. Nonlinearity27(4) (2014), 647] established fibrewise stability of random absolutely continuous invariant measures (acims) for cocycles of random Lasota–Yorke maps under a variety of perturbations, including ‘Ulam’s method’, a popular numerical method for approximating acims. The expansivity requirements of Froyland et al were that the cocycle (or powers of the cocycle) should be ‘expanding on average’ before applying a perturbation, such as Ulam’s method. In the present work, we make a significant theoretical and computational weakening of the expansivity hypotheses of Froyland et al, requiring only that the cocycle be eventually expanding on average, and importantly, allowing the perturbation to be applied after each single step of the cocycle. The family of random maps that generate our cocycle need not be close to a fixed map and our results can handle very general driving mechanisms. We provide a detailed numerical example of a random Lasota–Yorke map cocycle with expanding and contracting behaviour and illustrate the extra information carried by our fibred random acims, when compared to annealed acims or ‘physical’ random acims.

This paper is concerned with factors of independent and identically distributed processes on the $d$-regular tree for $d\geq 3$. We study the mutual information of values on two given vertices. If the vertices are neighbors (i.e. their distance is $1$), then a known inequality between the entropy of a vertex and the entropy of an edge provides an upper bound for the (normalized) mutual information. In this paper we obtain upper bounds for vertices at an arbitrary distance $k$, of order $(d-1)^{-k/2}$. Although these bounds are sharp, we also show that an interesting phenomenon occurs here: for any fixed process, the rate of mutual information decay is much faster, essentially of order $(d-1)^{-k}$.

We consider finite $2$-complexes $X$ that arise as quotients of Fuchsian buildings by subgroups of the combinatorial automorphism group, which we assume act freely and cocompactly. We show that locally CAT($-1$) metrics on $X$, which are piecewise hyperbolic and satisfy a natural non-singularity condition at vertices, are marked length spectrum rigid within certain classes of negatively curved, piecewise Riemannian metrics on $X$. As a key step in our proof, we show that the marked length spectrum function for such metrics determines the volume of $X$.

We consider group-valued cocycles over dynamical systems. The base system is a homeomorphism $f$ of a metric space satisfying a closing property, for example a hyperbolic dynamical system or a subshift of finite type. The cocycle $A$ takes values in the group of invertible bounded linear operators on a Banach space and is Hölder continuous. We prove that upper and lower Lyapunov exponents of $A$ with respect to an ergodic invariant measure $\unicode[STIX]{x1D707}$ can be approximated in terms of the norms of the values of $A$ on periodic orbits of $f$. We also show that these exponents cannot always be approximated by the exponents of $A$ with respect to measures on periodic orbits. Our arguments include a result of independent interest on construction and properties of a Lyapunov norm for the infinite-dimensional setting. As a corollary, we obtain estimates of the growth of the norm and of the quasiconformal distortion of the cocycle in terms of the growth at the periodic points of $f$.

Devaney and Krych showed that, for the exponential family $\unicode[STIX]{x1D706}e^{z}$, where $0\,<\,\unicode[STIX]{x1D706}\,<\,1/e$, the Julia set consists of uncountably many pairwise disjoint simple curves tending to $\infty$. Viana proved that these curves are smooth. In this article, we consider quasiregular counterparts of the exponential map, the so-called Zorich maps, and generalize Viana’s result to these maps.

In this paper we study quasi-symmetric conjugations of ${\mathcal{C}}^{2}$ weakly order-preserving circle maps with a flat interval. Under the assumption that the maps have the same rotation number of bounded type and that bounded geometry holds, we construct a quasi-symmetric conjugation between their non-wandering sets. Further, this conjugation is extended to a quasi-symmetric circle homeomorphism. Our proof techniques hinge on real-dynamic methods, allowing us to construct the conjugation under general and natural assumptions.

We define a new property called one-sided almost specification, which lies between the properties of specification and almost specification, and prove that it guarantees intrinsic ergodicity (i.e. uniqueness of the measure of maximal entropy) if the corresponding mistake function $g$ is bounded. We also show that uniqueness may fail for unbounded $g$ such as $\log \log n$. Our results have consequences for almost specification: we prove that almost specification with $g\equiv 1$ implies one-sided almost specification (with $g\equiv 1$) and hence uniqueness. On the other hand, the second author showed recently that almost specification with $g\equiv 4$ does not imply uniqueness.

Let $M$ be an analytic connected 2-manifold with empty boundary, over the ground field $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. Let $Y$ and $X$ denote differentiable vector fields on $M$. We say that $Y$ tracks $X$ if $[Y,X]=fX$ for some continuous function $f:\,M\rightarrow \mathbb{F}$. A subset $K$ of the zero set $\mathsf{Z}(X)$ is an essential block for $X$ if it is non-empty, compact and open in $\mathsf{Z}(X)$, and the Poincaré–Hopf index $\mathsf{i}_{K}(X)$ is non-zero. Let ${\mathcal{G}}$ be a finite-dimensional Lie algebra of analytic vector fields that tracks a non-trivial analytic vector field $X$. Let $K\subset \mathsf{Z}(X)$ be an essential block. Assume that if $M$ is complex and $\mathsf{i}_{K}(X)$ is a positive even integer, no quotient of ${\mathcal{G}}$ is isomorphic to $\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$. Then ${\mathcal{G}}$ has a zero in $K$ (main result). As a consequence, if $X$ and $Y$ are analytic, $X$ is non-trivial, and $Y$ tracks $X$, then every essential component of $\mathsf{Z}(X)$ meets $\mathsf{Z}(Y)$. Fixed-point theorems for certain types of transformation groups are proved. Several illustrative examples are given.

We give a new proof of Moser’s 1967 normal-form theorem for real analytic perturbations of vector fields possessing a reducible Diophantine invariant quasi-periodic torus. The proposed approach, based on an inverse function theorem in analytic class, is flexible and can be adapted to several contexts. This allows us to prove in a unified framework the persistence, up to finitely many parameters, of Diophantine quasi-periodic normally hyperbolic reducible invariant tori for vector fields originating from dissipative generalizations of Hamiltonian mechanics. As a byproduct, generalizations of Herman’s twist theorem and Rüssmann’s translated curve theorem are proved.

We show that on the four-symbol full shift, there is a finitely generated subgroup of the automorphism group whose action is (set-theoretically) transitive of all orders on the points of finite support, up to the necessary caveats due to shift-commutation. As a corollary, we obtain that there is a finite set of automorphisms whose centralizer is $\mathbb{Z}$ (the shift group), giving a finitary version of Ryan’s theorem (on the four-symbol full shift), suggesting an automorphism group invariant for mixing subshifts of finite type (SFTs). We show that any such set of automorphisms must generate an infinite group, and also show that there is also a group with this transitivity property that is a subgroup of the commutator subgroup and whose elements can be written as compositions of involutions. We ask many related questions and prove some easy transitivity results for the group of reversible Turing machines, topological full groups and Thompson’s $V$.

Lorenz maps are maps of the unit interval with one critical point of order $\unicode[STIX]{x1D70C}>1$ and a discontinuity at that point. They appear as return maps of sections of the geometric Lorenz flow. We construct real a priori bounds for renormalizable Lorenz maps with certain monotone combinatorics and a sufficiently flat critical point, and use these bounds to show existence of periodic points of renormalization, as well as existence of Cantor attractors for dynamics of infinitely renormalizable Lorenz maps.

We consider sequences of Davenport type or Gelfond type and prove that sequences of Davenport exponent larger than $\frac{1}{2}$ are good sequences of weights for the ergodic theorem, and that the ergodic sums weighted by a sequence of strong Gelfond property are well controlled almost everywhere. We prove that for any $q$-multiplicative sequence, the Gelfond property implies the strong Gelfond property and that sequences realized by dynamical systems can be fully oscillating and have the Gelfond property.

It is shown that each conservative non-singular Bernoulli shift is either of type $\mathit{II}_{1}$ or $\mathit{III}_{1}$. Moreover, in the latter case the corresponding Maharam extension of the shift is a $K$-automorphism. This extends earlier results obtained by Kosloff for equilibrial shifts. Non-equilibrial shifts of type $\mathit{III}_{1}$ are constructed. We further generalize (partly) the main results to non-singular Markov shifts.

Given an $\unicode[STIX]{x1D6FC}>1$ and a $\unicode[STIX]{x1D703}$ with unbounded continued fraction entries, we characterize new relations between Sturmian subshifts with slope $\unicode[STIX]{x1D703}$ with respect to (i) an $\unicode[STIX]{x1D6FC}$-Hölder regularity condition of a spectral metric, (ii) level sets defined in terms of the Diophantine properties of $\unicode[STIX]{x1D703}$, and (iii) complexity notions which we call $\unicode[STIX]{x1D6FC}$-repetitiveness, $\unicode[STIX]{x1D6FC}$-repulsiveness and $\unicode[STIX]{x1D6FC}$-finiteness—generalizations of the properties known as linear repetitiveness, repulsiveness and power freeness, respectively. We show that the level sets relate naturally to (exact) Jarník sets and prove that their Hausdorff dimension is $2/(\unicode[STIX]{x1D6FC}+1)$.

Pesin sets are measurable sets along which the behavior of a matrix cocycle above a measure-preserving dynamical system is explicitly controlled. In uniformly hyperbolic dynamics, we study how often points return to Pesin sets under suitable conditions on the cocycle: if it is locally constant, or if it admits invariant holonomies and is pinching and twisting, we show that the measure of points that do not return a linear number of times to Pesin sets is exponentially small. We discuss applications to the exponential mixing of contact Anosov flows and consider counterexamples illustrating the necessity of suitable conditions on the cocycle.

We show that there is an irrational rotation $Tx=x+\unicode[STIX]{x1D6FC}$ on the circle $\mathbb{T}$ and a continuous $\unicode[STIX]{x1D711}:\mathbb{T}\rightarrow \mathbb{R}$ such that for each (continuous) uniquely ergodic flow ${\mathcal{S}}=(S_{t})_{t\in \mathbb{R}}$ acting on a compact metric space $Y$, the automorphism $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}$ acting on $(X\times Y,\unicode[STIX]{x1D707}\otimes \unicode[STIX]{x1D708})$ by the formula $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}(x,y)=(Tx,S_{\unicode[STIX]{x1D711}(x)}(y))$, where $\unicode[STIX]{x1D707}$ stands for the Lebesgue measure on $\mathbb{T}$ and $\unicode[STIX]{x1D708}$ denotes the unique ${\mathcal{S}}$-invariant measure, has the property of asymptotically orthogonal powers. This gives a class of relatively weakly mixing extensions of irrational rotations for which Sarnak’s conjecture on the Möbius disjointness holds for all uniquely ergodic models of $T_{\unicode[STIX]{x1D711},{\mathcal{S}}}$. Moreover, we obtain a class of ‘random’ ergodic sequences $(c_{n})\subset \mathbb{Z}$ such that if $\boldsymbol{\unicode[STIX]{x1D707}}$ denotes the Möbius function, then

$$\begin{eqnarray}\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n\leq N}g(S_{c_{n}}y)\boldsymbol{\unicode[STIX]{x1D707}}(n)=0\end{eqnarray}$$

${\mathcal{S}}$

$g\in C(Y)$

$y\in Y$

We associate with every étale groupoid $\mathfrak{G}$ two normal subgroups $\mathsf{S}(\mathfrak{G})$ and $\mathsf{A}(\mathfrak{G})$ of the topological full group of $\mathfrak{G}$, which are analogs of the symmetric and alternating groups. We prove that if $\mathfrak{G}$ is a minimal groupoid of germs (e.g., of a group action), then $\mathsf{A}(\mathfrak{G})$ is simple and is contained in every non-trivial normal subgroup of the full group. We show that if $\mathfrak{G}$ is expansive (e.g., is the groupoid of germs of an expansive action of a group), then $\mathsf{A}(\mathfrak{G})$ is finitely generated. We also show that $\mathsf{S}(\mathfrak{G})/\mathsf{A}(\mathfrak{G})$ is a quotient of $H_{0}(\mathfrak{G},\mathbb{Z}/2\mathbb{Z})$.

Harmonic map theory is used to show that a convex cocompact surface group action on a $\text{CAT}(-1)$ metric space fixes a convex copy of the hyperbolic plane (i.e. the action is Fuchsian) if and only if the Hausdorff dimension of the limit set of the action is equal to 1. This provides another proof of a result of Bonk and Kleiner. More generally, we show that the limit set of every convex cocompact surface group action on a $\text{CAT}(-1)$ space has Hausdorff dimension $\geq 1$, where the inequality is strict unless the action is Fuchsian.

If $c$ is in the main cardioid of the Mandelbrot set, then the Julia set $J$ of the map $\unicode[STIX]{x1D719}_{c}:z\mapsto z^{2}+c$ is a Jordan curve of Hausdorff dimension $p\in [1,2)$. We provide a full proof of a formula for the Hausdorff measure on $J$ in terms of singular traces announced by the first named author in 1996.

In this paper, we study the ergodic theory of the geodesic flow on negatively curved geometrically finite manifolds. We prove that the measure-theoretical entropy is upper semicontinuous when there is no loss of mass. In the case where mass is lost, the critical exponents of parabolic subgroups of the fundamental group have a significant meaning. More precisely, the failure of upper-semicontinuity of the entropy is determined by the maximal parabolic critical exponent. We also study the pressure of positive Hölder-continuous potentials going to zero through the cusps. We prove that the pressure map $t\mapsto P(tF)$ is differentiable until it undergoes a phase transition, after which it becomes constant. This description allows us, in particular, to compute the entropy of the geodesic flow at infinity.