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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.
The topological entropy of a nonautonomous dynamical system given by a sequence of compact metric spaces and a sequence of continuous maps , , is defined. If all the spaces are compact real intervals and all the maps are piecewise monotone then, under some additional assumptions, a formula for the entropy of the system is obtained in terms of the number of pieces of monotonicity of . As an application we construct a large class of smooth triangular maps of the square of type and positive...
Let X be an uncountable compact metrizable space of topological dimension zero. Given any a ∈[0,∞] there is a homeomorphism on X whose topological entropy is a.
For a continuous map f preserving orbits of an aperiodic -action on a compact space, its displacement function assigns to x the “time” it takes to move x to f(x). We show that this function is continuous if the action is minimal. In particular, f is homotopic to the identity along the orbits of the action.
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