Speciality:
01.01.02 (Differential equations, dynamical systems, and optimal control)
Birth date:
27.02.1952
E-mail: ,
Keywords: dynamical systems; differential and topological dynamics of discrete dynamical systems in low dimensions; one-dimensional dynamics; chaotic dynamics.
Subject:
The problem of the coexistence of periods of periodic points of continuous maps of the circle was solved. Interdependence of arithmetic correlations between periods of periodic points with the degree of a continuous map of the circle is established. Criteria of the existence of homoclinic points of continuous endomorphisms of the circle and criteria of the disguishing of continuous endomorphisms of the circle with complicated dynamics (in the sense of A. N. Sharkovsky) are proved. The new concept of the investigation of skew products of interval maps based on the use of new set-valued functions (the $\Omega$-function and the $Bi$-function) of a skew product of interval maps was proposed. In the frames of this concept the dual nature of skew products of interval maps was explaned (it was established why some skew products of interval maps inherit the properties of interval maps, and others have the new properties which are not observed in interval maps). The criterion of the $\Omega$-stability of a skew product of interval maps in the space of $C^1$-smooth skew products of interval maps was proved. Nongenericity of $\Omega$-stability in the space of $C^1$-smooth skew products of interval maps was proved. The problem of the description of dynamics of the "most simple" continuous maps of dendrites with a closed set of brunch points of a finite order was formulated. A number of papers (joint with E. N. Makhrova) were devoted to the investigation of dynamics of monotone and piecewise monotone maps of dendrites with a closed set of periodic points. The possibility of the existence of piecewise monotone maps with fixed points and zero topological entropy possessing of the wandering homoclinic points; nonwandering, but not $\omega$-limit homoclinic points; $\omega$-limit homoclinic points on dendrites with a closed set of brunch points was determined.
Main publications:
L. S. Efremova, “Dynamics of skew products of interval maps”, Russian Math. Surveys, 72:1 (2017), 101–178
L. S. Efremova, E. N. Makhrova, “One-dimensional dynamical systems”, Russian Math. Surveys, 76:5 (2021), 821–881.
