We study 1) the slopes of central branches of iterates of S-unimodal maps, comparing them to the derivatives on the critical trajectory, 2) the hyperbolic structure of Collet-Eckmann maps estimating the exponents, and under a summability condition 3) the images of the density one under the iterates of the Perron-Frobenius operator, 4) the density of the absolutely continuous invariant measure.

A class of strictly ergodic Toeplitz flows with positive entropies and trivial topological centralizers is presented.

Let $T:L^p\left(\Omega \right)\to L^p\left(\Omega \right)$ be a positive contraction, with $1\<p\<\infty$. Assume that $T$ is analytic, that is, there exists a constant $K\ge 0$ such that $\parallel T^n-T^{n-1}\parallel \le K/n$ for any integer $n\ge 1$. Let $2\<q\<\infty$ and let $v^q$ be the space of all complex sequences with a finite strong $q$-variation. We show that for any $x\in L^p\left(\Omega \right)$, the sequence ${\left(\left[T^n\left(x\right)\right]\left(\lambda \right)\right)}_{n\ge 0}$ belongs to $v^q$ for almost every $\lambda \in \Omega$, with an estimate $\parallel {\left(T^n\left(x\right)\right)}_{n\ge 0}{\parallel }_{L^p\left(v^q\right)}\le C{\parallel x\parallel }_p$. If we remove the analyticity assumption, we obtain an estimate $\parallel {\left(M_n\left(T\right)x\right)}_{n\ge 0}{\parallel }_{L^p\left(v^q\right)}\le C{\parallel x\parallel }_p$, where $M_n\left(T\right)={(n+1)}^{-1}{\sum }_{k=0}^n{T}^k$ denotes the ergodic average of $T$. We also obtain similar results for strongly continuous semigroups ${\left(T_t\right)}_{t\ge 0}$ of positive...