Tomasz Nowicki

Spectral Theory, Zeta Functions and the Distribution of Periodic Points for Collet-Eckmann Maps

AbstractWe study unimodal interval mapsT with negative Schwarzian derivative satisfying the Collet-Eckmann condition |DTn(Tc)|≧Kλcn for some constantsK>0 and λc>1 (c is the critical point ofT). We prove exponential mixing properties of the unique invariant probability density ofT, describe the long term behaviour of typical (in the sense of Lebesgue measure) trajectories by Central Limit and Large Deviations Theorems for partial sum processes of the formn

Invariant measures exist under a summability condition for unimodal maps

S_n = Sigma _{i = 0}^{n - 1} f(T^i x)

Symmetric S-unimodal mappings and positive Liapunov exponents

n, and study the distribution of “typical” periodic orbits, also in the sense of a Central Limit Theorem and a Large Deviations Theorem.This is achieved by proving quasicompactness of the Perron Frobenius operator and of similar transfer operators for the Markov extension ofT and relating the isolated eigenvalues of these operators to the poles of the corresponding Ruelle zeta functions.

Some phase transitions in coupled map lattices

In this paper we shall show that there exists a polynomial unimodal map f: 0; 1] ! 0; 1] with so-called Fibonacci dynamics which is non-renormalizable and in particular, for each x from a residual set, !(x) is equal to an interval; (here !(x) is deened to be the set of accumulation points of the sequence x; f(x); f 2 (x); : : :); for which the closure of the forward orbit of the critical point c, i.e., !(c), is a Cantor set and for which !(x) = !(c) for Lebesgue almost all x. So the topological and the metric attractor of such a map do not coincide. This gives the answer to a question posed by Milnor Mil] in dimension one.

Absolutely continuous invariant measures forC2 unimodal maps satisfying the Collet-Eckmann conditions

SummaryFor unimodal maps with negative Schwarzian derivative a sufficient condition for the existence of an invariant measure, absolutely continuous with respect to Lebesgue measure, is given. Namely the derivatives of the iterations of the map in the (unique) critical value must be so large that the sum of (some root of) the inverses is finite.

Infinite clusters and critical values in two-dimensional circle percolation

A positive Liapunov exponent for the critical value of an S-unimodal mapping implies a positive Liapunov exponent of the backward orbit of the critical point, uniform hyperbolic structure on the set of periodic points and an exponential diminution of the length of the intervals of monotonicity. This is the proof of the Collet-Eckmann conjecture from 1981 in the general case.

Hyperbolicity properties of ² multi-modal Collet-Eckmann maps without Schwarzian derivative assumptions

Symmetric S-unimodal functions with positive Liapunov exponent of the critical value have an invariant measure absolutely continuous with respect to Lebesgue measure.

Statistical and deterministic methods for reverse engineering biological pathways

We study finite coupled map lattices of size d ⩾ 2 with individual maps τ: [0, 1] → [0, 1] and constant diffuse coupling. For τ (x) = 2x mod 1 we give sufficient conditions that the coupled system has a continuum of ergodic components. In the case d = 2 we determine the number of ergodic components for all coupling strengths. If τ is a mixing tent map close to the transition from mixing to a periodic interval with period 2, the uncoupled system is mixing, whereas numerical studies suggest that coupling with a suitable strength breaks up the phase space into domains which are interchanged with period 2. In case d = 2 we prove this rigorously.