We construct an increasing sequence of natural numbers $(m_{n})_{n=1}^{+\infty }$ with the property that $(m_{n}{\it\theta}[1])_{n\geq 1}$ is dense in $\mathbb{T}$ for any ${\it\theta}\in \mathbb{R}\setminus \mathbb{Q}$, and a continuous measure on the circle ${\it\mu}$ such that $\lim _{n\rightarrow +\infty }\int _{\mathbb{T}}\Vert m_{n}{\it\theta}\Vert \,d{\it\mu}({\it\theta})=0$. Moreover, for every fixed $k\in \mathbb{N}$, the set $\{n\in \mathbb{N}:k\nmid m_{n}\}$ is infinite. This is a sufficient condition for the existence of a rigid, weakly mixing dynamical system whose rigidity time is not a rigidity time for any system with a discrete part in its spectrum.

On a compact real interval, the spaces of all transitive maps, all piecewise monotone transitive maps and all piecewise linear transitive maps are considered with the uniform metric. It is proved that they are contractible and uniformly locally arcwise connected. Then the spaces of all piecewise monotone transitive maps with given number of pieces as well as various unions of such spaces are considered and their connectedness properties are studied.

It is known that the disconnected Julia set of any polynomial map does not contain buried Julia components. But such Julia components may arise for rational maps. The first example is due to Curtis T. McMullen who provided a family of rational maps for which the Julia sets are Cantor of Jordan curves. However, all known examples of buried Julia components, up to now, are points or Jordan curves and comes from rational maps of degree at least five. This paper introduces a family of hyperbolic rational maps with disconnected Julia set whose exchanging dynamics of postcritically separating Julia components is encoded by a weighted dynamical tree. Each of these Julia sets presents buried Julia components of several types: points, Jordan curves, but also Julia components which are neither points nor Jordan curves. Moreover, this family contains some rational maps of degree three with explicit formula that answers a question McMullen raised.

We define a random iterated function system (RIFS) to be a finite set of (deterministic) iterated function systems (IFSs) acting on the same metric space. For a given RIFS, there exists a continuum of random attractors corresponding to each sequence of deterministic IFSs. Much work has been done on computing the ‘almost sure’ dimensions of these random attractors. Here we compute the typical dimensions (in the sense of Baire) and observe that our results are in stark contrast to those obtained using the probabilistic approach. Furthermore, we examine the typical Hausdorff and packing measures of the random attractors and give examples to illustrate some of the strange phenomena that can occur. The only restriction we impose on the maps is that they are bi-Lipschitz and we obtain our dimension results without assuming any separation conditions.

We associate the topological entropy of monotone recurrence relations with the Aubry–Mather theory. If there exists an interval $[{\it\rho}_{0},{\it\rho}_{1}]$ such that, for each ${\it\omega}\in ({\it\rho}_{0},{\it\rho}_{1})$, all Birkhoff minimizers with rotation number ${\it\omega}$ do not form a foliation, then the diffeomorphism on the high-dimensional cylinder defined via the monotone recurrence relation has positive topological entropy.

In this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their corresponding Julia sets. In particular, we give the specific expressions of some rational maps whose Julia sets are Cantor circles, but they are not topologically conjugate to any McMullen maps on their Julia sets. Moreover, some non-hyperbolic rational maps whose Julia sets are Cantor circles are also constructed.

We consider the dynamics on the $C^{\ast }$-algebras of finite graphs obtained by lifting the gauge action to an action of the real line. Enomoto, Fujii and Watatani [KMS states for gauge action on ${\mathcal{O}}_{A}$. Math. Japon.29 (1984), 607–619] proved that if the vertex matrix of the graph is irreducible, then the dynamics on the graph algebra admits a single Kubo–Martin–Schwinger (KMS) state. We have previously studied the dynamics on the Toeplitz algebra, and explicitly described a finite-dimensional simplex of KMS states for inverse temperatures above a critical value. Here we study the KMS states for graphs with reducible vertex matrix, and for inverse temperatures at and below the critical value. We prove a general result which describes all the KMS states at a fixed inverse temperature, and then apply this theorem to a variety of examples. We find that there can be many patterns of phase transition, depending on the behaviour of paths in the underlying graph.

In this paper, we use abstract Lyapunov graphs as a combinatorial tool to obtain a complete classification of Smale flows on $\mathbb{S}^{2}\times \mathbb{S}^{1}$. This classification gives necessary and sufficient conditions that must be satisfied by an (abstract) Lyapunov graph in order for it to be associated to a Smale flow on $\mathbb{S}^{2}\times \mathbb{S}^{1}$.

For $r\geq 3$, $p\geq 2$, we classify all actions of the groups ${ \mathrm{Diff} }_{c}^{r} ( \mathbb{R} )$ and ${ \mathrm{Diff} }_{+ }^{r} ({S}^{1} )$ by ${C}^{p} $-diffeomorphisms on the line and on the circle. This is the same as describing all non-trivial group homomorphisms between groups of compactly supported diffeomorphisms on 1-manifolds. We show that all such actions have an elementary form, which we call topologically diagonal. As an application, we answer a question of Ghys in the 1-manifold case: if $M$ is any closed manifold, and ${\mathrm{Diff} }^{\infty } \hspace{-2.0pt} \mathop{(M)}\nolimits_{0} $ injects into the diffeomorphism group of a 1-manifold, must $M$ be one-dimensional? We show that the answer is yes, even under more general conditions. Several lemmas on subgroups of diffeomorphism groups are of independent interest, including results on commuting subgroups and flows.

In this paper we introduce the notion of parabolic-like mapping. Such an object is similar to a polynomial-like mapping, but it has a parabolic external class, i.e. an external map with a parabolic fixed point. We define the notion of parabolic-like mapping and we study the dynamical properties of parabolic-like mappings. We prove a straightening theorem for parabolic-like mappings which states that any parabolic-like mapping of degree two is hybrid conjugate to a member of the family

$$\begin{eqnarray}\mathit{Per}_{1}(1)=\left\{[P_{A}]\,\bigg|\,P_{A}(z)=z+\frac{1}{z}+A,~A\in \mathbb{C}\right\}\!,\end{eqnarray}$$

Let $f$ be a transitive homeomorphism of the two-dimensional torus in the homotopy class of the identity. We show that a lift of $f$ to the universal covering is transitive if and only if the rotation set of the lift contains the origin in its interior.

Let $0\lt a\lt b\lt 1$ and let $T$ be the doubling map. Set $ \mathcal{J} (a, b): = \{ x\in [0, 1] : {T}^{n} x\not\in (a, b), n\geq 0\} $. In this paper we completely characterize the holes $(a, b)$ for which any of the following scenarios hold: (i) $ \mathcal{J} (a, b)$ contains a point $x\in (0, 1)$; (ii) $ \mathcal{J} (a, b)\cap [\delta , 1- \delta ] $ is infinite for any fixed $\delta \gt 0$; (iii) $ \mathcal{J} (a, b)$ is uncountable of zero Hausdorff dimension; (iv) $ \mathcal{J} (a, b)$ is of positive Hausdorff dimension. In particular, we show that (iv) is always the case if

$$\begin{eqnarray*}b- a\lt \frac{1}{4} { \mathop{\prod }\nolimits}_{n= 1}^{\infty } (1- {2}^{- {2}^{n} } )\approx 0. 175\hspace{0.167em} 092\end{eqnarray*}$$

A Herman–Avila–Bochi type formula is obtained for the average sum of the top $d$ Lyapunov exponents over a one-parameter family of $\mathbb{G}$-cocycles, where $\mathbb{G}$ is the group that leaves a certain, non-degenerate Hermitian form of signature $(c,d)$ invariant. The generic example of such a group is the pseudo-unitary group $\text{U}(c,d)$ or, in the case $c=d$, the Hermitian-symplectic group $\text{HSp}(2d)$ which naturally appears for cocycles related to Schrödinger operators. In the case $d=1$, the formula for $\text{HSp}(2d)$ cocycles reduces to the Herman–Avila–Bochi formula for $\text{SL}(2,\mathbb{R})$ cocycles.

We study the long-term behavior of injective piecewise contractions of the interval. We prove that every injective piecewise contraction with $n-1$ discontinuities has at most $n$ periodic orbits and is topologically conjugate to a piecewise linear contraction.

Variational monotone recurrence relations arise in solid state physics as generalizations of the Frenkel–Kontorova model for a ferromagnetic crystal. For such problems, Aubry–Mather theory establishes the existence of ‘ground states’ or ‘global minimizers’ of arbitrary rotation number. A nearest neighbor crystal model is equivalent to a Hamiltonian twist map. In this case, the global minimizers have a special property: they can only cross once. As a non-trivial consequence, every one of them has the Birkhoff property. In crystals with a larger range of interaction and for higher order recurrence relations, the single crossing property does not hold and there can exist global minimizers that are not Birkhoff. In this paper we investigate the crossings of global minimizers. Under a strong twist condition, we prove the following dichotomy: they are either Birkhoff, and thus very regular, or extremely irregular and non-physical: they then grow exponentially and oscillate. For Birkhoff minimizers, we also prove certain strong ordering properties that are well known for twist maps.

We investigate a parametric extension of the classical $s$-dimensional Halton sequence where the bases are special Pisot numbers. In a one-dimensional setting the properties of such sequences have already been investigated by several authors. We use methods from ergodic theory in order to investigate the distribution behavior of multidimensional versions of such sequences. As a consequence it is shown that the Kakutani–Fibonacci transformation is uniquely ergodic.

It has been shown that sufficiently well mixing dynamical systems with positive entropy have extreme-value laws which in the limit converge to one of the three standard distributions known for independently and identically distributed processes, namely Gumbel, Fréchet and Weibull distributions. In this short note, we give an example which has a non-standard limiting distribution for its extreme values. Rotations of the circle by irrational numbers are used and it will be shown that the limiting distribution is a step function where the limit has to be taken along a suitable sequence given by the convergents.

In this paper we study the radial and orthogonal projections and the distance sets of the random Cantor sets $E\subset { \mathbb{R} }^{2} $, which are called Mandelbrot percolation or percolation fractals. We prove that the following assertion holds almost surely: if the Hausdorff dimension of $E$ is greater than $1$ then the orthogonal projection to every line, the radial projection with every centre, and the distance set from every point contain intervals.

In this paper we prove a multifractal formalism of Birkhoff averages for interval maps with countably many branches. Furthermore, we prove that under certain assumptions the Birkhoff spectrum is real analytic. We also show that new phenomena occur; indeed, the spectrum can be constant or it can have points where it is not analytic. Conditions for these to happen are obtained. Applications of these results to number theory are also given. Finally, we compute the Hausdorff dimension of the set of points for which the Birkhoff average is infinite.

In a non-compact setting, the notion of hyperbolicity, together with the associated structure of stable and unstable manifolds (for unbounded orbits), is highly dependent on the choice of metric used to define it. We consider the simplest version of this, the analogue for the plane of Anosov diffeomorphisms, studied earlier by White and Mendes. The two known topological conjugacy classes of such diffeomorphisms are linear hyperbolic automorphisms and translations. We show that if the structure of stable and unstable manifolds is required to be preserved by these conjugacies, the number of distinct equivalence classes of Anosov diffeomorphisms in the plane becomes infinite.