Self-Similar Groups and Holomorphic Dynamics: Renormalization, Integrability, and Spectrum

Self-similar fractals and arithmetic dynamics

‎The concept of self-similarity on subsets of algebraic varieties‎ ‎is defined by considering algebraic endomorphisms of the variety‎ ‎as `similarity' maps‎. ‎Self-similar fractals are subsets of algebraic varieties‎ ‎which can be written as a finite and disjoint union of‎ ‎`similar' copies‎. ‎Fractals provide a framework in which‎, ‎one can‎ ‎unite some results and conjectures in Diophantine g...

Symbolic Dynamics and Self-similar Groups

Self-similar groups and permutational bimodules are used to study combinatorics and symbolic dynamics of expanding self-coverings. We describe functors between the category of contracting self-similar groups and the category of expanding self-coverings (with appropriate morphisms). These functors transform some questions in dynamical systems to questions in algebra. As examples we show how some...

Kleinian Groups and Holomorphic Dynamics

In this work we explore some relations between iteration theory and theory of Kleinian groups. There is already work done in that direction. In the case of Fuchsian groups there is the result of Bowen and Series [1979] which guarantees the existence of a boundary map, orbit equivalent to the action of the Fuchsian group on the hyperbolic plane. We use this result in Sec. 3. Brooks and Matelski ...

مروری بر کتاب self-similar groups

خودسانی یا فرکتالی بودن پدیده ای مهم در طبیعت است که در بسیاری از وجوه تمدن جدید  انعکاس یافته است. این پدیده علاوه بر هنر، در فیزیک، شیمی علوم پزشکی و علوم کامپیوتر نیز رخ می نماید. خودسانی در مباحث مختلفی از ریاضیات و مدلسازی ریاضی از جمله دستگاههای دینامیکی و آشوب، فرآیندهای تصادفی و فیزیک آماری، توپولوژی و هندسه فرکتالی  نیز ظاهر می شود.

C-algebras and Self-similar Groups