One of the fundamental properties of the Mandelbrot set is that the set of postcritically finite parameters is structured like a tree. We extend this result to the set of quadratic kneading sequences and show that this space contains no irrational decorations. Along the way, we prove a combinatorial analogue to the correspondence principle of dynamic and parameter rays. Our key tool is to work simultaneously with the two equivalent combinatorial concepts of Hubbard trees and kneading sequences.

Over the last few years Pohl (partly jointly with coauthors) has developed dual ‘slow/fast’ transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ with cusps and all finite-dimensional unitary representations $\unicode[STIX]{x1D712}$ of $\unicode[STIX]{x1D6E4}$. The eigenfunctions with eigenvalue 1 of the fast transfer operators determine the zeros of the Selberg zeta function for $(\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D712})$. Further, if $\unicode[STIX]{x1D6E4}$ is cofinite and $\unicode[STIX]{x1D712}$ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue 1 of the slow transfer operators characterize Maass cusp forms for $\unicode[STIX]{x1D6E4}$. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. In this article we study, without relying on Selberg theory, the relation between the eigenspaces of these two types of transfer operators for any Hecke triangle surface $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ of finite or infinite area and any finite-dimensional unitary representation $\unicode[STIX]{x1D712}$ of the Hecke triangle group $\unicode[STIX]{x1D6E4}$. In particular, we provide explicit isomorphisms between relevant subspaces. This solves a conjecture by Möller and Pohl, characterizes some of the zeros of the Selberg zeta functions independently of the Selberg trace formula, and supports the previously mentioned conjectures.

We give an example of a $C^{\infty }$ vector field $X$, defined in a neighbourhood $U$ of $0\in \mathbb{R}^{8}$, such that $U-\{0\}$ is foliated by closed integral curves of $X$, the differential $DX(0)$ at $0$ defines a one-parameter group of non-degenerate rotations and $X$ is not orbitally equivalent to its linearization. Such a vector field $X$ has the first integral $I(x)=\Vert x\Vert ^{2}$, and its main feature is that its period function is locally unbounded near the stationary point. This proves in the $C^{\infty }$ category that the classical Poincaré centre theorem, true for planar non-degenerate centres, is not generalizable to multicentres. Such an example is obtained through a careful study and a suitable modification of a celebrated example by Sullivan [A counterexample to the periodic orbit conjecture. Publ. Math. Inst. Hautes Études Sci. 46 (1976), 5–14], by blowing up the stationary point at the origin and through the construction of a smooth one-parameter family of foliations by circles of $S^{7}$ whose orbits have unbounded lengths (equivalently, unbounded periods) for each value of the parameter and which smoothly converges to the Hopf fibration $S^{1}{\hookrightarrow}S^{7}\rightarrow \mathbb{CP}^{3}$.

We consider piecewise invertible systems exhibiting intermittency and establish a generalized variational principle adapted to a non-stationary process in the following sense; the supremum is attained by non-singular (not necessarily invariant) probability measures and if the system exhibits hyperbolicity, then it reduces to the usual variational principle for the pressure. Our method relies on Ruelle’s program in the study of non-equilibrium statistical mechanics to analyze dissipative phenomena. We show non-positivity of entropy production at weak Gibbs measures and clarify when it indeed vanishes. We also discuss a generalized variational principle in the context of σ-finite invariant measures.

Let ℱn be families of entire functions, holomorphically parameterized by a complex manifold M. We consider those parameters in M that correspond to non-escaping hyperbolic functions, i.e. those maps f∈ℱn for which the postsingular set P(f) is a compact subset of the Fatou set ℱ(f) of f. We prove that if ℱn→ℱ∞ in the sense of a certain dynamically sensible metric, then every non-escaping hyperbolic component in the parameter space of ℱ∞ is a kernel of a sequence of non-escaping hyperbolic components in the parameter spaces of ℱn. Parameters belonging to such a kernel do not always correspond to hyperbolic functions in ℱ∞. Nevertheless, we show that these functions must be J-stable. Using quasiconformal equivalences, we are able to construct many examples of families to which our results can be applied.

For any \varepsilon > 0, we construct an explicit smooth Riemannian metric on the sphere S^n, n \geq 3, that is within \varepsilon of the round metric and has a geodesic for which the corresponding orbit of the geodesic flow is \varepsilon-dense in the unit tangent bundle. Moreover, for any \varepsilon > 0, we construct a smooth Riemannian metric on S^n, n \geq 3, that is within \varepsilon of the round metric and has a geodesic for which the complement of the closure of the corresponding orbit of the geodesic flow has Liouville measure less than \varepsilon.

Let $F$ be an area-preserving homeomorphism of the two-dimensional torus homotopic to the identity and let $f$ be a lift to the plane which preserves the center of mass. We prove that $f$ has at least three fixed points whose images are distinct in the torus.

We characterize measure-theoretic sequence entropy pairs of continuous actions of abelian groups using mean sensitivity. This addresses an open question of Li and Yu [On mean sensitive tuples. J. Differential Equations 297 (2021), 175–200]. As a consequence of our results, we provide a simpler characterization of Kerr and Li’s independence sequence entropy pairs ($\mu $-IN-pairs) when the measure is ergodic and the group is abelian.

In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $\mu $ on M where the transition kernel ${\mathcal P}$ admits an eigenfunction $0\leq \eta \in L^1(M,\mu )$. We find conditions on the transition densities of ${\mathcal P}$ with respect to $\mu $ which ensure that $\eta (x) \mu (\mathrm {d} x)$ is a quasi-ergodic measure for $X_n$ and that the Yaglom limit converges to the quasi-stationary measure $\mu $-almost surely. We apply this result to the random logistic map $X_{n+1} = \omega _n X_n (1-X_n)$ absorbed at ${\mathbb R} \setminus [0,1],$ where $\omega _n$ is an independent and identically distributed sequence of random variables uniformly distributed in $[a,b],$ for $1\leq a <4$ and $b>4.$

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 $2$.

Furstenberg has associated to every topological group $G$ a universal boundary $\unicode[STIX]{x2202}(G)$. If we consider in addition a subgroup $H<G$, the relative notion of $(G,H)$-boundaries admits again a maximal object $\unicode[STIX]{x2202}(G,H)$. In the case of discrete groups, an equivalent notion was introduced by Bearden and Kalantar (Topological boundaries of unitary representations. Preprint, 2019, arXiv:1901.10937v1) as a very special instance of their constructions. However, the analogous universality does not always hold, even for discrete groups. On the other hand, it does hold in the affine reformulation in terms of convex compact sets, which admits a universal simplex $\unicode[STIX]{x1D6E5}(G,H)$, namely the simplex of measures on $\unicode[STIX]{x2202}(G,H)$. We determine the boundary $\unicode[STIX]{x2202}(G,H)$ in a number of cases, highlighting properties that might appear unexpected.

In this work, we study ergodic and dynamical properties of symbolic dynamical system associated to substitutions on an infinite countable alphabet. Specifically, we consider shift dynamical systems associated to irreducible substitutions which have well-established properties in the case of finite alphabets. Based on dynamical properties of a countable integer matrix related to the substitution, we obtain results on existence and uniqueness of shift invariant measures.

Cheng and Newhouse (Ergod. Th. & Dynam. Sys. 25 (2005), 1091–1113) proved a variational principle for topological preimage entropy $h_{\mathrm {pre}}(f)$:

$$ \begin{align*} h_{\mathrm{pre}}(f)=\sup_{\mu\in\mathcal{M}(X,f)}h_{\mathrm{pre},\mu}(f). \end{align*} $$

Unfortunately, we show in this note that this variational principle is not true.

Let T be an orientation-preserving Lipschitz expanding map of the circle . A pre-image selector is a map with finitely many discontinuities, each of which is a jump discontinuity, and such that τ(x)∈T−1(x) for all . The closure of the image of a pre-image selector is called a flower and a flower with p connected components is called a p-flower. We say that a real-valued Lipschitz function can be Lipschitz flattened on a flower whenever it is Lipschitz cohomologous to a constant on that flower. The space of Lipschitz functions which can be flattened on a given p-flower is shown to be of codimension p in the space of all Lipschitz functions, and the linear constraints determining this subspace are derived explicitly. If a Lipschitz function f has a maximizing measure S which is Sturmian (i.e. is carried by a 1-flower), it is shown that f can be Lipschitz flattened on some 1-flower carrying S.

In this paper escape rates and local escape rates for special flows are studied. In a general context the first result is that the escape rate depends monotonically on the ceiling function and fulfils certain scaling, invariance and continuity properties. In the context of metric spaces local escape rates are considered. If the base transformation is ergodic and exhibits an exponential convergence in probability of ergodic sums, then the local escape rate with respect to the flow is just the local escape rate with respect to the base transformation, divided by the integral of the ceiling function. Also, a reformulation with respect to induced pressure is presented. Finally, under additional regularity conditions higher order asymptotics for the local escape rate are established.

We develop the notion of the Rokhlin dimension for partial actions of finite groups, extending the well-established theory for global systems. The partial setting exhibits phenomena that cannot be expected for global actions, usually stemming from the fact that virtually all averaging arguments for finite group actions completely break down for partial systems. For example, fixed point algebras and crossed products are not in general Morita equivalent, and there is in general no local approximation of the crossed product $A\rtimes G$ by matrices over A. Using decomposition arguments for partial actions of finite groups, we show that a number of structural properties are preserved by formation of crossed products, including finite stable rank, finite nuclear dimension, and absorption of a strongly self-absorbing $C^*$ -algebra. Some of our results are new even in the global case.

We also study the Rokhlin dimension of globalizable actions: while in general it differs from the Rokhlin dimension of its globalization, we show that they agree if the coefficient algebra is unital. For topological partial actions on spaces of finite covering dimension, we show that finiteness of the Rokhlin dimension is equivalent to freeness, thus providing a large class of examples to which our theory applies.

We give three corrections to the paper [K. Schmidt and A. M. Vershik, Algebraic polymorphisms, Ergod. Th. & Dynam. Sys.28 (2008), 633–642].

The notion of an iterated extension of a flow is introduced and studied. In particular it is shown how eigenfunctions occur in a natural way. This is then exploited to produce an example of a weakly mixing minimal set with a non-weakly mixing quasi-factor.

We correct two technical errors in the original paper. The main result in the original paper remains valid without any changes.

We show that the Hausdorff dimension of any slice of the graph of the Takagi function is bounded above by the Assouad dimension of the graph minus one, and that the bound is sharp. The result is deduced from a statement on more general self-affine sets, which is of independent interest. We also prove that Marstrand’s slicing theorem on the graph of the Takagi function extends to all slices if and only if the upper pointwise dimension of every projection of the length measure on the x-axis lifted to the graph is at least one.