Return to Answer

Misha Lyubich

He graduated from Kharkiv University (Ukraine) in 1980 and now works at SUNY, Stony Brook, as the Director of the Math. Institute. His work belongs to holomorphic dynamics (iteration of holomorphic maps). He made major contributions to all parts of holomorphic dynamics (rational functions, entire functions and holomorphic maps in several dimensions). His work is nicely described in his web page

http://www.math.sunysb.edu/~mlyubich/

One of his principal results is easy to state. Consider the dynamical system $x\mapsto x^2+c$ on the real line. Then for almost every $c\in[−2,1/4]$, the quadratic map $f_c(x)=x^2+c$ is either regular or stochastic.

"Regular" means almost all orbits are attracted to an attracting cycle. "Stochastic" means that there exists an ergodic invariant measure which is absolutely continuous with respect to Lebesgue measure.

This is a complete qualitative description of the nature of chaos in the real quadratic family.

This result actually completes a long line of development in which many people participated.

There is a nice non-technical exposition of this fundamental result in his paper The quadratic family as a qualitatively solvable model of chaos. Notices Amer. Math. Soc. 47 (2000), no. 9, 1042–1052.

Misha Lyubich

He graduated from Kharkiv University (Ukraine) in 1980 and now works at SUNY, Stony Brook, as the Director of the Math. Institute. His work belongs to holomorphic dynamics (iteration of holomorphic maps). He made major contributions to all parts of holomorphic dynamics (rational functions, entire functions and holomorphic maps in several dimensions). His work is nicely described in his web page

http://www.math.sunysb.edu/~mlyubich/

One of his principal results is easy to state. Consider the dynamical system $x\mapsto x^2+c$ on the real line. Then for almost every $c\in[−2,1/4]$, the quadratic map $f_c(x)=x^2+c$ is either regular or stochastic.

"Regular" means almost all orbits are attracted to an attracting cycle. "Stochastic" means that there exists an ergodic invariant measure which is absolutely continuous with respect to Lebesgue measure.

This is a complete qualitative description of the nature of chaos in the real quadratic family.

This result actually completes a long line of development in which many people participated.

There is a nice non-technical exposition of this fundamental result in his paper The quadratic family as a qualitatively solvable model of chaos. Notices Amer. Math. Soc. 47 (2000), no. 9, 1042–1052.

Of his early famous results, I will mention existence and uniqueness of the measure of maximal entropy for rational maps of the Riemann sphere $P^1$, and discovery that the map $z\mapsto e^z$ of the complex plane is not ergodic with respect to the Lebesgue measure.

Misha Lyubich

He graduated from Kharkiv University (Ukraine) in 1980 and now works at SUNY, Stony Brook, as the Director of the Math. Institute. His work belongs to holomorphic dynamics (iteration of holomorphic maps). He made major contributions to all parts of holomorphic dynamics (rational functions, entire functions and holomorphic maps in several dimensions). His work is nicely described in his web page

http://www.math.sunysb.edu/~mlyubich/

One of his principal results is easy to state. For the dynamical system $x\mapsto x^2+c$ on the real line. Then for almost every $c\in[−2,1/4]$, the quadratic map $f_c(x)=x^2+c$ is either regular or stochastic.

"Regular" means almost all orbits are attracted to an attracting cycle. "Stochastic" means that there exists an ergodic invariant measure which is absolutely continuous with respect to Lebesgue measure.

This is a complete qualitative description of the nature of chaos in the real quadratic family.

This result actually completes a long line of development in which many people participated.

There is a nice non-technical exposition of this fundamental result in his paper The quadratic family as a qualitatively solvable model of chaos. Notices Amer. Math. Soc. 47 (2000), no. 9, 1042–1052.

Misha Lyubich

He graduated from Kharkiv University (Ukraine) in 1980 and now works at SUNY, Stony Brook, as the Director of the Math. Institute. His work belongs to holomorphic dynamics (iteration of holomorphic maps). He made major contributions to all parts of holomorphic dynamics (rational functions, entire functions and holomorphic maps in several dimensions). His work is nicely described in his web page

http://www.math.sunysb.edu/~mlyubich/

One of his principal results is easy to state. Consider the dynamical system $x\mapsto x^2+c$ on the real line. Then for almost every $c\in[−2,1/4]$, the quadratic map $f_c(x)=x^2+c$ is either regular or stochastic.

"Regular" means almost all orbits are attracted to an attracting cycle. "Stochastic" means that there exists an ergodic invariant measure which is absolutely continuous with respect to Lebesgue measure.

This is a complete qualitative description of the nature of chaos in the real quadratic family.

This result actually completes a long line of development in which many people participated.

There is a nice non-technical exposition of this fundamental result in his paper The quadratic family as a qualitatively solvable model of chaos. Notices Amer. Math. Soc. 47 (2000), no. 9, 1042–1052.

Misha Lyubich

He graduated from Kharkiv University (Ukraine) in 1980 and now works at SUNY, Stony Brook, as the Director of the Math. Institute. His work belongs to holomorphic dynamics (iteration of holomorphic maps). He made major contributions to all parts of holomorphic dynamics (rational functions, entire functions and holomorphic maps in several dimensions). His work is nicely described in his web page

http://www.math.sunysb.edu/~mlyubich/

One of his principal results is easy to state. For the dynamical system $x\mapsto x^2+c$ on the real line. Then for almost every $c\in[−2,1/4]$, the quadratic map $f_c(x)=x^2+c$ is either regular or stochastic.

"Regular" means almost all orbits are attracted to an attracting cycle. "Stochastic" means that there exists an ergodic invariant measure which is absolutely continuous with respect to Lebesgue measure.

This is a complete qualitative description of the nature of chaos in the real quadratic family.

This result actually completes a long line of development in which many people participated.

There is a nice non-technical exposition of this fundamental result in his paper The quadratic family as a qualitatively solvable model of chaos. Notices Amer. Math. Soc. 47 (2000), no. 9, 1042–1052.

Misha Lyubich

He graduated from Kharkiv University (Ukraine) in 1980 and now works at SUNY, Stony Brook, as the Director of the Math. Institute. His work belongs to holomorphic dynamics (iteration of holomorphic maps). He made major contributions to all parts of holomorphic dynamics (rational functions, entire functions and holomorphic maps in several dimensions). His work is nicely described in his web page

http://www.math.sunysb.edu/~mlyubich/

One of his principal results is easy to state. For the dynamical system $x\mapsto x^2+c$ on the real line. Then for almost every $c\in[−2,1/4]$, the quadratic map $f_c(x)=x^2+c$ is either regular or stochastic.

"Regular" means almost all orbits are attracted to an attracting cycle. "Stochastic" means that there exists an ergodic invariant measure which is absolutely continuous with respect to Lebesgue measure.

This is a complete qualitative description of the nature of chaos in the real quadratic family.

This result actually completes a long line of development in which many people participated.

There is a nice non-technical exposition of this fundamental result in his paper The quadratic family as a qualitatively solvable model of chaos. Notices Amer. Math. Soc. 47 (2000), no. 9, 1042–1052.