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We introduce some notions that are useful for studying the behavior of periodic orbits of maps of one-dimensional spaces. We use them to characterize the set of periods of periodic orbits for continuous maps of Y = (z e C: z3 e [0,1]} into itself having zero as a fixed point. We also obtain new proofs of some known results for maps of an interval into… (More)
For a family of dynamical systems we define sensitive dependence on parameters in a way resembling Guckenheimer's definition of sensitive dependence on initial conditions. While sensitive dependence on initial conditions tells us that if we know the initial condition only approximately then we cannot make determin-istic predictions, sensitive dependence on… (More)
Rotation Theory has its roots in the theory of rotation numbers for circle homeomorphisms, developed by Poincaré. It is particularly useful for the study and classification of periodic orbits of dynamical systems. It deals with ergodic averages and their limits, not only for almost all points, like in Ergodic Theory, but for all points. We present the… (More)
We prove that if an interval map of positive entropy is perturbed to a compact multidimensional map then the topological entropy cannot drop down a lot if the perturbation is small.
We consider a problem in Mathematical Biology that leads to a question in Graph Theory, which can be solved using an old but not widely known upper estimate of the spectral radius of a nonnegative matrix. We provide a new proof of this estimate.
Combinatorial Dynamics has its roots in Sharkovsky's Theorem. This beautiful theorem describes the possible sets of periods of all cycles of a continuous map of an interval (or the real line) into itself. Here by a cycle I mean a periodic orbit, and by a period its minimal period. Consider the following Sharkovsky's ordering < s of the set N of natural… (More)