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M. Malkin

Gagarin Pr. 23, Nizhny Novgorod, 603950 Russia
Department of Mathematics and Mechanics, Nizhni Novgorod State University


Li M., Malkin M. I.
Approximation of entropy on hyperbolic sets for one-dimensional maps and their multidimensional perturbations
2010, vol. 15, no. 2-3, pp.  210-221
We consider piecewise monotone (not necessarily, strictly) piecewise $C^2$ maps on the interval with positive topological entropy. For such a map $f$ we prove that its topological entropy $h_{top}(f)$ can be approximated (with any required accuracy) by restriction on a compact strictly $f$-invariant hyperbolic set disjoint from some neighborhood of prescribed set consisting of periodic attractors, nonhyperbolic intervals and endpoints of monotonicity intervals. By using this result we are able to generalize main theorem from [1] on chaotic behavior of multidimensional perturbations of solutions for difference equations which depend on two variables at nonperturbed value of parameter.
Keywords: chaotic dynamics, difference equations, one-dimensional maps, topological entropy, hyperbolic orbits
Citation: Li M., Malkin M. I.,  Approximation of entropy on hyperbolic sets for one-dimensional maps and their multidimensional perturbations, Regular and Chaotic Dynamics, 2010, vol. 15, no. 2-3, pp. 210-221
Du B., Li M., Malkin M. I.
Topological horseshoes for Arneodo–Coullet–Tresser maps
2006, vol. 11, no. 2, pp.  181-190
In this paper, we study the family of Arneodo–Coullet–Tresser maps $F(x,y,z)=(ax-b(y-z)$, $bx+a(y-z)$, $cx-dxk+e z)$ where $a$, $b$, $c$, $d$, $e$ are real parameters with $bd \ne 0$ and $k>1$ is an integer. We find regions of parameters near anti-integrable limits and near singularities for which there exist hyperbolic invariant sets such that the restriction of $F$ to these sets is conjugate to the full shift on two or three symbols.
Keywords: topological horseshoe, full shift, polynomial maps, generalized Hénon maps, nonwandering set, inverse limit, topological entropy
Citation: Du B., Li M., Malkin M. I.,  Topological horseshoes for Arneodo–Coullet–Tresser maps , Regular and Chaotic Dynamics, 2006, vol. 11, no. 2, pp. 181-190
DOI: 10.1070/RD2006v011n02ABEH000344

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