The Gauss map on a class of interval translation mappings

We study the dynamics of a class of interval translation map on three intervals. We show that in this class the typical ITM is of finite type (reduce to an interval exchange transformation) and that the complement contains a Cantor set. We relate our maps to substitution subshifts. Results on Hausdorff dimension of the attractor and on unique ergodicity are obtained

On `observable' Li-Yorke tuples for interval maps

In this paper we study the set of Li-Yorke $d$-tuples and its $d$-dimensional Lebesgue measure for interval maps $T\colon [0,1] \to [0,1]$. If a topologically mixing $T$ preserves an absolutely continuous probability measure 9with respect to Lebesgue), then the $d$-tuples have Lebesgue full measure, but if $T$ preserves an infinite absolutely continuous measure, the situation becomes more interesting. Taking the family of Manneville-Pomeau maps as example, we show that for any $d \ge 2$, it is possible that the set of Li-Yorke $d$-tuples has full Lebesgue measure, but the set of Li-Yorke $d+1$-tuples has zero Lebesgue measure

Natural equilibrium states for multimodal maps

This paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps. This class contains, but it is larger than, Collet-Eckmann. For a map in this class, we prove existence and uniqueness of equilibrium states for the geometric potentials $-t \log|Df|$, for the largest possible interval of parameters $t$. We also study the regularity and convexity properties of the pressure function, completely characterising the first order phase transitions. Results concerning the existence of absolutely continuous invariant measures with respect to the Lebesgue measure are also obtained

On the Lebesgue measure of Li-Yorke pairs for interval maps

We investigate the prevalence of Li-Yorke pairs for $C^2$ and $C^3$ multimodal maps $f$ with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points. If $f$ is topologically mixing and has no Cantor attractor, then typical (w.r.t. two-dimensional Lebesgue measure) pairs are Li-Yorke; if additionally $f$ admits an absolutely continuous invariant probability measure (acip), then typical pairs have a dense orbit for $f \times f$. These results make use of so-called nice neighborhoods of the critical set of general multimodal maps, and hence uniformly expanding Markov induced maps, the existence of either is proved in this paper as well. For the setting where $f$ has a Cantor attractor, we present a trichotomy explaining when the set of Li-Yorke pairs and distal pairs have positive two-dimensional Lebesgue measure.Comment: 41 pages, 3 figure

Equilibrium states for potentials with \sup\phi - \inf\phi < \htop(f)

In the context of smooth interval maps, we study an inducing scheme approach to prove existence and uniqueness of equilibrium states for potentials $\phi$ with he `bounded range' condition \sup \phi - \inf \phi < \htop, first used by Hofbauer and Keller. We compare our results to Hofbauer and Keller's use of Perron-Frobenius operators. We demonstrate that this `bounded range' condition on the potential is important even if the potential is H\"older continuous. We also prove analyticity of the pressure in this context.Comment: Added Lemma 6 to deal with the disparity between leading eigenvalues and operator norms. Added extra references and corrected some typo

Complex maps without invariant densities

We consider complex polynomials $f(z) = z^\ell+c_1$ for $\ell \in 2\N$ and $c_1 \in \R$, and find some combinatorial types and values of $\ell$ such that there is no invariant probability measure equivalent to conformal measure on the Julia set. This holds for particular Fibonacci-like and Feigenbaum combinatorial types when $\ell$ sufficiently large and also for a class of `long-branched' maps of any critical order.Comment: Typos corrected, minor changes, principally to Section

Costs and benefits of adapting to climate change at six meters below sea level

Climate change increases the vulnerability of low-lying coastal areas. Careful spatial planning can reduce this vulnerability. An assessment framework aimed at reducing vulnerability to climate change enables decision-makers to make better informed decisions about investments in adaptation to climate change through spatial planning. This paper presents and evaluates an approach to assess adaptation options, with the use of cost-benefit analysis

Top-Down Composition of Software Architectures

This paper discusses an approach for top-down composition of software architectures. First, an architecture is derived that addresses functional requirements only. This architecture contains a number of variability points which are next filled in to address quality concerns. The quality requirements and associated architectural solution fragments are captured in a so-called Feature-Solution (FS) graph. The solution fragments captured in this graph are used to iteratively compose an architecture. Our versatile composition technique allows for pre- and post-refinements, and refinements that involve multiple variability points. In addition, the usage of the FS graph supports Aspect-Oriented Programming (AOP) at the architecture level

The Dolgopyat inequality in bounded variation for non-Markov maps

This is the author accepted manuscript. The final version is available from World Scientific via the DOI in this record.Let F be a (non-Markov) countably piecewise expanding interval map satisfying certain regularity conditions, and Ł˜Ł̃ the corresponding transfer operator. We prove the Dolgopyat inequality for the twisted operator Ł˜s(v)=Ł˜s(esφv)Ł̃s(v)=Ł̃s(esφv) acting on the space BV of functions of bounded variation, where φφ is a piecewise C1C1 roof function.We are also grateful for the support the Erwin Schrodinger Institute in Vienna, where this paper was completed