Let $Q\rightarrow X$ be a continuous principal bundle whose group $G$ is reductive. A flow ${\it\phi}$ of automorphisms of $Q$ endowed with an ergodic probability measure on the compact base space $X$ induces two decompositions of the flag bundles associated to $Q$: a continuous one given by the finest Morse decomposition and a measurable one furnished by the multiplicative ergodic theorem. The second is contained in the first. In this paper we find necessary and sufficient conditions so that they coincide. The equality between the two decompositions implies continuity of the Lyapunov spectra under perturbations leaving unchanged the flow on the base space.

In this paper we extend Bourgain’s double recurrence result to the Wiener–Wintner averages. Let $(X,{\mathcal{F}},{\it\mu},T)$ be a standard ergodic system. We will show that for any $f_{1},f_{2}\in L^{\infty }(X)$, the double recurrence Wiener–Wintner average

$$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)e^{2{\it\pi}int}\end{eqnarray}$$

$X$

$t$

$N\rightarrow \infty$

$f_{1}$

$f_{2}$

$t$

$0$

We show that a partially hyperbolic $C^{1}$-diffeomorphism $f:M\rightarrow M$ with a uniformly compact $f$-invariant center foliation ${\mathcal{F}}^{c}$ is dynamically coherent. Further, the induced homeomorphism $F:M/{\mathcal{F}}^{c}\rightarrow M/{\mathcal{F}}^{c}$ on the quotient space of the center foliation has the shadowing property, i.e. for every ${\it\epsilon}>0$ there exists ${\it\delta}>0$ such that every ${\it\delta}$-pseudo-orbit of center leaves is ${\it\epsilon}$-shadowed by an orbit of center leaves. Although the shadowing orbit is not necessarily unique, we prove the density of periodic center leaves inside the chain recurrent set of the quotient dynamics. Other interesting properties of the quotient dynamics are also discussed.

We show that a large class of i.c.c., countable, discrete groups satisfying a weak negative curvature condition are not inner amenable. By recent work of Hull and Osin [Groups with hyperbolically embedded subgroups. Algebr. Geom. Topol.13 (2013), 2635–2665], our result recovers that mapping class groups and $\text{Out}(\mathbb{F}_{n})$ are not inner amenable. We also show that the group-measure space constructions associated to free, strongly ergodic p.m.p. actions of such groups do not have property Gamma of Murray and von Neumann [On rings of operators IV. Ann. of Math. (2) 44 (1943), 716–808].

We study one-parameter families of quasi-periodically forced monotone interval maps and provide sufficient conditions for the existence of a parameter at which the respective system possesses a non-uniformly hyperbolic attractor. This is equivalent to the existence of a sink-source orbit, that is, an orbit with positive Lyapunov exponent both forwards and backwards in time. The attractor itself is a non-continuous invariant graph with negative Lyapunov exponent, often referred to as ‘SNA’. In contrast to former results in this direction, our conditions are ${\mathcal{C}}^{2}$-open in the fibre maps. By applying a general result about saddle-node bifurcations in skew-products, we obtain a conclusion on the occurrence of non-smooth bifurcations in the respective families. Explicit examples show the applicability of the derived statements.

We consider the space of complex polynomials of degree $n\geq 3$ with $n-1$ distinct marked periodic orbits of given periods. We prove that this space is irreducible and the multipliers of the marked periodic orbits, considered as algebraic functions on that space, are algebraically independent over $\mathbb{C}$. Equivalently, this means that at its generic point the moduli space of degree-$n$ polynomial maps can be locally parameterized by the multipliers of $n-1$ arbitrary distinct periodic orbits. We also prove a similar result for a certain class of affine subspaces of the space of complex polynomials of degree $n$.

D’après un célèbre lemme de John Franks, toute perturbation de la différentielle d’un difféomorphisme $f$ le long d’une orbite périodique est réalisée par une $C^{1}$-perturbation $g$ du difféomorphisme sur un petit voisinage de ladite orbite. On n’a cependant aucune information sur le comportement des variétés invariantes de l’orbite périodique après perturbation. Nous montrons que si la perturbation de la dérivée est obtenue par une isotopie le long de laquelle existent les variétés stables/instables fortes de certaines dimensions, alors on peut faire la perturbation ci-dessus en préservant les variétés stables/instables semi-locales correspondantes. Ce résultat a de nombreuses applications en systèmes dynamiques de classes $C^{1}$. Nous en démontrons quelques unes.

A classical result in ergodic theory says that there always exists a topological model for any factor map ${\it\pi}:(X,{\mathcal{X}},{\it\mu},T)\rightarrow (Y,{\mathcal{Y}},{\it\nu},S)$ of ergodic systems. That is, there are some topological factor map $\hat{{\it\pi}}:(\hat{X},\hat{T})\rightarrow ({\hat{Y}},{\hat{S}})$ and invariant measures $\hat{{\it\mu}}$, $\hat{{\it\nu}}$ such that the diagram

$$\begin{eqnarray}\displaystyle & & \displaystyle \nonumber\end{eqnarray}$$

${\it\phi}$

${\it\psi}$

$\hat{{\it\pi}}$

We study a problem that arises from the study of Lorentz surfaces and Anosov flows. For a non-decreasing map of degree one $h:\mathbb{S}^{1}\rightarrow \mathbb{S}^{1}$, we are interested in groups of circle diffeomorphisms that act on the complement of the graph of $h$ in $\mathbb{S}^{1}\times \mathbb{S}^{1}$ by preserving a volume form. We show that such groups are semiconjugate to subgroups of $\text{PSL}(2,\mathbb{R})$ and that, when $h\in \text{Homeo}(\mathbb{S}^{1})$, we have a topological conjugacy. We also construct examples where $h$ is not continuous, for which there is no such conjugacy.

Let $1,-1,-1,1,-1,1,1,-1,-1,1,1,\ldots$ be the $\{-1,1\}$-valued Thue–Morse sequence. Its correlation dimension is $D_{2}$, satisfying

$$\begin{eqnarray}\mathop{\sum }_{k=0}^{K-1}|{\it\gamma}(k)|^{2}\asymp K^{1-D_{2}}\end{eqnarray}$$

$\infty$

$K\rightarrow \infty$

${\it\gamma}$

$$\begin{eqnarray}D_{2}=1-\log \frac{1+\sqrt{17}}{4}\bigg/\log 2=0.64298\ldots .\end{eqnarray}$$

${\it\mu}$

$[0,1)$

$T$

$Tx=2x$

$\text{mod}~1$

$D_{2}\log 2$

$T^{-1}$

$$\begin{eqnarray}P_{1}((1/2)x+(1/2)j\mid x)=(1/2)(1-\cos ({\it\pi}(x+j)))\quad (j=0,1).\end{eqnarray}$$

${\it\mu}$

$$\begin{eqnarray}\lim _{N\rightarrow \infty }\int P_{N}((x-{\it\varepsilon},x+{\it\varepsilon})|x)\,d{\it\mu}(x)\asymp {\it\varepsilon}^{D_{2}}\quad (\text{as}~{\it\varepsilon}\rightarrow 0),\end{eqnarray}$$

$P_{N}(\cdot \mid \cdot )$

$N$

It is known that a polynomial on $\mathbb{C}$ is holomorphically conjugate to its term of highest degree near infinity. By assigning suitable weights, we generalize this fact to polynomial skew products on $\mathbb{C}^{2}$.

We apply the tools of continued fractions to tackle the Diophantine approximation, including the classic Jarník–Besicovitch theorem, localized Jarník–Besicovitch theorem and its several generalizations. As is well known, the classic Jarník–Besicovitch sets, expressed in terms of continued fractions, can be written as

$$\begin{eqnarray}\{x\in [0,1):a_{n+1}(x)\geq e^{{\it\tau}(\log |T^{\prime }x|+\cdots +\log |T^{\prime }(T^{n-1}x)|)}~\text{for infinitely many}~n\in \mathbb{N}\},\end{eqnarray}$$

$T$

$a_{n}(x)$

$n$

$x$

$$\begin{eqnarray}\{x\in [0,1):a_{n+1}(x)\geq e^{{\it\tau}(x)(f(x)+\cdots +f(T^{n-1}x))}~\text{for infinitely many}~n\in \mathbb{N}\},\end{eqnarray}$$

${\it\tau}(x)$

$f(x)$

$[0,1]$

A Lattès map $f:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ is a rational map that is obtained from a finite quotient of a conformal torus endomorphism. We characterize Lattès maps by their combinatorial expansion behavior.