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Journal of Modern Dynamics

2007 , Volume 1 , Issue 3

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Self-similar groups, operator algebras and Schur complement
Rostislav Grigorchuk and Volodymyr Nekrashevych
2007, 1(3): 323-370 doi: 10.3934/jmd.2007.1.323 +[Abstract](366) +[PDF](522.4KB)
Abstract:
In the first part of the article we introduce $C$*-algebras associated to self-similar groups and study their properties and relations to known algebras. The algebras are constructed as subalgebras of the Cuntz-Pimsner algebra (and its homomorphic images) associated with the self-similarity of the group. We study such properties as nuclearity, simplicity and Morita equivalence with algebras related to solenoids.
    The second part deals with Schur complement transformations of elements of self-similar algebras. We study the properties of such transformations and apply them to the spectral problem for Markov type elements in self-similar $C$*-algebras. This is related to the spectral problem of the discrete Laplace operator on groups and graphs. Application of the Schur complement method in many situations reduces the spectral problem to study of invariant sets (very often of the type of a "strange attractor'') of a multidimensional rational transformation. A number of illustrating examples is provided. Finally, we observe a relation between Schur complement transformations and Bartholdi-Kaimanovich-Virag transformations of random walks on self-similar groups.
Unbounded orbits for outer billiards I
Richard Evan Schwartz
2007, 1(3): 371-424 doi: 10.3934/jmd.2007.1.371 +[Abstract](302) +[PDF](501.4KB)
Abstract:
The question of B.H. Neumann, which dates back to the 1950s, asks if there exists an outer billiards system with an unbounded orbit. We prove that outer billiards for the Penrose kite, the convex quadrilateral from the Penrose tiling, has an unbounded orbit. We also analyze some finer properties of the orbit structure, and in particular produce an uncountable family of unbounded orbits. Our methods relate outer billiards on the Penrose kite to polygon exchange maps, arithmetic dynamics, and self-similar tilings.
Global rigidity of certain Abelian actions by toral automorphisms
Federico Rodriguez Hertz
2007, 1(3): 425-442 doi: 10.3934/jmd.2007.1.425 +[Abstract](308) +[PDF](220.9KB)
Abstract:
We prove global rigidity results for some linear abelian actions on tori. The type of actions we deal with includes in particular maximal rank semisimple actions on $\mathbb T^N$.
Fixed points of Abelian actions
John Franks, Michael Handel and Kamlesh Parwani
2007, 1(3): 443-464 doi: 10.3934/jmd.2007.1.443 +[Abstract](367) +[PDF](260.8KB)
Abstract:
We prove that if $\mathfrak{F}$ is an abelian group of $C^1$ diffeomorphisms isotopic to the identity of a closed surface $S$ of genus at least two, then there is a common fixed point for all elements of $\mathfrak{F}$. If $\mathfrak{F}$ is an abelian group of $C^1$ diffeomorphisms (not necessarily isotopic to the identity) of a closed surface $S$ of genus at least two, then $\mathfrak{F}$ has a subgroup of finite index all of whose elements share a common fixed point.
Quasi-states and the Poisson bracket on surfaces
Frol Zapolsky
2007, 1(3): 465-475 doi: 10.3934/jmd.2007.1.465 +[Abstract](309) +[PDF](145.7KB)
Abstract:
We present a convexity-type result concerning simple quasi-states on closed manifolds. As a corollary, an inequality emerges which relates the Poisson bracket to the measure of non-additivity of a simple quasi-state on a closed surface equipped with an area form. In addition, we prove that the uniform norm of the Poisson bracket of two functions on a surface is stable from below under $C^0$-perturbations.
Renormalization and central limit theorem for critical dynamical systems with weak external noise
Oliver Díaz-Espinosa and Rafael de la Llave
2007, 1(3): 477-543 doi: 10.3934/jmd.2007.1.477 +[Abstract](338) +[PDF](707.7KB)
Abstract:
We study the effect of weak noise on critical one-dimensional maps; that is, maps with a renormalization theory.
    We establish a one-dimensional central limit theorem for weak noise and obtain Berry--Esseen estimates for the rate of this convergence.
    We analyze in detail maps at the accumulation of period doubling and critical circle maps with golden mean rotation number. Using renormalization group methods, we derive scaling relations for several features of the effective noise after long periods. We use these scaling relations to show that the central limit theorem for weak noise holds in both examples.
    We note that, for the results presented here, it is essential that the maps have parabolic behavior. They are false for hyperbolic orbits.

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