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Dynamical Systems Generated by Linear Maps

Ćemal B. Dolićanin and Anatolij B. Antonevich
Publication Date: 
Number of Pages: 
[Reviewed by
Russell Jay Hendel
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This is an excellent book that can be used for a third or fourth year graduate course in dynamical systems or a related field. The prerequisites seem to be linear algebra, discrete mathematics, and real analysis (including vector spaces and measure theory). The book has no exercises so I would not recommend using this text for an undergraduate course or for a first/second year graduate course. However, there is a good solid bibliography and pertinent references (both books and journal articles) making the book ideal for a seminar type presentation where students participate in the course with presentation of papers.

The book deals with dynamical systems, generated by linear mappings of finite dimensional spaces and their applications, say \(A:\mathbb{C}^m \to \mathbb{C}^m\), where \(A\) is a linear mapping and \(\mathbb{C}^m\) is the standard m-dimensional vector space over the complex numbers. The book studies three broad classes of problems:

  • Trajectory (and limit) of an arbitrary vector: \(\lim A^n \mathbf{v}\).
  • Trajectory (and limit) of an arbitrary direction: \(\lim A^n \mathbf{v}/\| A^n\mathbf{v}\|\).
  • Trajectory (and limit) of an arbitrary d-dimensional subspace: \(\lim A^n V\).

Appealing features of the book include the following:

  • Multiple Methods: The problems are approached through multiple methods, something expected in a graduate course. More specifically, both algebraic and geometric approaches are presented (using embeddings of the Grassman manifold, the space of d-dimensional subspaces of \(\mathbb{C}^m\), into the projective space generated by the exterior power of the initial vector space, and linear combinations of vectors \(A^n\mathbf{v}\) that have non-zero limits, respectively.)
  • Applications: The book has many applications including i) spectral properties of weighted shift operators; ii) an explicit description of properties and spectrum of \(B-rI\) (for spectral values r); and a new derivation of principal facts on exterior powers of vector spaces and Grassman manifolds.
  • New: Many of the results or methods of obtaining them are new. For example, previously there has been no complete investigation of the trajectory of \(A^nV\).

Russell Jay Hendel, [email protected], holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, applications of technology to education, problem writing, actuarial science and the interaction between mathematics, art and poetry.