Let $ \mathbb{B} $ be a $p$-uniformly convex Banach space, with $p\geq 2$. Let $T$ be a linear operator on $ \mathbb{B} $, and let ${A}_{n} x$ denote the ergodic average $(1/ n){\mathop{\sum }\nolimits}_{i\lt n} {T}^{n} x$. We prove the following variational inequality in the case where $T$ is power bounded from above and below: for any increasing sequence $\mathop{({t}_{k} )}\nolimits_{k\in \mathbb{N} } $ of natural numbers we have ${\mathop{\sum }\nolimits}_{k} \mathop{\Vert {A}_{{t}_{k+ 1} } x- {A}_{{t}_{k} } x\Vert }\nolimits ^{p} \leq C\mathop{\Vert x\Vert }\nolimits ^{p} $, where the constant $C$ depends only on $p$ and the modulus of uniform convexity. For $T$ a non-expansive operator, we obtain a weaker bound on the number of $\varepsilon $-fluctuations in the sequence. We clarify the relationship between bounds on the number of $\varepsilon $-fluctuations in a sequence and bounds on the rate of metastability, and provide lower bounds on the rate of metastability that show that our main result is sharp.

We use an Ulam-type discretization scheme to provide pointwise approximations for invariant densities of interval maps with a neutral fixed point. We prove that the approximate invariant density converges pointwise to the true density at a rate ${C}^{\ast } \cdot (\ln m)/ m$, where ${C}^{\ast } $ is a computable fixed constant and ${m}^{- 1} $ is the mesh size of the discretization.

In this paper we consider quadratic polynomials on the complex plane ${f}_{c} (z)= {z}^{2} + c$ and their associated Julia sets, ${J}_{c} $. Specifically, we consider the case that the kneading sequence is periodic and not an $n$-tupling. In this case ${J}_{c} $ contains subsets that are homeomorphic to the unit circle, usually infinitely many disjoint such subsets. We prove that ${f}_{c} : {J}_{c} \rightarrow {J}_{c} $ has shadowing, and we classify all $\omega $-limit sets for these maps by showing that a closed set $R\subseteq {J}_{c} $ is internally chain transitive if, and only if, there is some $z\in {J}_{c} $ with $\omega (z)= R$.

Let $G$ be a locally compact group acting properly, by type-preserving automorphisms on a locally finite thick Euclidean building $\Delta $, and $K$ be the stabilizer of a special vertex in $\Delta $. It is known that $(G, K)$ is a Gelfand pair as soon as $G$ acts strongly transitively on $\Delta $; in particular, this is the case when $G$ is a semi-simple algebraic group over a local field. We show a converse to this statement, namely that if $(G, K)$ is a Gelfand pair and $G$ acts cocompactly on $\Delta $, then the action is strongly transitive. The proof uses the existence of strongly regular hyperbolic elements in $G$ and their peculiar dynamics on the spherical building at infinity. Other equivalent formulations are also obtained, including the fact that $G$ is strongly transitive on $\Delta $ if and only if it is strongly transitive on the spherical building at infinity.

We consider one-dimensional systems in the presence of a quasi-periodic perturbation, in the analytical setting, and study the problem of existence of quasi-periodic solutions which are resonant with the frequency vector of the perturbation. We assume that the unperturbed system is locally integrable and anisochronous, and that the frequency vector of the perturbation satisfies the Bryuno condition. Existence of resonant solutions is related to the zeros of a suitable function, called the Melnikov function—by analogy with the periodic case. We show that, if the Melnikov function has a zero of odd order and under some further condition on the sign of the perturbation parameter, then there exists at least one resonant solution which continues an unperturbed solution. If the Melnikov function is identically zero then one can push perturbation theory up to the order where a counterpart of Melnikov function appears and does not vanish identically: if such a function has a zero of odd order and a suitable positiveness condition is met, again the same persistence result is obtained. If the system is Hamiltonian, then the procedure can be indefinitely iterated and no positiveness condition must be required: as a byproduct, the result follows that at least one resonant quasi-periodic solution always exists with no assumption on the perturbation. Such a solution can be interpreted as a (parabolic) lower-dimensional torus.

We study the notions of weak rational ergodicity and rational weak mixing as defined by J. Aaronson [Rational ergodicity and a metric invariant for Markov shifts. Israel J. Math. 27(2) (1977), 93–123; Rational weak mixing in infinite measure spaces. Ergod. Th. & Dynam. Sys. 2012, to appear. http://arxiv.org/abs/1105.3541]. We prove that various families of infinite measure-preserving rank-one transformations possess or do not posses these properties, and consider their relation to other notions of mixing in infinite measure.

We first prove that topological fiber entropy potential in a relatively symbolic extension (topological joining with a subshift) of a topological dynamical system (which is a non-negative, non-decreasing, upper semicontinuous and subadditive potential $\mathfrak{H}$) yields topological pressure equal to the topological entropy of the extended system. The terms occurring on the other side of the variational principle for pressure are equal to the extension entropies of the invariant measures. Thus the variational principle for pressure reduces to the usual variational principle (for entropy) applied to the extended system. Next we prove our main theorem saying that every non-negative, upper semicontinuous and subadditive potential $\mathfrak{F}$ (we drop the monotonicity assumption) is ‘nearly equal’ to the fiber entropy potential H in some relatively symbolic extension of the system, in the sense that all terms occurring in the variational principle for pressure are the same for both potentials. This gives a new interpretation of all such potentials $\mathfrak{F}$ as a kind of additional information function enhancing the natural information arising from the dynamical system, and provides a new proof of the variational principle for pressure. At the end of the paper we provide examples showing that both assumptions, continuity and additivity, under which so-called lower pressure (defined with the help of spanning sets) equals the pressure, are essential, already in the class of non-negative, upper semicontinuous, subadditive potentials.

In this paper, we develop a variational approach to study the dynamic of a homeomorphism on a compact metric space. In particular, we describe orbits along which any Lipschitz Lyapunov function has to be constant via a non-negative Lipschitz semi-distance. We give the link with Auslander’s notion of generalized recurrence, and recover in a different way some parts of a more recent work of Akin and Auslander.

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

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

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.

We study the digits of $\beta $-expansions in the case where $\beta $ is a Salem number. We introduce new upper bounds for the numbers of occurrences of consecutive 0s in the expansion of 1. We also give lower bounds for the numbers of non-zero digits in the $\beta $-expansions of algebraic numbers. As applications, we give criteria for transcendence of the values of power series at certain algebraic points.

We study the Peierls barrier $P_{\omega }(\xi )$ for a broad class of monotone variational problems. These problems arise naturally in solid state physics and from Hamiltonian twist maps. We start by deriving an estimate for the difference $\vert P_{\omega }(\xi ) - P_{q/p}(\xi ) \vert $ of the Peierls barriers of rotation numbers $\omega \in {{\mathbb{R}}}$ and $q/p\in {\mathbb{Q}}$. A similar estimate was obtained by Mather [Modulus of continuity for Peierls’s barrier. Proc. NATO Advanced Research Workshop on Periodic Solutions of Hamiltonian Systems and Related Topics (Il Ciocco, Italy, 13–18 October 1986) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 209). Eds. P. H. Rabinowitz, A. Ambrosetti and I. Eckeland. D. Reidel, Dordrecht, 1987, pp. 177–202] in the context of twist maps, but our proof is different and applies more generally. It follows from the estimate that $\omega \mapsto P_{\omega }(\xi )$ is continuous at irrational points. Moreover, we show that the Peierls barrier depends continuously on parameters and hence that the property that a monotone variational problem admits a lamination of minimizers of rotation number $\omega \in {{\mathbb{R}}}\delimiter "026E30F {\mathbb{Q}}$ is open in the $C^1$-topology.

We show that every dendrite $X$ satisfying the condition that no subtree of $X$ contains all free arcs admits a transitive, even exactly Devaney chaotic map with arbitrarily small entropy. This gives a partial answer to a question of Baldwin from 2001.

The enumeration of combinatorial classes of the complex polynomial vector fields in $ \mathbb{C} $ presented by K. Dias [Enumerating combinatorial classes of the complex polynomial vector fields in $ \mathbb{C} $. Ergod. Th. & Dynam. Sys. 33 (2013), 416–440] is extended here to a closed form enumeration of combinatorial classes for degree $d$ polynomial vector fields up to rotations of the $2(d- 1)\mathrm{th} $ roots of unity. The main tool in the proof of this result is based on a general method of enumeration developed by V. A. Liskovets [Reductive enumeration under mutually orthogonal group actions. Acta Appl. Math. 52 (1998), 91–120].