N.A. Sidorov. Bijective and General Arithmetic Codings for Pisot Toral Automorphisms. J. Dynam. Control Systems 7 (2001), no. 4, 447--472.
Let T be an algebraic automorphism of \T^m having
the following property: the characteristic polynomial of its
matrix is irreducible over \Q, and a Pisot number
\beta is one of its roots. We define the mapping
\varphi_t acting from the two-sided \beta-compactum
onto \T^m as follows:
\varphi_t (\bar{\varepsilon})=\sum_{k\in{\Z}}\varepsilon_{k}T^{-k} t,
where t is a fundamental homoclinic point for T,
i.e., a point homoclinic to 0 such that the linear span
of its orbit is the whole homoclinic group (provided that such a point
exists). We call such a mapping an arithmetic coding of T. This
paper is aimed to show that under some natural hypothesis on
\beta (which is apparently satisfied for all Pisot units) the
mapping \varphi_t is bijective a.e. with respect to
the Haar measure on the torus. Moreover, we study the case of more
general parameters t, not necessarily fundamental, and
relate the number of preimages of \varphi_t to
certain number-theoretic quantities. We also give several full
criteria for T to admit a bijective arithmetic coding and
consider some examples of arithmetic codings of Cartan actions.
This work continues the study begun in \cite{SV98} for the special
case m=2.
A fundamental result of sub-Riemannian geometry, the ball-box theorem, states that small sub-Riemannian balls look like boxes [-\eps^{w_1},\eps^{w_1}] \times \cdots \times [-\eps^{w_n},\eps^{w_n}] in privileged coordinates. This description is not uniform in general. Thus, it does allow us neither to compute Hausdorff measures and dimensions nor to prove the convergence of certain motion planning algorithms. In this paper, we present a description of the shape of small sub-Riemannian balls depending uniformly on their center and their radius. This result is a generalization of the ball-box theorem. The proof is based on the one hand on a lifting method, which replaces the sub-Riemannian manifold by an extended equiregular one (where the ball-box theorem is uniform); and on the other hand, it based on an estimate of sets defined by families of vector fields, which allows us to project the balls in suitable coordinates.
Given a system of linear differential equations with a pole, say at z=0, it is well known that the system has a formal fundamental solution which is the product of a formal power series in a root of z, a matrix power of z, and the exponential of a polynomial in a root of z^{-1}. Suppose that the system depends analytically upon several parameters in a neighborhood of some point in the parameter space. Then the question arises whether there exists an analytic formal fundamental solution, i.e., a formal fundamental solution whose coefficients are analytic in the parameters in a possibly smaller neighborhood of the given point. In 1985, Babbitt and Varadarajan treated this problem together with that of the deformation of nilpotent matrices over rings. They assume that the exponential parts of the formal fundamental solutions are well behaved in some precise sense. In the present paper I will provide constructive proofs of their theorems on formal fundamental solutions and in this way also improve them slightly. Furthermore I will give a condition for well behaved exponential parts -- and hence for the existence of an analytic formal fundamental solution -- which can be expressed solely in terms of the given equation.
Let V be a finite set, S be an infinite countable commutative semigroup,
{\tau_{\bi},\bi \in S} be the semigroup of translations in the function
space X=V^S, \ca={A_n} be a sequence of finite sets in S, f be a
continuous function on X with values in a separable real Banach space B,
and let \alpha \in B. We introduce in X a ``scale metric" generating
the product topology. Under some assumptions on f and \ca, we evaluate the
Hausdorff dimension of the set \x defined by the following formula:
\x=\Bigl\{x:x\in X,
\underset {n\to\infty}{\lim}\frac1{|A_n|}\sum\limits_{\bi\in A_n}
f(\tau_{\bi}x)={\alpha}\Bigr\}.
It turns out that this dimension does not depend on the choice of a
Folner ``pointwise averaging" sequence \ca and is completely specified by
the ``scale index" of the metric in X. This general model includes the
important special cases where S=\dz or \Bbb {Z_+}^d, d\ge 1,
and the sets A_n are infinitely increasing cubes; if B = R^m then
f(x)=(f_1(x),\ldots,f_m(x)), \alpha=(\alpha_1,\dots,\alpha_m), and
\x =\Bigl\{x:x\in X,
\underset {n\to\infty}{\lim}\frac1{|A_n|}\sum\limits_{\bi\in A_n}
f_1(\tau_{\bi}x)=\\&{}=\alpha_1, \dots,\underset
{n\to\infty}{\lim}\frac1{|A_n|}
\sum\limits_{\bi\in A_n}f_m(\tau_{\bi}x)=\alpha_m\Bigr\}.
Thus the multifractal analysis of the ergodic averages of several
continuous functions is a special case of our results; in particular,
in Examples \ref{Eggl} and \ref{Bill}
we generalize the well-known theorems due to Eggleston and Billingsley.
We consider the class V_n of germs of holomorphic vector fields in (C^2,0) with vanishing (n-1)-jet at zero. We prove that the formal equivalence of two generic germs in V_n implies their analytic equivalence. This result is analogous to the one obtained in \cite{V} for the case of orbital analytic equivalence.