To acquire a solid knowledge of some advanced problems in mathematical physics
Curriculum
teacher profile teaching materials
Partitions. Frequencies of visit and symbolic motions.
Quasi-periodic motions and ergodic properties. Birkhoff's theorem.
Ergodic and mixing systems. Potentials and energies.
Gibbs measures: existence and uniqueness. Gibbs measures on Z_+.
Variational properties of Gibbs measures. Expansive maps on the interval.
Part II. Hyperbolic systems.
Hyperbolic systems. Anosov systems. Arnold's cat. Markov pavements.
Symbolic dynamics for Anosov systems. Codes for the volume measure and its
restriction to Z_+. Stable and unstable foliations. SRM measure.
Structural stability and perturbations of Arnold's cat. Perturbation series and diagrammatic techniques
for the conjugation and for the expansion and contraction coefficients. Coupled Arnold's cats.
Part III. Synchronization in chaotic systems.
Partially hyperbolic systems in the presence of dissipative interactions.
Construction of a local attractor, conjugation to the linearized system
and computation of the Lyapunov exponents. Study of the correlations.
Aspects of the ergodic, qualitative and statistical theory of motion
Springer, Berlin, 2004.
Programme
Part I. Elements of ergodic theory.Partitions. Frequencies of visit and symbolic motions.
Quasi-periodic motions and ergodic properties. Birkhoff's theorem.
Ergodic and mixing systems. Potentials and energies.
Gibbs measures: existence and uniqueness. Gibbs measures on Z_+.
Variational properties of Gibbs measures. Expansive maps on the interval.
Part II. Hyperbolic systems.
Hyperbolic systems. Anosov systems. Arnold's cat. Markov pavements.
Symbolic dynamics for Anosov systems. Codes for the volume measure and its
restriction to Z_+. Stable and unstable foliations. SRM measure.
Structural stability and perturbations of Arnold's cat. Perturbation series and diagrammatic techniques
for the conjugation and for the expansion and contraction coefficients. Coupled Arnold's cats.
Part III. Synchronization in chaotic systems.
Partially hyperbolic systems in the presence of dissipative interactions.
Construction of a local attractor, conjugation to the linearized system
and computation of the Lyapunov exponents. Study of the correlations.
Core Documentation
G. Gallavotti, F. Bonetto, G. GentileAspects of the ergodic, qualitative and statistical theory of motion
Springer, Berlin, 2004.
Type of delivery of the course
Lectures and exercises.Attendance
Attendance is not compulsory but is strongly recommended.Type of evaluation
The exam consists of an oral interview, in which the student will have to discuss the topics treated in class, with reference to the texts that have been used. teacher profile teaching materials
Partitions. Frequencies of visit and symbolic motions.
Quasi-periodic motions and ergodic properties. Birkhoff's theorem.
Ergodic and mixing systems. Potentials and energies.
Gibbs measures: existence and uniqueness. Gibbs measures on Z_+.
Variational properties of Gibbs measures. Expansive maps on the interval.
Part II. Hyperbolic systems.
Hyperbolic systems. Anosov systems. Arnold's cat. Markov pavements.
Symbolic dynamics for Anosov systems. Codes for the volume measure and its
restriction to Z_+. Stable and unstable foliations. SRM measure.
Structural stability and perturbations of Arnold's cat. Perturbation series and diagrammatic techniques
for the conjugation and for the expansion and contraction coefficients. Coupled Arnold's cats.
Part III. Synchronization in chaotic systems.
Partially hyperbolic systems in the presence of dissipative interactions.
Construction of a local attractor, conjugation to the linearized system
and computation of the Lyapunov exponents. Study of the correlations.
Aspects of the ergodic, qualitative and statistical theory of motion
Springer, Berlin, 2004.
Programme
Part I. Elements of ergodic theory.Partitions. Frequencies of visit and symbolic motions.
Quasi-periodic motions and ergodic properties. Birkhoff's theorem.
Ergodic and mixing systems. Potentials and energies.
Gibbs measures: existence and uniqueness. Gibbs measures on Z_+.
Variational properties of Gibbs measures. Expansive maps on the interval.
Part II. Hyperbolic systems.
Hyperbolic systems. Anosov systems. Arnold's cat. Markov pavements.
Symbolic dynamics for Anosov systems. Codes for the volume measure and its
restriction to Z_+. Stable and unstable foliations. SRM measure.
Structural stability and perturbations of Arnold's cat. Perturbation series and diagrammatic techniques
for the conjugation and for the expansion and contraction coefficients. Coupled Arnold's cats.
Part III. Synchronization in chaotic systems.
Partially hyperbolic systems in the presence of dissipative interactions.
Construction of a local attractor, conjugation to the linearized system
and computation of the Lyapunov exponents. Study of the correlations.
Core Documentation
G. Gallavotti, F. Bonetto, G. GentileAspects of the ergodic, qualitative and statistical theory of motion
Springer, Berlin, 2004.
Type of delivery of the course
Lectures and exercises.Attendance
Attendance is not compulsory but is strongly recommended.Type of evaluation
The exam consists of an oral interview, in which the student will have to discuss the topics treated in class, with reference to the texts that have been used. teacher profile teaching materials
Partitions. Frequencies of visit and symbolic motions.
Quasi-periodic motions and ergodic properties. Birkhoff's theorem.
Ergodic and mixing systems. Potentials and energies.
Gibbs measures: existence and uniqueness. Gibbs measures on Z_+.
Variational properties of Gibbs measures. Expansive maps on the interval.
Part II. Hyperbolic systems.
Hyperbolic systems. Anosov systems. Arnold's cat. Markov pavements.
Symbolic dynamics for Anosov systems. Codes for the volume measure and its
restriction to Z_+. Stable and unstable foliations. SRM measure.
Structural stability and perturbations of Arnold's cat. Perturbation series and diagrammatic techniques
for the conjugation and for the expansion and contraction coefficients. Coupled Arnold's cats.
Part III. Synchronization in chaotic systems.
Partially hyperbolic systems in the presence of dissipative interactions.
Construction of a local attractor, conjugation to the linearized system
and computation of the Lyapunov exponents. Study of the correlations.
Aspects of the ergodic, qualitative and statistical theory of motion
Springer, Berlin, 2004.
Mutuazione: 20410878 FM440 - FISICA MATEMATICA in Matematica LM-40 GENTILE GUIDO, CORSI LIVIA
Programme
Part I. Elements of ergodic theory.Partitions. Frequencies of visit and symbolic motions.
Quasi-periodic motions and ergodic properties. Birkhoff's theorem.
Ergodic and mixing systems. Potentials and energies.
Gibbs measures: existence and uniqueness. Gibbs measures on Z_+.
Variational properties of Gibbs measures. Expansive maps on the interval.
Part II. Hyperbolic systems.
Hyperbolic systems. Anosov systems. Arnold's cat. Markov pavements.
Symbolic dynamics for Anosov systems. Codes for the volume measure and its
restriction to Z_+. Stable and unstable foliations. SRM measure.
Structural stability and perturbations of Arnold's cat. Perturbation series and diagrammatic techniques
for the conjugation and for the expansion and contraction coefficients. Coupled Arnold's cats.
Part III. Synchronization in chaotic systems.
Partially hyperbolic systems in the presence of dissipative interactions.
Construction of a local attractor, conjugation to the linearized system
and computation of the Lyapunov exponents. Study of the correlations.
Core Documentation
G. Gallavotti, F. Bonetto, G. GentileAspects of the ergodic, qualitative and statistical theory of motion
Springer, Berlin, 2004.
Type of delivery of the course
Lectures and exercises.Attendance
Attendance is not compulsory but is strongly recommended.Type of evaluation
The exam consists of an oral interview, in which the student will have to discuss the topics treated in class, with reference to the texts that have been used. teacher profile teaching materials
Partitions. Frequencies of visit and symbolic motions.
Quasi-periodic motions and ergodic properties. Birkhoff's theorem.
Ergodic and mixing systems. Potentials and energies.
Gibbs measures: existence and uniqueness. Gibbs measures on Z_+.
Variational properties of Gibbs measures. Expansive maps on the interval.
Part II. Hyperbolic systems.
Hyperbolic systems. Anosov systems. Arnold's cat. Markov pavements.
Symbolic dynamics for Anosov systems. Codes for the volume measure and its
restriction to Z_+. Stable and unstable foliations. SRM measure.
Structural stability and perturbations of Arnold's cat. Perturbation series and diagrammatic techniques
for the conjugation and for the expansion and contraction coefficients. Coupled Arnold's cats.
Part III. Synchronization in chaotic systems.
Partially hyperbolic systems in the presence of dissipative interactions.
Construction of a local attractor, conjugation to the linearized system
and computation of the Lyapunov exponents. Study of the correlations.
Aspects of the ergodic, qualitative and statistical theory of motion
Springer, Berlin, 2004.
Mutuazione: 20410878 FM440 - FISICA MATEMATICA in Matematica LM-40 GENTILE GUIDO, CORSI LIVIA
Programme
Part I. Elements of ergodic theory.Partitions. Frequencies of visit and symbolic motions.
Quasi-periodic motions and ergodic properties. Birkhoff's theorem.
Ergodic and mixing systems. Potentials and energies.
Gibbs measures: existence and uniqueness. Gibbs measures on Z_+.
Variational properties of Gibbs measures. Expansive maps on the interval.
Part II. Hyperbolic systems.
Hyperbolic systems. Anosov systems. Arnold's cat. Markov pavements.
Symbolic dynamics for Anosov systems. Codes for the volume measure and its
restriction to Z_+. Stable and unstable foliations. SRM measure.
Structural stability and perturbations of Arnold's cat. Perturbation series and diagrammatic techniques
for the conjugation and for the expansion and contraction coefficients. Coupled Arnold's cats.
Part III. Synchronization in chaotic systems.
Partially hyperbolic systems in the presence of dissipative interactions.
Construction of a local attractor, conjugation to the linearized system
and computation of the Lyapunov exponents. Study of the correlations.
Core Documentation
G. Gallavotti, F. Bonetto, G. GentileAspects of the ergodic, qualitative and statistical theory of motion
Springer, Berlin, 2004.
Type of delivery of the course
Lectures and exercises.Attendance
Attendance is not compulsory but is strongly recommended.Type of evaluation
The exam consists of an oral interview, in which the student will have to discuss the topics treated in class, with reference to the texts that have been used.