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Register a new userIntermediate dimensions of infinitely generated attractors
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We study the dimension theory of limit sets of iterated function systems consisting of a countably infinite number of contractions. Our primary focus is on the intermediate dimensions: a family of dimensions depending on a parameter $theta in [0,1]$ which interpolate between the Hausdorff and box dimensions. Our main results are in the case when all the contractions are conformal. Under a natural separation condition we prove that the intermediate dimensions of the limit set are the maximum of the Hausdorff dimension of the limit set and the intermediate dimensions of the set of fixed points of the contractions. This builds on work of Mauldin and Urbanski concerning the Hausdorff and upper box dimension. We give several (often counter-intuitive) applications of our work to dimensions of projections, fractional Brownian images, and general Holder images. These applications apply to well-studied examples such as sets of numbers which have real or complex continued fraction expansions with restricted entries. We also obtain several results without assuming conformality or any separation conditions. We prove general upper bounds for the Hausdorff, box and intermediate dimensions of infinitely generated attractors in terms of a topological pressure function. We also show that the limit set of a generic infinite iterated function system has box and intermediate dimensions equal to the ambient spatial dimension, where generic can refer to any one of (i) full measure; (ii) prevalent; or (iii) comeagre.
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Given a non-empty bounded subset of hyperbolic space and a Kleinian group acting on that space, the orbital set is the orbit of the given set under the action of the group. We may view orbital sets as bounded (often fractal) subsets of Euclidean space. We prove that the upper box dimension of an orbital set is given by the maximum of three quantities: the upper box dimension of the given set; the Poincare exponent of the Kleinian group; and the upper box dimension of the limit set of the Kleinian group. Since we do not make any assumptions about the Kleinian group, none of the terms in the maximum can be removed in general. We show by constructing an explicit example that the (hyperbolic) boundedness assumption on $C$ cannot be removed in general.
We study equilibrium measures (Kaenmaki measures) supported on self-affine sets generated by a finite collection of diagonal and anti-diagonal matrices acting on the plane and satisfying the strong separation property. Our main result is that such measures are exact dimensional and the dimension satisfies the Ledrappier-Young formula, which gives an explicit expression for the dimension in terms of the entropy and Lyapunov exponents as well as the dimension of the important coordinate projection of the measure. In particular, we do this by showing that the Kaenmaki measure is equal to the sum of (the pushforwards) of two Gibbs measures on an associated subshift of finite type.
It is observed that the conjugacy growth series of the infinite fini-tary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed, for other generating sets, for restricted permutational wreath products of finite groups by the finitary symmetric group, and for alternating groups. Similar methods are used to compute usual growth polynomials and conjugacy growth polynomials for finite symmetric groups and alternating groups, with respect to various generating sets of transpositions. Computations suggest a class of finite graphs, that we call partition-complete, which generalizes the class of semi-hamiltonian graphs, and which is of independent interest. The coefficients of a series related to the finitary alternating group satisfy congruence relations analogous to Ramanujan congruences for the partition function. They follow from partly conjectural generalized Ramanujan congruences, as we call them, for which we give numerical evidence in Appendix C.
Global random attractors and random point attractors for random dynamical systems have been studied for several decades. Here we introduce two intermediate concepts: $Delta$-attractors are characterized by attracting all deterministic compact sets of Hausdorff dimension at most $Delta$, where $Delta$ is a non-negative number, while cc-attractors attract all countable compact sets. We provide two examples showing that a given random dynamical system may have various different $Delta$-attractors for different values of $Delta$. It seems that both concepts are new even in the context of deterministic dynamical systems.
We prove the existence of Veech groups having a critical exponent strictly greater than any elementary Fuchsian group (i.e. $>frac{1}{2}$) but strictly smaller than any lattice (i.e. $<1$). More precisely, every affine covering of a primitive L-shaped Veech surface $X$ ramified over the singularity and a non-periodic connection point $Pin X$ has such a Veech group. Hubert and Schmidt showed that these Veech groups are infinitely generated and of the first kind. We use a result of Roblin and Tapie which connects the critical exponent of the Veech group of the covering with the Cheeger constant of the Schreier graph of $mathrm{SL}(X)/mathrm{Stab}_{mathrm{SL}(X)}(P)$. The main task is to show that the Cheeger constant is strictly positive, i.e. the graph is non-amenable. In this context, we introduce a measure of the complexity of connection points that helps to simplify the graph to a forest for which non-amenability can be seen easily.
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