Thermodynamic formalism for transient dynamics on the real line

We develop a new thermodynamic formalism to investigate the transient behaviour of maps on the real line which are skew-periodic Z -extensions of expanding interval maps. Our main focus lies in the dimensional analysis of the recurrent and transient sets as well as in determining the full dimension spectrum with respect to α-escaping sets. Our results provide a one-dimensional model for the phenomenon of a dimension gap occurring for limit sets of Kleinian groups. In particular, we show that a dimension gap occurs if and only if we have non-zero drift and we are able to precisely quantify its width as an application of our new formalism.

On α-Limit Sets in Lorenz Maps

The aim of this paper is to show that α-limit sets in Lorenz maps do not have to be completely invariant. This highlights unexpected dynamical behavior in these maps, showing gaps existing in the literature. Similar result is obtained for unimodal maps on [0,1]. On the basis of provided examples, we also present how the performed study on the structure of α-limit sets is closely connected with the calculation of the topological entropy.

Theoretical Calculation of Wind (Or Water) Turbine: Extending the Betz Limit

The Betz limit sets a theoretical upper limit for the power production by turbines expressed as a maximum power coefficient of 16/27. Betz’s theory is accurate and it is based on the calculation of kinetic energy.

Packing dimensions of basins generated by distributions on a finite alphabet

We consider a space of infinite signals composed of letters from a finite alphabet. Each signal generates a sequence of empirical measures on the alphabet and the limit set corresponding to this sequence. The space of signals is partitioned into narrow basins consisting of signals with identical limit sets for the sequence of empirical measures and for each narrow basin its packing dimension is computed. Furthermore, we compute packing dimensions for two other types of basins defined in terms of limit behaviour of the empirical measures.