We consider skew-product maps over circle rotations $x\mapsto x+\alpha \;(\mod 1)$ with factors that take values in ${\textrm {SL}}(2,{\mathbb {R}})$ . In numerical experiments, with $\alpha $ the inverse golden mean, Fibonacci iterates of maps from the almost Mathieu family exhibit asymptotic scaling behavior that is reminiscent of critical phase transitions. In a restricted setup that is characterized by a symmetry, we prove that critical behavior indeed occurs and is universal in an open neighborhood of the almost Mathieu family. This behavior is governed by a periodic orbit of a renormalization transformation. An extension of this transformation is shown to have a second periodic orbit as well, and we present some evidence that this orbit attracts supercritical almost Mathieu maps.

We consider a class of skew product maps of interval diffeomorphisms over the doubling map. The interval maps fix the end points of the interval. It is assumed that the system has zero fiber Lyapunov exponent at one endpoint and zero or positive fiber Lyapunov exponent at the other endpoint. We prove the appearance of on–off intermittency. This is done using the equivalent description of chaotic walks: random walks driven by the doubling map. The analysis further relies on approximating the chaotic walks by Markov random walks, that are constructed using Markov partitions for the doubling map.

Consider a component ${\cal Q}$ of a stratum in the moduli space of area-one abelian differentials on a surface of genus g. Call a property ${\cal P}$ for periodic orbits of the Teichmüller flow on ${\cal Q}$ typical if the growth rate of orbits with property ${\cal P}$ is maximal. We show that the following property is typical. Given a continuous integrable cocycle over the Teichmüller flow with values in a vector bundle $V\to {\cal Q}$ , the logarithms of the eigenvalues of the matrix defined by the cocycle and the orbit are arbitrarily close to the Lyapunov exponents of the cocycle for the Masur–Veech measure.

An IP set in ℕ is a subset of ℕ which coincides with the set of finite sums taken from an infinite sequence in ℕ with not necessarily distinct terms. It has been established by H. Furstenberg and others that there is a rich connection between IP sets and idempotents in the enveloping semigroup E(X) of a compact metric dynamical system. Our aim in this paper is to further develop this program by considering special elements of E(X) with various IP properties.

We describe the asymptotic dynamics of homeomorphisms obtained as restrictions of generic C2 endomorphisms of an interval with finitely many critical points, all of which are non-flat, and with all periodic points hyperbolic. The ω -limit set of such a restricted endomorphism cannot be infinite, except when the restriction of the endomorphism to the closure of the orbit of some critical point is a minimal homeomorphism of an infinite set.

We use a new method to construct a class of strongly uniformly distributed D-sequences. Such sequences are used to determine ergodic measure-preserving transformations.

Let $\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ from the unit interval $\mathbf{I}= [\!0,1\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if

$$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$

$[T_{\unicode[STIX]{x1D703}},\text{Id}]$

$\unicode[STIX]{x1D703}$

$[T_{\unicode[STIX]{x1D703}},\text{Id}]$

$$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$

$$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}|<+\infty ,\end{eqnarray}$$

$[0;a_{1},a_{2},\ldots ]$

$(p_{n}/q_{n})_{n\geq 0}$

$\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$

$\unicode[STIX]{x1D703}$

According to a theorem of Moser, every monotone twist map $\varphi$ on the cylinder ${\Bbb S}^1\times {\Bbb R}$, which is integrable outside a compact set, is the time-1-map $\varphi_H^1$ of a fibrewise convex Hamiltonian $H$. In this paper we prove that if this particular flow $\varphi_H^t$ is also integrable outside a compact set, then $\varphi$ has to be integrable on the whole cylinder (and vice versa, of course). From this dynamical point of view, integrable twist maps appear to be quite rigid.

As is shown in the appendix, an analogous rigidity result becomes trivial in higher dimensions.

We construct a homeomorphism from the filled Julia set of a given polynomial into the connectedness locus of the family of all polynomials of higher degree by using intertwining surgery. A polynomial in the image of such a homeomorphism has a renormalization of ‘capture’ type and the inverse map is given by the ‘marking’ of the captured critical orbit.

We consider the notion of the graph product of actions of discrete groups $\{G_{v}\}$ on a $C^{\ast }$-algebra ${\mathcal{A}}$ and show that under suitable commutativity conditions the graph product action $\star _{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D6FC}_{v}:\star _{\unicode[STIX]{x1D6E4}}G_{v}\curvearrowright {\mathcal{A}}$ has the Haagerup property if each action $\unicode[STIX]{x1D6FC}_{v}:G_{v}\curvearrowright {\mathcal{A}}$ possesses the Haagerup property. This generalizes the known results on graph products of groups with the Haagerup property. To accomplish this, we introduce the graph product of multipliers associated to the actions and show that the graph product of positive-definite multipliers is positive definite. These results have impacts on left-transformation groupoids and give an alternative proof of a known result for coarse embeddability. We also record a cohomological characterization of the Haagerup property for group actions.

We study spectral properties of the flow $\dot x =1/F(x,y)$, $\dot y = 1/\lambda F(x,y)$ on the 2-torus. We show that, in general, the speed of approximation in cyclic approximation gives an upper bound on the Hausdorff dimension of the supports of spectral measures. We use this to prove that for generic pairs $(F,\lambda)$ the spectrum of the flow on the torus is singular continuous with all spectral measures supported on sets of zero Hausdorff dimension.

Let $R$ be a ring of characteristic $0$ with field of fractions $K$ and let $m\geq 2$. The Böttcher coordinate of a power series $\unicode[STIX]{x1D711}(x)\in x^{m}+x^{m+1}R\unicode[STIX]{x27E6}x\unicode[STIX]{x27E7}$ is the unique power series $f_{\unicode[STIX]{x1D711}}(x)\in x+x^{2}K\unicode[STIX]{x27E6}x\unicode[STIX]{x27E7}$ satisfying $\unicode[STIX]{x1D711}\circ f_{\unicode[STIX]{x1D711}}(x)=f_{\unicode[STIX]{x1D711}}(x^{m})$. In this paper we study the integrality properties of the coefficients of $f_{\unicode[STIX]{x1D711}}(x)$, partly for their intrinsic interest and partly for potential applications to $p$-adic dynamics. Results include: (1) if $p$ is prime and $R=\mathbb{Z}_{p}$ and $\unicode[STIX]{x1D711}(x)\in x^{p}+px^{p+1}R\unicode[STIX]{x27E6}x\unicode[STIX]{x27E7}$, then $f_{\unicode[STIX]{x1D711}}(x)\in R\unicode[STIX]{x27E6}x\unicode[STIX]{x27E7}$. (2) If $\unicode[STIX]{x1D711}(x)\in x^{m}+mx^{m+1}R\unicode[STIX]{x27E6}x\unicode[STIX]{x27E7}$, then $f_{\unicode[STIX]{x1D711}}(x)=x\sum _{k=0}^{\infty }a_{k}x^{k}/k!$ with all $a_{k}\in R$. (3) In (2), if $m=p^{2}$, then $a_{k}\equiv -1~\text{(mod}~p\text{)}$ for all $k$ that are powers of $p$.

Given a ${\mathcal{C}}^{\infty }$ expanding map $T$ of the circle, we construct a Hilbert space ${\mathcal{H}}$ of smooth functions on which the transfer operator ${\mathcal{L}}$ associated to $T$ acts as a compact operator. This result is made quantitative (in terms of singular values of the operator ${\mathcal{L}}$ acting on ${\mathcal{H}}$) using the language of Denjoy–Carleman classes. Moreover, the nuclear power decomposition of Baladi and Tsujii can be performed on the space ${\mathcal{H}}$, providing a bound on the growth of the dynamical determinant associated to ${\mathcal{L}}$.

We consider the dynamics of complex rational maps on $\widehat{\mathbb{C}}$. We prove that, after reducing their orbits to a fixed number of positive values representing the Fubini–Study distances between finitely many initial elements of the orbit and the origin, ergodic properties of the rational map are preserved.

Del Junco–Lemańczyk [Generic spectral properties of measure-preserving maps and applications. Proc. Amer. Math. Soc., 115 (3) (1992)] showed that a generic measure-preserving transformation satisfies certain orthogonality conditions. More precisely, there is a dense $G_{\unicode[STIX]{x1D6FF}}$ subset of measure preserving transformations such that, for every $T\in G$ and $k(1),k(2),\ldots ,k(l)\in \mathbb{Z}^{+}$, $k^{\prime }(1),k^{\prime }(2),\ldots ,k^{\prime }(l^{\prime })\in \mathbb{Z}^{+}$, the convolutions

$$\begin{eqnarray}\unicode[STIX]{x1D70E}_{T^{k(1)}}\ast \cdots \ast \unicode[STIX]{x1D70E}_{T^{k(l)}}\quad \text{and}\quad \unicode[STIX]{x1D70E}_{T^{k^{\prime }(1)}}\ast \cdots \ast \unicode[STIX]{x1D70E}_{T^{k^{\prime }(l^{\prime })}},\end{eqnarray}$$

$\unicode[STIX]{x1D70E}_{T^{k}}$

$T^{k}$

$(k(1),k(2),\ldots ,k(l))$

$(k^{\prime }(1),k^{\prime }(2),\ldots ,k^{\prime }(l^{\prime }))$

$L^{0}(\unicode[STIX]{x1D707},\mathbb{T})$

$L^{0}(\unicode[STIX]{x1D707},\mathbb{T})$

$\{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D705}}\}$

$\unicode[STIX]{x1D705}$

$\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D705}}$

Let $f:M\rightarrow M$ be a biholomorphism on a two-dimensional complex manifold, and let $X\subseteq M$ be a compact $f$-invariant set such that $f|_{X}$ is asymptotically dissipative and without periodic sinks. We introduce a solely dynamical obstruction to dominated splitting, namely critical point. Critical point is a dynamical object and captures many of the dynamical properties of a one-dimensional critical point.

We show that every isometric action on a Cantor set is conjugate to an inverse limit of actions on finite sets; and that every faithful isometric action by a finitely generated amenable group is residually finite.

Let (B,T) be a fibred system, where B is a compact and convex subset of \mathbb{R}^{d} and T:B\mapsto B is a map. Under adequate norm the space \mathcal{L} (\mathcal{H}) (respectively \mathcal{C} (\mathcal{H})) of Lipschitz continuous (respectively continuous) functions on every cell H of a certain finite partition \mathcal{H} of B is a Banach space. We delimit by Conditions \rm (A),\dotsc,(E),(F)' a class of fibred systems, including multidimensional continued fractions. We first prove that the Perron–Frobenius operator A associated with T under the Lebesgue measure \lambda considered on <formula form="inline" disc="math" id="eq014"><formtex notation="amstex">\mathcal{L} (\mathcal{H}) is weak-mixing. Using the ergodic theorem of Ionescu–Tulcea and Marinescu we then prove that there exist positive constants q >1 and M for which

\vert\mspace{-2mu}\vert\mspace{-2mu}\vert A^{n}\varphi -h\int_{B}\varphi \,d\lambda\vert\mspace{-2mu}\vert\mspace{-2mu}\vert \leq q^{n}M\vert\mspace{-2mu}\vert\mspace{-2mu}\vert\varphi \vert\mspace{-2mu}\vert\mspace{-2mu}\vert, \quad n\in \mathbb{N}, \varphi \in \mathcal{L} (\mathcal{H}),

where \vert\mspace{-2mu}\vert\mspace{-2mu}\vert\cdot\vert\mspace{-2mu}\vert\mspace{-2mu}\vert is the norm in \mathcal{L} (\mathcal{H}) and h is the density of the (unique) T-invariant probability \mu on the Borel \sigma-algebra \Sigma in B. The analogous result for the Perron–Frobenius operator U of T under \mu is also proved.

In this paper we prove the following rigidity result: let ${\it\varphi}$ be a $C^{\infty }$ topologically mixing transversely symplectic Anosov flow. If (i) its weak stable and weak unstable distributions are $C^{\infty }$ and (ii) its Hamenstädt metrics are sub-Riemannian, then up to finite covers and a constant change of time scale, ${\it\varphi}$ is $C^{\infty }$ flow conjugate to the geodesic flow of a closed locally symmetric Riemannian space of rank one.

We propose to study the renormalization operator acting on critical $\mathcal{C}^r$ circle mappings. (More precisely, the operator acts on critical commuting pairs.) Assuming that there is a Banach manifold of critical analytic commuting pairs on which the renormalization operator acts hyperbolically (with non-trivial hyperbolic attractor), we prove that, for r > 2, the operator remains hyperbolic with the same expanding subspaces when acting on $\mathcal{C}^r$ commuting pairs. By this we mean that the tangent renormalization operator admits a hyperbolic splitting with the same unstable subbundle.