Neil Dobbs

Renormalisation-Induced Phase Transitions for Unimodal Maps

The thermodynamical formalism is studied for renormalisable maps of the interval and the natural potential −t log | Df |. Multiple and indeed infinitely many phase transitions at positive t can occur for some quadratic maps. All unimodal quadratic maps with positive topological entropy exhibit a phase transition in the negative spectrum.

Measures with positive Lyapunov exponent and conformal measures in rational dynamics

Ergodic properties of rational maps are studied, generalising the work of F. Ledrappier. A new construction allows for simpler proofs of stronger results. Very general conformal measures are considered. Equivalent conditions are given for an ergodic invariant probability measure with positive Lyapunov exponent to be absolutely continuous with respect to a general conformal measure. If they hold, we can construct an induced expanding Markov map with integrable return time which generates the invariant measure.

On cusps and flat tops

We develop non-invertible Pesin theory for a new class of maps called cusp maps. These maps may have unbounded derivative, but nevertheless verify a property analogous to

Pesin theory and equilibrium measures on the interval

C^{1+\epsilon}

Lattice-free sets, multi-branch split disjunctions, and mixed-integer programming

. We do not require the critical points to verify a non-flatness condition, so the results are applicable to

Hyperbolic dimension for interval maps

C^{1+\epsilon}

Perturbing Misiurewicz Parameters in the Exponential Family

maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.

Nice sets and invariant densities in complex dynamics

We use Pesin theory to study possible equilibrium measures for piecewise monotone maps of the interval. The maps may have unbounded derivative.

Quasistatic dynamical systems

In this paper we study the relationship between valid inequalities for mixed-integer sets, lattice-free sets associated with these inequalities and the multi-branch split cuts introduced by Li and Richard (Discret Optim 5:724–734, 2008). By analyzing