Toral Endomorphisms Research Articles

On the speed of convergence in the ergodic theorem for shift operators

Abstract Given a probability space $(X,\mu )$ , a square integrable function f on such space and a (unilateral or bilateral) shift operator T, we prove under suitable assumptions that the ergodic means $N^{-1}\sum _{n=0}^{N-1} T^nf$ converge pointwise almost everywhere to zero with a speed of convergence which, up to a small logarithmic transgression, is essentially of the order of $N^{-1/2}$ . We also provide a few applications of our results, especially in the case of shifts associated with toral endomorphisms.

New classes of C 1-robustly transitive maps with persistent critical points

Recently, the authors proved in [C. Lizana and W. Ranter, Topological obstructions for robustly transitive endomorphisms on surfaces, Adv. Math. 390 (2021), pp. 107901] that every C 1 -robustly transitive toral endomorphism displaying critical points must be homotopic to a linear endomorphism having at least one eigenvalue with modulus greater than one. Here, we exhibit some examples of C 1 -robustly transitive surface endomorphisms displaying critical points in certain homotopy classes.

Uniformly distributed orbits in $$\mathbb {T}^d$$ and singular substitution dynamical systems

We find sufficient conditions for the singularity of a substitution $\mathbb{Z}$-action's spectrum, which generalize the conditions given in arXiv:2003.11287, Theorem 2.4, and we also obtain a similar statement for a collection of substitution $\mathbb{R}$-actions, including the self-similar one. To achieve this, we first study the distribution of related toral endomorphism orbits. In particular, given a toral endomorphism and a vector $\mathbf{v}\in\mathbb{Q}^d$, we find necessary and sufficient conditions for the orbit of $\omega\mathbf{v}$ to be uniformly distributed modulo $1$ for almost every $\omega\in\mathbb{R}$. We use our results to find new examples of singular substitution $\mathbb{Z}$- and $\mathbb{R}$-actions.

On codimension one special Anosov endomorphisms

We show that if a closed smooth n-manifold M admits a u-dimension one special Anosov endomorphism f, then M is homeomorphic to and f is topologically conjugate to a hyperbolic toral endomorphism. Moreover, any u-dimension one special Anosov endomorphism is transitive and the stable foliation is minimal.

On the Density of Pre-Orbits under Linear Toral Endomorphisms

It is known that if each pre-orbit of a non-injective endomorphism is dense, the endomorphism is transitive (i.e., a dense orbit exists). However, it is still unknown whether the pre-orbits of an Anosov map are dense, and the conditions necessary for all pre-orbits to be dense are also unknown. Using the properties of integral lattices, we construct our proof by considering the pre-orbits of linear endomorphisms. We introduce a class of hyperbolic linear endomorphisms characterized by the property of absolute hyperbolicity to prove that if A : Tm → Tm is an absolutely hyperbolic endomorphism, the pre-orbit of any point is dense in Tm.

On the density of pre-orbits under linear toral endomorphisms

It is well known for non-injective endomorphisms that if for every point the set of preimages is dense in the manifold then the endomorphism is transitive (i. e. there exists a point that its orbit is dense in the manifold). But it has not yet been completely investigated that if the pre-orbit of points are dense under Anosov endomorphisms or what are the necessary conditions that make the pre-orbits of each point dense. By making a great use of the integral lattice properties, we construct our proof on the pre-image sets of points under the iterations of the linear dynamical system. We introduce a class of hyperbolic linear endomorphism that is called absolutely hyperbolic and show that if A : Tm → Tm is an absolutely hyperbolic linear endomorphism of degree more than 1 then the pre-orbit of each point is dense in Tm.

On periodic points of torus endomorphisms

It is a well-known fact that Anosov endomorphisms of n-torus which are different from automorphisms and expanding endomorphisms are not structurally stable and, in general, are not conjugated to algebraic endomorphisms. Nevertheless, hyperbolic algebraic endomorphisms of torus are conjugated with their C1 perturbations on the set of periodic points. Therefore the study of algebraic toral endomorphisms is very important. This paper is devoted to study of the structure of the sets of periodic and pre-periodic points of algebraic toral endomorphisms. Various group properties of this sets of points are studied. The density of periodic points for algebraic endomorphisms of n-torus is proved; it is clarifief how the number of periodic and pre-periodic points with a fixed denominator depends on the properties of the characteristic polynomial. The Theorem 1.1 is the main result of this paper. It contains an algorithm that allows to split the sets of periodic and pre-periodic points of a given algebraic endomorphism of two-dimensional torus.

Partially hyperbolic surface endomorphisms

We prove that a class of weakly partially hyperbolic endomorphisms on $\mathbb{T}^{2}$ are dynamically coherent and leaf conjugate to linear toral endomorphisms. Moreover, we give an example of a partially hyperbolic endomorphism on $\mathbb{T}^{2}$ which does not admit a centre foliation.

On the Hyperbolicity of Toral Endomorphisms

Nonsingular endomorphisms of the m-torus $$\mathbb{T}$$ m, m ≥ 2, which are C1 perturbations of linear hyperbolic endomorphisms are considered. Sufficient conditions for such maps to be hyperbolic (i.e., belong to the class of Anosov endomorphisms) are found.

Simultaneous dense and non-dense orbits for certain partially hyperbolic diffeomorphisms

Let$g:M\rightarrow M$be a$C^{1+\unicode[STIX]{x1D6FC}}$-partially hyperbolic diffeomorphism preserving an ergodic normalized volume on$M$. We show that, if$f:M\rightarrow M$is a$C^{1+\unicode[STIX]{x1D6FC}}$-Anosov diffeomorphism such that the stable subspaces of$f$and$g$span the whole tangent space at some point on$M$, the set of points that equidistribute under$g$but have non-dense orbits under$f$has full Hausdorff dimension. The same result is also obtained when$M$is the torus and$f$is a toral endomorphism whose center-stable subspace does not contain the stable subspace of$g$at some point.

Transitivity of conservative toral endomorphisms

It is shown that if a non-invertible area preserving local homeomorphism on is homotopic to a linear expanding or hyperbolic endomorphism, then it must be topologically transitive. This gives a complete characterization, in any smoothness category, of those homotopy classes of conservative endomorphisms that consist entirely of transitive maps.

Period sets of linear toral endomorphisms on [formula omitted

The period set of a dynamical system is defined as the subset of all integers n such that the system has a periodic orbit of length n. Based on known results on the intersection of period sets of torus maps within a homotopy class, we give a complete classification of the period sets of (not necessarily invertible) toral endomorphisms on the 2-dimensional torus T2.

Scaling scenery of [formula omitted] invariant measures

We study the scaling scenery and limit geometry of invariant measures for the non-conformal toral endomorphism (x,y)↦(mxmod1,nymod1) that are Bernoulli measures for the natural Markov partition. We show that the statistics of the scaling can be described by an ergodic CP-chain in the sense of Furstenberg. Invoking the machinery of CP-chains yields a projection theorem for Bernoulli measures, which generalises in part earlier results by Hochman–Shmerkin and Ferguson–Jordan–Shmerkin. We also give an ergodic theoretic criterion for the dimension part of Falconer's distance set conjecture for general sets with positive length using CP-chains and hence verify it for various classes of fractals such as self-affine carpets of Bedford–McMullen, Lalley–Gatzouras and Barański class and all planar self-similar sets.

Jarník-type inequalities

It is well known due to Jarník that the set Bad R 1 of badly approximable numbers is of Hausdorff dimension 1. If Bad R 1 ( c ) denotes the subset of x ∈ Bad R 1 for which the approximation constant c ( x ) ⩾ c , then Jarník was in fact more precise and gave non-trivial lower and upper bounds on the Hausdorff dimension of Bad R 1 ( c ) in terms of the parameter c > 0 . Our aim is to determine simple conditions on a framework which allow one to extend ’Jarník's inequality’ to further examples. For many dynamical examples, these extensions are related to the Hausdorff dimension of the set of orbits that avoid a suitable given neighborhood of an obstacle. Among the applications, we discuss the set Bad R n r ¯ of badly approximable vectors in R n with weights r ¯ , the set of orbits in the Bernoulli shift that avoid a neighborhood of a periodic orbit, the set of geodesics in the hyperbolic space H n that avoid a suitable collection of convex sets, and the set of orbits of a toral endomorphism that avoid neighborhoods of a separated set.

Higher dimensional expanding maps and toral extensions

We prove that expanding endomorphisms on arbitrary tori are 1-sided Bernoulli with respect to their corresponding measure of maximal entropy and are thus, measurably, as far from invertible as possible. This applies in particular to expanding linear toral endomorphisms and their smooth perturbations. Then we study toral extensions of expanding toral endomorphisms, in particular probabilistic systems on skew products, and prove that under certain not too restrictive conditions on the extension cocycle, these skew products are 1-sided Bernoulli too. We also give a large class of examples of group extensions of expanding maps in higher dimensions, for which we check the conditions on the extension cocycle.

Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices

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.

Central Limit Theorem for $\mathbb{Z}_{+}^d$-actions by toral endomorphisms

In this paper we prove the central limit theorem for the following multisequence $$\sum_{n_1=1}^{N_1} ... \sum_{n_d=1}^{N_d} f(A_1^{n_1}...A_d^{n_d} {\bf x} )$$ where $f$ is a Hölder's continue function, $A_1,\ldots,A_d$ are $s\times s$ partially hyperbolic commuting integer matrices, and $\bf x$ is a uniformly distributed random variable in $[0,1]^s$. Next we prove the functional central limit theorem, and the almost sure central limit theorem. The main tool is the $S$-unit theorem.

On some coding and mixing properties for a class of chaotic systems

We study certain ergodic properties of equilibrium measures of hyperbolic non-invertible maps f on basic sets with overlaps Λ. We prove that if the equilibrium measure \({\mu_\phi}\) of a Holder potential \({\phi}\) , is 1-sided Bernoulli, then f is expanding from the point of view of a pointwise section dimension of \({\mu_\phi}\) . If the measure of maximal entropy μ0 is 1-sided Bernoulli, then f is shown to be distance expanding on Λ; and if \({\mu_\phi}\) is 1-sided Bernoulli for f expanding, then \({\mu_\phi}\) must be the measure of maximal entropy. These properties are very different from the case of hyperbolic diffeomorphisms. Another result is about the non 1-sided Bernoullicity for certain equilibrium measures for hyperbolic toral endomorphisms. We also prove the non-existence of generating Rokhlin partitions for measure-preserving endomorphisms in several cases, among which the case of hyperbolic non-expanding toral endomorphisms with Haar measure. Nevertheless the system \({(\Lambda, f, \mu_\phi)}\) is shown to have always exponential decay of correlations on Holder observables and to be mixing of any order.

Approximations for Gibbs states of arbitrary Hölder potentials on hyperbolic folded sets

We give approximations for the Gibbsstates of arbitrary Hölder potentials $\phi$,with the help of weighted sums of atomic measures on preimage sets, in the case of smooth non-invertible maps hyperbolic onfolded basic sets $\Lambda$. The endomorphism may have also stable directions on$\Lambda$ and is non-expanding in general. Folding of the phase space means that we do not have a foliation structure for the local unstable manifolds(instead they depend on the whole past and may intersect each other both inside and outside $\Lambda$). We consider here simultaneously all $n$-preimages in $\Lambda$ of a point, instead of the usual way of taking only the consecutive preimages from some given prehistory. We thus obtain the weighted distribution of consecutive preimage sets, with respect to various equilibrium measures on the saddle-type folded set $\Lambda$. In particular we obtain the distribution of preimage sets on $\Lambda$, with respect to the measure of maximal entropy. Our result is not a direct application of Birkhoff Ergodic Theorem on the inverse limit $\hat \Lambda$, since the set of prehistories of a point is uncountable in general, and the speed of convergence may vary for different prehistories in $\hat \Lambda$. For hyperbolic toral endomorphisms, we obtain the distribution of the consecutive preimage sets towards an inverse SRB measure, for Lebesgue-almost all points.

Schmidt’s game, fractals, and orbits of toral endomorphisms

AbstractGiven an integer matrix M∈GLn(ℝ) and a point y∈ℝn/ℤn, consider the set S. G. Dani showed in 1988 that whenever M is semisimple and y∈ℚn/ℤn, the set $ \tilde E(M,y)$ has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary M∈GLn(ℝ)∩Mn×n(ℤ) and y∈ℝn/ℤn, and in fact replacing the sequence of powers of M by any lacunary sequence of (not necessarily integer) m×n matrices. Furthermore, we show that sets of the form $ \tilde E(M,y)$ and their generalizations always intersect with ‘sufficiently regular’ fractal subsets of ℝn. As an application, we give an alternative proof of a recent result [M. Einsiedler and J. Tseng. Badly approximable systems of affine forms, fractals, and Schmidt games. Preprint, arXiv:0912.2445] on badly approximable systems of affine forms.