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Suppose has a 2-dimensional expanding subspace , satisfies a regularity condition, called “good star”, and has , where is an oriented compound of . A morphism of the free group on is called a non-abelianization of if it has structure matrix . We show that there is a tiling substitution whose “boundary substitution” is a non-abelianization of . Such a tiling substitution leads to a self-affine tiling of with as its expansion. In the last section we find conditions on so...
Deterministic and stochastic approach to modeling common trends has been applied to time series of horizontal coordinates of the permanent GPS station Modra – Piesky (recorded weekly during the period of 4 years).
Phenotypic evolution of two-element populations with proportional selection and normally distributed mutation is considered. Trajectories of the expected location of the population in the space of population states are investigated. The expected location of the population generates a discrete dynamical system. The study of its fixed points, their stability and time to convergence is presented. Fixed points are located in the vicinity of optima and saddles. For large values of the standard deviation...
For invertible transformations we introduce various notions of topological entropy. For compact invariant sets these notions are all the same and equal the usual topological entropy. We show that for non-invariant sets these notions are different. They can be used to detect the direction in time in which the system evolves to highest complexity.
We shed some light on the inter-connections between different characterizations leading to the classical Meixner family. This allows us to give free analogs of both Sheffer's and Al-Salam and Chihara's characterizations in the classical case by the use of the free derivative operator. The paper closes with a discussion of the q-deformed case, |q| < 1.
This article presents a theorem about the topological conjugacy of a gradient dynamical system with a constant time step and the cascade generated by its Euler method. It is shown that on the two-dimensional sphere S² the gradient dynamical flow is, under some natural assumptions, correctly reproduced by the Euler method for a sufficiently small time step. This means that the time-map of the induced dynamical system is globally topologically conjugate to the discrete dynamical system obtained via...
Let ℝ be the real line and let Homeo₊(ℝ) be the orientation preserving homeomorphism group of ℝ. Then a subgroup G of Homeo₊(ℝ) is called tightly transitive if there is some point x ∈ X such that the orbit Gx is dense in X and no subgroups H of G with |G:H| = ∞ have this property. In this paper, for each integer n > 1, we determine all the topological conjugation classes of tightly transitive subgroups G of Homeo₊(ℝ) which are isomorphic to ℤⁿ and have countably many nontransitive points.
We investigate the connections between Ramsey properties of Fraïssé classes and the universal minimal flow of the automorphism group of their Fraïssé limits. As an extension of a result of Kechris, Pestov and Todorcevic (2005) we show that if the class has finite Ramsey degree for embeddings, then this degree equals the size of . We give a partial answer to a question of Angel, Kechris and Lyons (2014) showing that if is a relational Ramsey class and is amenable, then admits a unique invariant...
On the background of a brief survey panorama of results on the topic in the title, one new theorem is presented concerning a positive topological entropy (i.e. topological chaos) for the impulsive differential equations on the Cartesian product of compact intervals, which is positively invariant under the composition of the associated Poincaré translation operator with a multivalued upper semicontinuous impulsive mapping.
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