Toral automorphisms, represented by unimodular integer matrices, are investigated with respect to their symmetries and reversing symmetries. We characterize the symmetry groups of GL(n,Z) matrices with simple spectrum through their connection with unit groups in orders of algebraic number fields. For the question of reversibility, we derive necessary conditions in terms of the characteristic polynomial and the polynomial invariants. We also briefly discuss extensions to (reversing) symmetries within affine transformations, to PGL(n,Z) matrices, and to the more general setting of integer matrices beyond the unimodular ones.
We study various aspects of the dynamics induced by integer matrices on the invariant rational lattices of the torus in dimension 2 and greater. Firstly, we investigate the orbit structure when the toral endomorphism is not invertible on the lattice, characterising the pretails of eventually periodic orbits. Next we study the nature of the symmetries and reversing symmetries of toral automorphisms on a given lattice, which has particular relevance to (quantum) cat maps.
We show that for every linear toral automorphism, especially the non-hyperbolic ones, the entropies of ergodic measures form a dense set on the interval from zero to the topological entropy.
We show in prime dimension that for two non-commuting totally irreducible toral automorphisms the set of points that equidistribute under the first map but have non-dense orbit under the second has full Hausdorff dimension. In non-prime dimension the argument fails only if the automorphisms have strong algebraic relations.
A dynamical Mertens' theorem for ergodic toral automorphisms with error term O(N^{-1}) is found, and the influence of resonances among the eigenvalues of unit modulus is examined. Examples are found with many more, and with many fewer, periodic orbits than expected.
We introduce a relation of block conjugacy for irreducible toral automorphism, and prove that block conjugacy is equivalent to weak equivalence of the ideals associated to the automorphisms. We characterize when block conjugate automorphisms are actually conjugate in terms of a group action on invariant and invariantly complemented subtori, and detail the relation of block conjugacy with a Galois group. We also investigate the nature of the relationship between ideals associated to non-block conjugate irreducible automorphisms.
We show that a collection of generalized Bowen-Franks group, what we call the principal Bowen-Franks $R$-modules, form a complete set of $R$-module invariants for the equivalence relation of profinite conjugacy for similar hyperbolic toral automorphisms. We also show that these principal generalized Bowen-Franks $R$-modules are the principal invariants in a large class of generalized Bowen-Franks $R$-module invariants for similar hyperbolic automorphisms. We further show how the conjugacy invariant of cyclicity, when applied to hyperbolic toral automorphisms, is realized for profinite conjugac…
This survey gives an account of an algebraic construction of symbolic covers and representations of ergodic automorphisms of compact abelian groups, initiated by A.M. Vershik around 1992 for hyperbolic automorphisms of finite-dimensional tori. The key ingredient in this approach, which was subsequently extended to arbitrary expansive automorphisms of compact abelian groups, is the use of homoclinic points of the automorphism. Although existence and abundance of homoclinic points is intimately connected to expansiveness of the automorphism, it is nevertheless possible to extend certain aspects …
We study the regularity of a conjugacy $H$ between a hyperbolic toral automorphism $A$ and its smooth perturbation $f$ We show that if $H$ is weakly differentiable then it is $C^{1+Hölder}$ and, if $A$ is also weakly irreducible, then $H$ is $C^\infty$. As a part of the proof, we establish results of independent interest on Hölder continuity of a measurable conjugacy between linear cocycles over a hyperbolic system. As a corollary, we improve regularity of the conjugacy to $C^\infty$ in prior local rigidity results.
We prove sharp estimates in a shrinking target problem for the action of an arbitrary subgroup $Γ$ of $SL_2(\mathbb{Z})$ on the 2-torus. This can also be viewed as a non-commutative Diophantine approximation problem. The methods require constructing spectrally optimal random walks on groups acting properly cocompactly on Gromov hyperbolic spaces. Additionally, using Fourier analysis we give estimates for the same problem in higher dimensions.
We consider a hyperbolic toral automorphism $L$ and its $C^1$-small perturbation $f$. It is well-known that $f$ is Anosov and topologically conjugate to $L$, but a conjugacy $H$ is only Hölder continuous in general. We discuss conditions for smoothness of $H$, such as conjugacy of the periodic data of $f$ and $L$, coincidence of their Lyapunov exponents, and weaker regularity of $H$, and we summarize questions, results, and techniques in this area. Then we introduce our new results: if $H$ is weakly differentiable then it is $C^{1+\text{Hölder}}$ and, if $L$ is also weakly irreducible, then $H…
We show that if $r\geq 3$ and $α$ is a faithful $Z^r$-Cartan action on a torus $T^d$ by automorphisms, then any closed subset of $(T^d)^2$ which is invariant and topologically transitive under the diagonal $\bZ^r$-action by $α$ is homogeneous, in the sense that it is either the full torus $(T^d)^2$, or a finite set of rational points, or a finite disjoint union of parallel translates of some d-dimensional invariant subtorus. A counterexample is constructed for the rank 2 case.
We study the regularity of the conjugacy between an Anosov automorphism $L$ of a torus and its small perturbation. We assume that $L$ has no more than two eigenvalues of the same modulus and that $L^4$ is irreducible over $\mathbb Q$. We consider a volume-preserving $C^1$-small perturbation $f$ of $L$. We show that if Lyapunov exponents of $f$ with respect to the volume are the same as Lyapunov exponents of $L$, then $f$ is $C^{1+\text{Hölder}}$ conjugate to $L$. Further, we establish a similar result for irreducible partially hyperbolic automorphisms with two-dimensional center bundle.
Berend gives necessary and sufficient conditions on a $Z^r$-action $α$ on a torus $T^d$ by toral automorphisms in order for every orbit be either finite or dense. One of these conditions is that on every common eigendirection of the $Z^r$-action there is an element $\mathbf n \in Z^r$ so that $α^\mathbf n$ expands this direction. In this paper, we investigate what happens when this condition is removed; more generally, we consider a partial orbit $\{α^\mathbf n.x : \mathbf n \in Ω\}$ where $Ω$ is a set of elements which acts approximately as the identity on a given set of eigendirections. This…
For a hyperbolic toral automorphism, we construct a profinite completion of an isomorphic copy of the homoclinic group of its right action using isomorphic copies of the periodic data of its left action. The resulting profinite group has a natural module structure over a ring determined by the right action of the hyperbolic toral automorphism. This module is an invariant of conjugacy that provides means in which to characterize when two similar hyperbolic toral automorphisms are conjugate or not. In particular, this shows for two similar hyperbolic toral automorphisms with module isomorphic le…
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 $\T^N$.
We show that the Voiculescu-Brown entropy of a noncommutative toral automorphism arising from a matrix S in GL(d,Z) is at least half the value of the topological entropy of the corresponding classical toral automorphism. We also obtain some information concerning the positivity of local Voiculescu-Brown entropy with respect to single unitaries. In particular we show that if S has no roots of unity as eigenvalues then the local Voiculescu-Brown entropy with respect to every product of canonical unitaries is positive, and also that in the presence of completely positive CNT entropy the unital ve…
We consider pairs of toral automorphisms (A,B) satisfying an invariant cone property. At each iteration, A acts with probability p and B with probability 1-p. We prove exponential decay of correlations for a class of Holder continuous observables.
We study the $C^*$-algebras of the étale groupoids defined by the asymptotic equivalence relations for hyperbolic automorphisms on the two-dimensional torus. The algebras are proved to be four-dimensional non-commutative tori by an explicit numerical computation. The ranges of the unique tracial states of its $K_0$-groups of the $C^*$-algebras are described in terms of the hyperbolic matrices of the automorphisms on the torus.
In this paper we study $C^*$-algebra version of Sarnak Conjecture for noncommutative toral automorphisms. Let $A_Θ$ be a noncommutative torus and $α_Θ$ be the noncommutative toral automorphism arising from a matrix $S\in GL(d,\mathbb{Z})$. We show that if the Voiculescu-Brown entropy of $α_Θ$ is zero, then the sequence $\{ρ(α_Θ^nu)\}_{n\in \mathbb{Z}}$ is a sum of a nilsequence and a zero-density-sequence, where $u\in A_Θ$ and $ρ$ is any state on $A_Θ$. Then by a result of Green and Tao, this sequence is linearly disjoint from the Möbius function.
Toral automorphisms are widely used (discrete) dynamical systems, the perhaps most prominent example (in 2D) being Arnold's cat map. Given such an automorphism M, its symmetries (i.e. all automorphisms that commute with M) and reversing symmetries (i.e. all automorphisms that conjugate M into its inverse) can be determined by means of number theoretic tools. Here, the case of GL(2,Z) is presented and the possible (reversing) symmetry groups are completely classified. Extensions to affine mappings and to k-(reversing) symmetries (i.e. (reversing) symmetries of the k-th power of M), and applicat…
We call a Markov partition of a two dimensional hyperbolic toral automorphism a Berg partition if it contains just two rectangles. We describe all Berg partitions for a given hyperbolic toral automorphism. In particular there are exactly (k + n + l + m)/2 nonequivalent Berg partitions with the same connectivity matrix (k, l, m, n).
If $α$ is an irreducible nonexpansive ergodic automorphism of a compact abelian group $X$ (such as an irreducible nonhyperbolic ergodic toral automorphism), then $α$ has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts in $X$. In spite of this we are able to construct a symbolic space $V$ and a class of shift-invariant probability measures on $V$ each of which corresponds to an $α$-invariant probability measure on $X$. Moreover, every $α$-invariant probability measure on $X$ arises essentially in this way. The last part of the pa…
Let $\Cal S$ be an abelian finitely generated semigroup of endomorphisms of a probability space $(Ω, {\Cal A}, μ)$, with $(T_1, ..., T_d)$ a system of generators in ${\Cal S}$. Given an increasing sequence of domains $(D_n) \subset \N^d$, a question is the convergence in distribution of the normalized sequence $|D_n|^{-\frac12} \sum_{{\k} \, \in D_n} \, f \circ T^{\,{\k}}$, for $f \in L^2_0(μ)$, where $T^{\k}= T_1^{k_1} ... T_d^{k_d}$, ${\k}= (k_1, ..., k_d) \in {\N}^d$. After a preliminary spectral study when the action of $\Cal S$ has a Lebesgue spectrum, we consider $\N^d$- or $\Z^d$-action…
We establish a quenched Central Limit Theorem (CLT) for a smooth observable of random sequences of iterated linear hyperbolic maps on the torus. To this end we also obtain an annealed CLT for the same system. We show that, almost surely, the variance of the quenched system is the same as for the annealed system. Our technique is the study of the transfer operator on an anisotropic Banach space specifically tailored to use the cone condition satisfied by the maps.
This paper is dedicated to the conjugacy problem in $GL(n,\Z)$ and its connection with algebraic number theory. This connection may be summed up in the Latimer-MacDuffee-Taussky Theorem, which, in a very broad sense, identifies the conjugacy relation in $GL(n,\Z)$ with the relation of arithmetic equivalence between certain ideals of algebraic numbers. The main purpouse is to clarify the link between a weaker relation between ideals, weak equivalence, and local conjugacy in $GL(n,\Z)$, i.e., the conjugacy of $GL(n,\Z)$ matrices in the groups $GL(n,\Z_{p})$ of invertible matrices over the $p$-ad…
It is well-known that the Artin-Mazur dynamical zeta function of a hyperbolic or quasi-hyperbolic toral automorphism is a rational function, which can be calculated in terms of the eigenvalues of the corresponding integer matrix. We give an elementary proof of this fact that extends to the case of general toral endomorphisms without change. The result is a closed formula that can be calculated by integer arithmetic only. We also address the functional equation and the relation between the Artin-Mazur and Lefschetz zeta functions.
The polynomial automorphisms of the affine plane over a field K form a group which has the structure of an amalgamated free product. This well-known algebraic structure can be used to determine some key results about the symmetry and reversing symmetry groups of a given polynomial automorphism.
For every irreducible automorphism $ϕ\in\text{SL}_3({\mathbb Z})$ of the $3$-torus, for which the product of the expanding eigenvalues is positive, we construct a pseudo-Anosov mapping $f$ of an associated surface, semi-conjugate and almost-isomorphic to $ϕ$, whose stretch factor is the product of the expanding eigenvalues of $ϕ$. This shows that any norm-$1$ cubic Pisot number occurs as the stretch factor of a pseudo-Anosov mapping, proving a conjecture of Fried in degree $3$. A similar construction works for the $4$-torus on condition that $ϕ$ has exactly two eigenvalues outside the unit cir…
Let ϕ be an automorphism on a connected Lie group G. Through several G-subgroups associated to the dynamics of ϕ we analyze their topological entropy. Assume that G belongs to the class of finite semisimple center Lie groups which admits a ϕ invariant Levi subgroup. Then we prove that the topological entropy information of ϕ is contained in the toral component of the unstable subgroup of ϕ in the radical of G. We specialize the main result in a couple of interesting situations.
We determine the symmetries and reversing symmetries within G, the group of real planar polynomial automorphisms, of area-preserving nonlinear polynomial maps L in generalised standard form, L: x'=x+p(y), y'=y+q(x'), where p and q are polynomials. We do this by using the amalgamated free product structure of G. Our results lead to normal forms for polynomial maps in generalised standard form and to a classification of the group structures of the reversing symmetry groups for such maps.
We consider hyperbolic toral automorphisms which are reversible with respect to a linear area-preserving involution. We will prove that within this context reversibility is linked to a generalized Pell equation whose solutions we will analyze. Additionally, we will verify to what extent reversibility is a common feature and characterize the generic setting.
The reversing symmetry group is a well-studied extension of the symmetry group of a dynamical system, the latter being defined by the action of a single homeomorphism on a topological space. While it is traditionally considered in nonlinear dynamics, where the space is simple but the map is complicated, it has an interesting counterpart in symbolic dynamics, where the map is simple but the space is not. Moreover, there is an interesting extension to the case of higher-dimensional shifts, where a similar concept can be introduced via the centraliser and the normaliser of the acting group in the…
Let $J$ be a semisimple Lie group with all simple factors of real rank at least two. Let $Γ<J$ be a lattice. We prove a very general local rigidity result about actions of $J$ or $Γ$. This shows that almost all so-called "standard actions" are locally rigid. As a special case, we see that any action of $Γ$ by toral automorphisms is locally rigid. More generally, given a manifold $M$ on which $Γ$ acts isometrically and a torus $\Ta^n$ on which it acts by automorphisms, we show that the diagonal action on $\Ta^n{\times}M$ is locally rigid. This paper is the culmination of a series of papers and …
