The Fischer, Krieger, and fiber product covers of sofic beta-shifts are constructed and used to show that every strictly sofic beta-shift is 2-sofic. Flow invariants based on the covers are computed, and shown to depend only on a single integer that can easily be determined from the $\unicode[STIX]{x1D6FD}$-expansion of 1. It is shown that any beta-shift is flow equivalent to a beta-shift given by some $1<\unicode[STIX]{x1D6FD}<2$, and concrete constructions lead to further reductions of the flow classification problem. For each sofic beta-shift, there is an action of $\mathbb{Z}/2\mathbb{Z}$ on the edge shift given by the fiber product, and it is shown precisely when there exists a flow equivalence respecting these $\mathbb{Z}/2\mathbb{Z}$-actions. This opens a connection to ongoing efforts to classify general irreducible 2-sofic shifts via flow equivalences of reducible shifts of finite type (SFTs) equipped with $\mathbb{Z}/2\mathbb{Z}$-actions.

Consider the skew product $F:\mathbb{T}^{2}\rightarrow \mathbb{T}^{2}$ , $F(x,y)=(f(x),y+\unicode[STIX]{x1D70F}(x))$ , where $f:\mathbb{T}^{1}\rightarrow \mathbb{T}^{1}$ is a piecewise $\mathscr{C}^{1+\unicode[STIX]{x1D6FC}}$ expanding map on a countable partition and $\unicode[STIX]{x1D70F}:\mathbb{T}^{1}\rightarrow \mathbb{R}$ is piecewise $\mathscr{C}^{1}$ . It is shown that if $\unicode[STIX]{x1D70F}$ is not Lipschitz-cohomologous to a piecewise constant function on the joint partition of $f$ and $\unicode[STIX]{x1D70F}$ , then $F$ is mixing at a stretched-exponential rate.

A necessary and sufficient condition is given for the existence of an embedding of an irreducible subshift of finite type into the Fibonacci–Dyck shift.

In this paper we give complete analytic invariants for the set of germs of holomorphic foliations in $(\mathbb{C}^{2},0)$ that become regular after a single blow-up. Some of the invariants describe the holonomy pseudogroup of the germ and are called transverse invariants. The other invariants lie in a finite dimensional complex vector space. Such singularities admit separatrices tangentially to any direction at the origin. When enough separatrices are leaves of a radial foliaton (a condition that can always be attained if the multiplicity of the germ at the origin is at most four) we are able to describe and realize all the analytical invariants geometrically and provide analytic normal forms. As a consequence, we prove that any two such germs sharing the same transverse invariants are conjugated by a very particular type of birational transformation. We also provide explicit examples of universal equisingular unfoldings of foliations that cannot be produced by unfolding functions. With these at hand we are able to explicitly parametrize families of analytically distinct foliations that share the same transverse invariants.

We introduce the notion of a quasistatic dynamical system, which generalizes that of an ordinary dynamical system. Quasistatic dynamical systems are inspired by the namesake processes in thermodynamics, which are idealized processes where the observed system transforms (infinitesimally) slowly due to external influence, tracing out a continuous path of thermodynamic equilibria over an (infinitely) long time span. Time evolution of states under a quasistatic dynamical system is entirely deterministic, but choosing the initial state randomly renders the process a stochastic one. In the prototypical setting where the time evolution is specified by strongly chaotic maps on the circle, we obtain a description of the statistical behavior as a stochastic diffusion process, under surprisingly mild conditions on the initial distribution, by solving a well-posed martingale problem. We also consider various admissible ways of centering the process, with the curious conclusion that the ‘obvious’ centering suggested by the initial distribution sometimes fails to yield the expected diffusion.

This note concerns stationary solutions of the Euler equations for an ideal fluid on a closed 3-manifold. We prove that if the velocity field of such a solution has no zeroes and real analytic Bernoulli function, then it can be rescaled to the Reeb vector field of a stable Hamiltonian structure. In particular, such a vector field has a periodic orbit unless the 3-manifold is a torus bundle over the circle. We provide a counterexample showing that the correspondence breaks down without the real analyticity hypothesis.

In 1985, Boshernitzan showed that a minimal (sub)shift satisfying a linear block growth condition must have a bounded number of ergodic probability measures. Recently, this bound was shown to be sharp through examples constructed by Cyr and Kra. In this paper, we show that under the stronger assumption of eventually constant growth, an improved bound exists. To this end, we introduce special Rauzy graphs. Variants of the well-known Rauzy graphs from symbolic dynamics, these graphs provide an explicit description of how a Rauzy graph for words of length $n$ relates to the one for words of length $n+1$ for each $n=1,2,3,\ldots \,$.

Let $G$ be a finitely generated amenable group. We study the space of shifts on $G$ over a given finite alphabet $A$. We show that the zero entropy shifts are generic in this space, and that, more generally, the shifts of entropy $c$ are generic in the space of shifts with entropy at least $c$. The same is shown to hold for the space of transitive shifts and for the space of weakly mixing shifts. As applications of this result, we show that, for every entropy value $c\in [0,\log |A|]$, there is a weakly mixing subshift of $A^{G}$ with entropy $c$. We also show that the set of strongly irreducible shifts does not form a $G_{\unicode[STIX]{x1D6FF}}$ in the space of shifts, and that all non-trivial, strongly irreducible shifts are non-isolated points in this space.

We say that an action of a countable discrete inverse semigroup on a locally compact Hausdorff space is amenable if its groupoid of germs is amenable in the sense of Anantharaman-Delaroche and Renault. We then show that for a given inverse semigroup ${\mathcal{S}}$, the action of ${\mathcal{S}}$ on its spectrum is amenable if and only if every action of ${\mathcal{S}}$ is amenable.

We consider the construction and classification of some new mathematical objects, called ergodic spatial stationary processes, on locally compact abelian groups. These objects provide a natural and very general setting for studying diffraction and the famous inverse problems associated with it. In particular, we can construct complete families of solutions to the inverse problem from any given positive pure point measure that is chosen to be the diffraction. In this case these processes can be classified by the dual of the group of relators based on the set of Bragg peaks, and this gives an abstract solution to the homometry problem for pure point diffraction.

We consider arbitrary orderings of the edges entering each vertex of the (downward directed) Pascal graph. Each ordering determines an adic (Bratteli–Vershik) system, with a transformation that is defined on most of the space of infinite paths that begin at the root. We prove that for every ordering the coding of orbits according to the partition of the path space determined by the first three edges is essentially faithful, meaning that it is one-to-one on a set of paths that has full measure for every fully supported invariant probability measure. We also show that for every $k$ the subshift that arises from coding orbits according to the first $k$ edges is topologically weakly mixing. We give a necessary and sufficient condition for any adic system to be topologically conjugate to an odometer and use this condition to determine the probability that a random order on a fixed diagram, or a diagram constructed at random in some way, is topologically conjugate to an odometer. We also show that the closure of the union over all orderings of the subshifts arising from codings of the Pascal adic by the first edge has superpolynomial complexity, is not topologically transitive, and has no periodic points besides the two fixed points, while the intersection over all orderings consists of just four orbits.

Consider a Riccati foliation whose monodromy representation is non-elementary and parabolic and consider a non-invariant section of the fibration whose associated developing map is onto. We prove that any holonomy germ from any non-invariant fibre to the section can be analytically continued along a generic Brownian path. To prove this theorem, we prove a dual result about complex projective structures. Let $\unicode[STIX]{x1D6F4}$ be a hyperbolic Riemann surface of finite type endowed with a branched complex projective structure: such a structure gives rise to a non-constant holomorphic map ${\mathcal{D}}:\tilde{\unicode[STIX]{x1D6F4}}\rightarrow \mathbb{C}\mathbb{P}^{1}$, from the universal cover of $\unicode[STIX]{x1D6F4}$ to the Riemann sphere $\mathbb{C}\mathbb{P}^{1}$, which is $\unicode[STIX]{x1D70C}$-equivariant for a morphism $\unicode[STIX]{x1D70C}:\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6F4})\rightarrow \mathit{PSL}(2,\mathbb{C})$. The dual result is the following. If the monodromy representation $\unicode[STIX]{x1D70C}$ is parabolic and non-elementary and if ${\mathcal{D}}$ is onto, then, for almost every Brownian path $\unicode[STIX]{x1D714}$ in $\tilde{\unicode[STIX]{x1D6F4}}$, ${\mathcal{D}}(\unicode[STIX]{x1D714}(t))$ does not have limit when $t$ goes to $\infty$. If, moreover, the projective structure is of parabolic type, we also prove that, although ${\mathcal{D}}(\unicode[STIX]{x1D714}(t))$ does not converge, it converges in the Cesàro sense.

We define weaker forms of topological and measure-theoretical equicontinuity for topological dynamical systems, and we study their relationships with sequence entropy and systems with discrete spectrum. We show that for topological systems equipped with ergodic measures having discrete spectrum is equivalent to $\unicode[STIX]{x1D707}$-mean equicontinuity. In the purely topological category we show that minimal subshifts with zero topological sequence entropy are strictly contained in diam-mean equicontinuous systems; and that transitive almost automorphic subshifts are diam-mean equicontinuous if and only if they are regular (i.e. the maximal equicontinuous factor map is one–one on a set of full Haar measure). For both categories we find characterizations using stronger versions of the classical notion of sensitivity. As a consequence, we obtain a dichotomy between discrete spectrum and a strong form of measure-theoretical sensitivity.

We consider the dynamics of holomorphic polynomials in $\mathbb{C}$ . We show that the ergodic properties of the map can be seen already from the real parts of the orbits.

We propose a new criterion for randomness of a word $x_{1}x_{2}\cdots x_{n}\in \mathbb{A}^{n}$ over a finite alphabet $\mathbb{A}$ defined by

$$\begin{eqnarray}\unicode[STIX]{x1D6EF}^{n}(x_{1}x_{2}\cdots x_{n})=\mathop{\sum }_{\unicode[STIX]{x1D709}\in \mathbb{A}^{+}}\unicode[STIX]{x1D713}(|x_{1}x_{2}\cdots x_{n}|_{\unicode[STIX]{x1D709}}),\end{eqnarray}$$

$\mathbb{A}^{+}=\bigcup _{k=1}^{\infty }\mathbb{A}^{k}$

$\mathbb{A}$

$\unicode[STIX]{x1D709}\in \mathbb{A}^{k}$

$$\begin{eqnarray}|x_{1}x_{2}\cdots x_{n}|_{\unicode[STIX]{x1D709}}=\#\{i;~1\leq i\leq n-k+1,~x_{i}x_{i+1}\cdots x_{i+k-1}=\unicode[STIX]{x1D709}\},\end{eqnarray}$$

$t\geq 0$

$\unicode[STIX]{x1D713}(0)=0$

$\unicode[STIX]{x1D713}(t)=t\log t~(t>0)$

$x_{1}x_{2}\cdots x_{n}$

$$\begin{eqnarray}Q(x_{1}x_{2}\cdots x_{n})=(1/2)(n\log n)^{2}/\unicode[STIX]{x1D6EF}^{n}(x_{1}x_{2}\cdots x_{n}).\end{eqnarray}$$

$$\begin{eqnarray}\lim _{n\rightarrow \infty }(1/n)Q(X_{1}X_{2}\cdots X_{n})=h(X)\end{eqnarray}$$

$X_{1}X_{2}\cdots \,$

$\mathbb{A}$

$h(X)=0$

$h(X)$

$t^{2}$

$t\log t$

We consider general transformations of random walks on groups determined by Markov stopping times and prove that the asymptotic entropy (respectively, rate of escape) of the transformed random walks is equal to the asymptotic entropy (respectively, rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov formula from ergodic theory; its particular cases were established earlier by Kaimanovich [Differential entropy of the boundary of a random walk on a group. Uspekhi Mat. Nauk38(5(233)) (1983), 187–188] and Hartman et al [An Abramov formula for stationary spaces of discrete groups. Ergod. Th. & Dynam. Sys.34(3) (2014), 837–853].

We study the problem of the existence of wild attractors for critical circle coverings with Fibonacci dynamics. This is known to be related to the drift for the corresponding fixed points of renormalization. The fixed point depends only on the order of the critical point $\ell$ and its drift is a number $\unicode[STIX]{x1D717}(\ell )$ which is finite for each finite $\ell$. We show that the limit $\unicode[STIX]{x1D717}(\infty ):=\lim _{\ell \rightarrow \infty }\unicode[STIX]{x1D717}(\ell )$ exists and is finite. The finiteness of the limit is in a sharp contrast with the case of Fibonacci unimodal maps. Furthermore, $\unicode[STIX]{x1D717}(\infty )$ is expressed as a contour integral in terms of the limit of the fixed points of renormalization when $\ell \rightarrow \infty$. There is a certain paradox here, since this dynamical limit is a circle homeomorphism with the golden mean rotation number whose own drift is $\infty$ for topological reasons.

Let $\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}>1$ and $T_{i}(x,y)=((x+i)/\unicode[STIX]{x1D6FD}_{1},(y+i)/\unicode[STIX]{x1D6FD}_{2}),i\in \{\pm 1\}$ . Let $A:=A_{\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}}$ be the unique compact set satisfying $A=T_{1}(A)\cup T_{-1}(A)$ . In this paper, we give a detailed analysis of $A$ and the parameters $(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$ where $A$ satisfies various topological properties. In particular, we show that if $\unicode[STIX]{x1D6FD}_{1}<\unicode[STIX]{x1D6FD}_{2}<1.202$ , then $A$ has a non-empty interior, thus significantly improving the bound from Dajani et al [Self-affine sets with positive Lebesgue measure. Indag. Math. (N.S.)25 (2014), 774–784]. In the opposite direction, we prove that the connectedness locus for this family studied in Solomyak [Connectedness locus for pairs of affine maps and zeros of power series. Preprint, 2014, arXiv:1407.2563] is not simply connected. We prove that the set of points of $A$ which have a unique address has positive Hausdorff dimension for all $(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$ . Finally, we investigate simultaneous $(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$ -expansions of reals, which were the initial motivation for studying this family in Güntürk [Simultaneous and hybrid beta-encodings. Information Sciences and Systems, 2008. CISS 2008. 42nd Annual Conference 2008, pp. 743–748].

We consider ergodic series of the form $\sum _{n=0}^{\infty }a_{n}f(T^{n}x)$, where $f$ is an integrable function with zero mean value with respect to a $T$-invariant measure $\unicode[STIX]{x1D707}$. Under certain conditions on the dynamical system $T$, the invariant measure $\unicode[STIX]{x1D707}$ and the function $f$, we prove that the series converges $\unicode[STIX]{x1D707}$-almost everywhere if and only if $\sum _{n=0}^{\infty }|a_{n}|^{2}<\infty$, and that in this case the sum of the convergent series is exponentially integrable and satisfies a Khintchine-type inequality. We also prove that the system $\{f\circ T^{n}\}$ is a Riesz system if and only if the spectral measure of $f$ is absolutely continuous with respect to the Lebesgue measure and the Radon–Nikodym derivative is bounded from above as well as from below by a constant. We check the conditions for Gibbs measures $\unicode[STIX]{x1D707}$ relative to hyperbolic dynamics $T$ and for Hölder functions $f$. An application is given to the study of differentiability of the Weierstrass-type functions $\sum _{n=0}^{\infty }a_{n}f(3^{n}x)$.

Let $G$ be a topological group, let $\unicode[STIX]{x1D719}$ be a continuous endomorphism of $G$ and let $H$ be a closed $\unicode[STIX]{x1D719}$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is,

$$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$

$\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$

$\unicode[STIX]{x1D719}$

$G$

$H$

$\unicode[STIX]{x1D719}H=H$

$\ker (\unicode[STIX]{x1D719})\leq H$

$s(\unicode[STIX]{x1D719})$

$\log s(\unicode[STIX]{x1D719})$

$\unicode[STIX]{x1D719}$

$\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$