We derive upper and lower bounds for the Assouad and lower dimensions of self-affine measures in $\mathbb {R}^d$ generated by diagonal matrices and satisfying suitable separation conditions. The upper and lower bounds always coincide for $d=2,3$, yielding precise explicit formulae for those dimensions. Moreover, there are easy-to-check conditions guaranteeing that the bounds coincide for $d \geqslant 4$. An interesting consequence of our results is that there can be a ‘dimension gap’ for such self-affine constructions, even in the plane. That is, we show that for some self-affine carpets of ‘Barański type’ the Assouad dimension of all associated self-affine measures strictly exceeds the Assouad dimension of the carpet by some fixed $\delta>0$ depending only on the carpet. We also provide examples of self-affine carpets of ‘Barański type’ where there is no dimension gap and in fact the Assouad dimension of the carpet is equal to the Assouad dimension of a carefully chosen self-affine measure.

We give an example of an FIID vertex-labeling of ${\mathbb T}_3$ whose marginals are uniform on $[0,1]$, and if we delete the edges between those vertices whose labels are different, then some of the remaining clusters are infinite. We also show that no such process can be finitary.

The ergodic properties of two uncoupled oscillators, one horizontal and one vertical, residing in a class of non-rectangular star-shaped polygons with only vertical and horizontal boundaries and impacting elastically from its boundaries are studied. We prove that the iso-energy level sets topology changes non-trivially; the flow on level sets is always conjugated to a translation flow on a translation surface, yet, for some segments of partial energies the genus of the surface is strictly greater than $1$ . When at least one of the oscillators is unharmonic, or when both are harmonic and non-resonant, we prove that for almost all partial energies, including the impacting ones, the flow on level sets is uniquely ergodic. When both oscillators are harmonic and resonant, we prove that there exist intervals of partial energies on which periodic ribbons and additional ergodic components coexist. We prove that for almost all partial energies in such segments the motion is uniquely ergodic on the part of the level set that is not occupied by the periodic ribbons. This implies that ergodic averages project to piecewise smooth weighted averages in the configuration space.

Let $k\geq 2$ and $(X_{i}, \mathcal {T}_{i}), i=1,\ldots ,k$, be $\mathbb {Z}^{d}$-actions topological dynamical systems with $\mathcal {T}_i:=\{T_i^{\textbf {g}}:X_i{\rightarrow } X_i\}_{\textbf {g}\in \mathbb {Z}^{d}}$, where $d\in \mathbb {N}$ and $f\in C(X_{1})$. Assume that for each $1\leq i\leq k-1$, $(X_{i+1}, \mathcal {T}_{i+1})$ is a factor of $(X_{i}, \mathcal {T}_{i})$. In this paper, we introduce the weighted topological pressure $P^{\textbf {a}}(\mathcal {T}_{1},f)$ and weighted measure-theoretic entropy $h_{\mu }^{\textbf {a}}(\mathcal {T}_{1})$ for $\mathbb {Z}^{d}$-actions, and establish a weighted variational principle as

$$ \begin{align*} P^{\textbf{a}}(\mathcal{T}_{1},f)=\sup\bigg\{h_{\mu}^{\textbf{a}}(\mathcal{T}_{1})+\int_{X_{1}}f\,d\mu:\mu\in\mathcal{M}(X_{1}, \mathcal{T}_{1})\bigg\}. \end{align*} $$

This result not only generalizes some well-known variational principles about topological pressure for compact or non-compact sets, but also improves the variational principle for weighted topological pressure in [16] from $\mathbb {Z}_{+}$-action topological dynamical systems to $\mathbb {Z}^{d}$-actions topological dynamical systems.

We construct one-dimensional foliations which are subfoliations of two-dimensional foliations in $3$ -manifolds. The subfoliation is by quasigeodesics in each two-dimensional leaf, but it is not funnel: not all quasigeodesics share a common ideal point in most leaves.

We define positively expansive semigroups of linear operators on Banach spaces. We characterize these semigroups in terms of the point spectrum of the infinitesimal generator. In particular, we prove that a positively expansive semigroup is neither uniformly bounded nor equicontinuous. We apply our results to the Lasota equation.

We consider families of random products of close-by Anosov diffeomorphisms, and show that statistical stability and linear response hold for the associated families of equivariant and stationary measures. Our analysis relies on the study of the top Oseledets space of a parametrized transfer operator cocycle, as well as ad-hoc abstract perturbation statements. As an application, we show that, when the quenched central limit theorem (CLT) holds, under the conditions that ensure linear response for our cocycle, the variance in the CLT depends differentiably on the parameter.

We introduce a class of rank-one transformations, which we call extremely elevated staircase transformations. We prove that they are measure-theoretically mixing and, for any $f : \mathbb {N} \to \mathbb {N}$ with $f(n)/n$ increasing and $\sum 1/f(n) < \infty $ , that there exists an extremely elevated staircase with word complexity $p(n) = o(f(n))$ . This improves the previously lowest known complexity for mixing subshifts, resolving a conjecture of Ferenczi.

Although detailed descriptions of the possible types of behaviour inside periodic Fatou components have been known for over 100 years, a classification of wandering domains has only recently been given. Recently, simply connected wandering domains were classified into nine possible types and examples of escaping wandering domains of each of these types were constructed. Here we consider the case of oscillating wandering domains, for which only six of these types are possible. We use a new technique based on approximation theory to construct examples of all six types of oscillating simply connected wandering domains. This requires delicate arguments since oscillating wandering domains return infinitely often to a bounded part of the plane. Our technique is inspired by that used by Eremenko and Lyubich to construct the first example of an oscillating wandering domain, but with considerable refinements which enable us to show that the wandering domains are bounded, to specify the degree of the mappings between wandering domains and to give precise descriptions of the dynamical behaviour of these mappings.

We study local biholomorphisms with finite orbits in some neighborhood of the origin since they are intimately related to holomorphic foliations with closed leaves. We describe the structure of the set of periodic points in dimension 2. As a consequence we show that given a finite-orbits local biholomorphism F, in dimension 2, there exists an analytic curve passing through the origin and contained in the fixed-point set of some non-trivial iterate of $F.$ As an application we obtain that at least one eigenvalue of the linear part of F at the origin is a root of unity. Moreover, we show that such a result is sharp by exhibiting examples of finite-orbits local biholomorphisms such that exactly one of the eigenvalues is a root of unity. These examples are subtle since we show they cannot be embedded in one-parameter groups.

Recall that two geodesics in a negatively curved surface S are of the same type if their free homotopy classes differ by a homeomorphism of the surface. In this note we study the distribution in the unit tangent bundle of the geodesics of fixed type, proving that they are asymptotically equidistributed with respect to a certain measure ${\mathfrak {m}}^S$ on $T^1S$ . We study a few properties of this measure, showing for example that it distinguishes between hyperbolic surfaces.

A parameter $c_{0}\in {\mathbb {C}}$ in the family of quadratic polynomials $f_{c}(z)=z^{2}+c$ is a critical point of a period n multiplier if the map $f_{c_{0}}$ has a periodic orbit of period n, whose multiplier, viewed as a locally analytic function of c, has a vanishing derivative at $c=c_{0}$ . We study the accumulation set ${\mathcal X}$ of the critical points of the multipliers as $n\to \infty $ . This study complements the equidistribution result for the critical points of the multipliers that was previously obtained by the authors. In particular, in the current paper, we prove that the accumulation set ${\mathcal X}$ is bounded, connected, and contains the Mandelbrot set as a proper subset. We also provide a necessary and sufficient condition for a parameter outside of the Mandelbrot set to be contained in the accumulation set ${\mathcal X}$ and show that this condition is satisfied for an open set of parameters. Our condition is similar in flavor to one of the conditions that define the Mandelbrot set. As an application, we get that the function that sends c to the Hausdorff dimension of $f_{c}$ does not have critical points outside of the accumulation set ${\mathcal X}$ .

For a polynomial progression

$$ \begin{align*}(x,\; x+P_1(y),\ldots,\; x+P_{t}(y)),\end{align*} $$

$P_1, \ldots ,\!P_t$

$$ \begin{align*}(x,\; x+y^2,\; x+2y^2,\; x+y^3,\; x+2y^3),\end{align*} $$

$\mathbb {Z}/N\mathbb {Z}$

$\mathbb {Z}/N\mathbb {Z}$

We are interested in dendrites for which all invariant measures of zero-entropy mappings have discrete spectrum, and we prove that this holds when the closure of the endpoint set of the dendrites is countable. This solves an open question which has been around for awhile, and almost completes the characterization of dendrites with this property.

It is shown that each locally compact second countable non-(T) group G admits non-strongly ergodic weakly mixing IDPFT Poisson actions of any possible Krieger type. These actions are amenable if and only if G is amenable. If G has the Haagerup property, then (and only then) these actions can be chosen of 0-type. If G is amenable, then G admits weakly mixing Bernoulli actions of arbitrary Krieger type.

We extend the concept of a Hubbard tree, well established and useful in the theory of polynomial dynamics, to the dynamics of transcendental entire functions. We show that Hubbard trees in the strict traditional sense, as invariant compact trees embedded in $\mathbb {C}$ , do not exist even for post-singularly finite exponential maps; the difficulty lies in the existence of asymptotic values. We therefore introduce the concept of a homotopy Hubbard tree that takes care of these difficulties. Specifically for the family of exponential maps, we show that every post-singularly finite map has a homotopy Hubbard tree that is unique up to homotopy, and that post-singularly finite exponential maps can be classified in terms of homotopy Hubbard trees, using a transcendental analogue of Thurston’s topological characterization theorem of rational maps.

The ergodic decomposition theorem is a cornerstone result of dynamical systems and ergodic theory. It states that every invariant measure on a dynamical system is a mixture of ergodic ones. Here, we formulate and prove the theorem in terms of string diagrams, using the formalism of Markov categories. We recover the usual measure-theoretic statement by instantiating our result in the category of stochastic kernels. Along the way, we give a conceptual treatment of several concepts in the theory of deterministic and stochastic dynamical systems. In particular, ergodic measures appear very naturally as particular cones of deterministic morphisms (in the sense of Markov categories), and the invariant $\sigma $-algebra of a dynamical system can be seen as a colimit in the category of Markov kernels. In line with other uses of category theory, once the necessary structures are in place, our proof of the main theorem is much more intuitive than traditional approaches. In particular, it does not use any quantitative limiting arguments, and it does not rely on the cardinality of the group or monoid indexing the dynamics. We hope that this result paves the way for further applications of category theory to dynamical systems, ergodic theory, and information theory.

We extract quantitative information (specifically, a rate of metastability in the sense of Terence Tao) from a proof due to Kazuo Kobayasi and Isao Miyadera, which shows strong convergence for Cesàro means of non-expansive maps on Banach spaces.

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 generalize a result of Lindenstrauss on the interplay between measurable and topological dynamics which shows that every separable ergodic measurably distal dynamical system has a minimal distal model. We show that such a model can, in fact, be chosen completely canonically. The construction is performed by going through the Furstenberg–Zimmer tower of a measurably distal system and showing that at each step there is a simple and canonical distal minimal model. This hinges on a new characterization of isometric extensions in topological dynamics.