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Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Artur Avila, Mikhail Lyubich, Weixiao Shen (2011)

Journal of the European Mathematical Society

We study the parameter space of unicritical polynomials f c : z z d + c . For complex parameters, we prove that for Lebesgue almost every c , the map f c is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every c , the map f c is either hyperbolic, or Collet–Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the “principal nest” of parapuzzle pieces.

Partial hyperbolicity and homoclinic tangencies

Sylvain Crovisier, Martin Sambarino, Dawei Yang (2015)

Journal of the European Mathematical Society

We show that any diffeomorphism of a compact manifold can be C 1 approximated by diffeomorphisms exhibiting a homoclinic tangency or by diffeomorphisms having a partial hyperbolic structure.

Partial variational principle for finitely generated groups of polynomial growth and some foliated spaces

Andrzej Biś (2008)

Colloquium Mathematicae

We generalize the notion of topological pressure to the case of a finitely generated group of continuous maps and introduce group measure entropy. Also, we provide an elementary proof that any finitely generated group of polynomial growth admits a group invariant measure and show that for a group of polynomial growth its measure entropy is less than or equal to its topological entropy. The dynamical properties of groups of polynomial growth are reflected in the dynamics of some foliated spaces.

Partition sensitivity for measurable maps

C. A. Morales (2013)

Mathematica Bohemica

We study countable partitions for measurable maps on measure spaces such that, for every point x , the set of points with the same itinerary as that of x is negligible. We prove in nonatomic probability spaces that every strong generator (Parry, W., Aperiodic transformations and generators, J. London Math. Soc. 43 (1968), 191–194) satisfies this property (but not conversely). In addition, measurable maps carrying partitions with this property are aperiodic and their corresponding spaces are nonatomic....

Path formulation for multiparameter 𝔻 3 -equivariant bifurcation problems

Jacques-Élie Furter, Angela Maria Sitta (2010)

Annales de l’institut Fourier

We implement a singularity theory approach, the path formulation, to classify 𝔻 3 -equivariant bifurcation problems of corank 2, with one or two distinguished parameters, and their perturbations. The bifurcation diagrams are identified with sections over paths in the parameter space of a 𝔻 3 -miniversal unfolding F 0 of their cores. Equivalence between paths is given by diffeomorphisms liftable over the projection from the zero-set of F 0 onto its unfolding parameter space. We apply our results to degenerate...

Pattern Formation of Competing Microorganisms in Sediments

Y. Schmitz, M. Baurmann, B. Engelen, U. Feudel (2010)

Mathematical Modelling of Natural Phenomena

We present a three species model describing the degradation of substrate by two competing populations of microorganisms in a marine sediment. Considering diffusion to be the main transport process, we obtain a reaction diffusion system (RDS) which we study in terms of spontaneous pattern formation. We find that the conditions for patterns to evolve are likely to be fulfilled in the sediment. Additionally, we present simulations that are consistent with experimental data from the literature. We...

Peano type theorem for random fuzzy initial value problem

Marek T. Malinowski (2011)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we consider the random fuzzy differential equations and show their application by an example. Under suitable conditions the Peano type theorem on existence of solutions is proved. For our purposes, a notion of ε-solution is exploited.

Performance and stochastic stability of the adaptive fading extended Kalman filter with the matrix forgetting factor

Cenker Biçer, Levent Özbek, Hasan Erbay (2016)

Open Mathematics

In this paper, the stability of the adaptive fading extended Kalman filter with the matrix forgetting factor when applied to the state estimation problem with noise terms in the non–linear discrete–time stochastic systems has been analysed. The analysis is conducted in a similar manner to the standard extended Kalman filter’s stability analysis based on stochastic framework. The theoretical results show that under certain conditions on the initial estimation error and the noise terms, the estimation...

Period doubling, entropy, and renormalization

Jun Hu, Charles Tresser (1998)

Fundamenta Mathematicae

We show that in any family of stunted sawtooth maps, the set of maps whose set of periods is the set of all powers of 2 has no interior point. Similar techniques then allow us to show that, under mild assumptions, smooth multimodal maps whose set of periods is the set of all powers of 2 are infinitely renormalizable with the diameters of all periodic intervals going to zero as the period goes to infinity.

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