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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.

Periodic billiard orbits in right triangles

Serge Troubetzkoy (2005)

Annales de l’institut Fourier

There is an open set of right triangles such that for each irrational triangle in this set (i) periodic billiards orbits are dense in the phase space, (ii) there is a unique nonsingular perpendicular billiard orbit which is not periodic, and (iii) the perpendicular periodic orbits fill the corresponding invariant surface.

Periodic harmonic functions on lattices and points count in positive characteristic

Mikhail Zaidenberg (2009)

Open Mathematics

This survey deals with pluri-periodic harmonic functions on lattices with values in a field of positive characteristic. We mention, as a motivation, the game “Lights Out” following the work of Sutner [20], Goldwasser- Klostermeyer-Ware [5], Barua-Ramakrishnan-Sarkar [2, 19], Hunzikel-Machiavello-Park [12] e.a.; see also [22, 23] for a more detailed account. Our approach uses harmonic analysis and algebraic geometry over a field of positive characteristic.

Periodic orbits and chain-transitive sets of C1-diffeomorphisms

Sylvain Crovisier (2006)

Publications Mathématiques de l'IHÉS

We prove that the chain-transitive sets of C1-generic diffeomorphisms are approximated in the Hausdorff topology by periodic orbits. This implies that the homoclinic classes are dense among the chain-recurrence classes. This result is a consequence of a global connecting lemma, which allows to build by a C1-perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by C1-generic diffeomorphisms: any pseudo-orbit is approximated in the Hausdorff topology...

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