Toral Endomorphisms Research Articles

Asymptotic distributions of preimages for endomorphisms

Attractors for hyperbolic diffeomorphisms are known to possess unique Sinai–Ruelle–Bowen measures with interesting properties. In this paper we investigate the case of non-invertible maps (endomorphisms) which have repellers Λ. We work with preimages of points in a neighbourhood of the repeller (assumed to be non-expanding); the situation here is different than the one for diffeomorphisms or positive iterates of endomorphisms. We give two methods to obtain invariant measures from local inverse iterates. We show that if Λ is a hyperbolic s-conformal repeller for f, not necessarily expanding, and if f is d-to-1 on Λ then for Lebesgue almost every x in the repelling basin of Λ there are histories of x asymptotically distributed like the equilibrium measure μs of the Hölder continuous potential Φs, with Φs (y):=log ∣Dfs (y)∣ for y∈Λ. The measure μs plays the role of an inverse Sinai–Ruelle–Bowen measure on the non-invertible repeller. We prove also that there exists a set A⊂Wuε(Λ) with λ(A)=λ(Wuε(Λ)) (where λ(⋅) is the Lebesgue measure) such that for any z∈A and any real continuous function g, with In particular, we obtain the asymptotic distribution of preimages of Lebesgue almost all points for a class of hyperbolic toral endomorphisms on 𝕋m,m≥2 .

The limit of a pedal sequence of triangles

Several authors have studied the dynamics of the pedal sequence in which a triangle ABC is replaced by its pedal triangle (whose vertices are the feet of the perpendiculars from each vertex to the opposite side) and this procedure is repeated indefinitely. They found cases where the shape of the triangle repeats periodically and showed that the map is ergodic and modelled by the full shift on four symbols. We show here that the parameter space of shapes of triangles is a tetrahedron and that this has as a double cover (branched over four points) the two-torus where the dynamics is multiplication by 2. The pedal sequence thus inherits its dense set of periodic points and also the mixing property from this toral endomorphism. We find an expression for the limit point of a pedal sequence and show that it depends continuously but not differentiably on the shape of the original triangle.

A note on the dynamical zeta function of general toral endomorphisms

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.

Periodic orbits of linear endomorphisms on the 2-torus and its lattices

Counting periodic orbits of endomorphisms on the 2-torus is considered, with special focus on the relation between global and local aspects and between the dynamical zeta function on the torus and its analogue on finite lattices. The situation on the lattices, up to local conjugacy, is completely determined by the determinant, the trace and a third invariant of the matrix defining the toral endomorphism.

Degree-growth of monomial maps

AbstractFor projectivizations of rational maps, Bellon and Viallet defined the notion of algebraic entropy using the exponential growth rate of the degrees of iterates. We want to call this notion to the attention of dynamicists by computing algebraic entropy for certain rational maps of projective spaces (Theorem 6.2) and comparing it with topological entropy (Theorem 5.1). The particular rational maps we study are monomial maps (Definition 1.2), which are closely related to toral endomorphisms. Theorems 5.1 and 6.2 that imply that the algebraic entropy of a monomial map is always bounded above by its topological entropy, and that the inequality is strict if the defining matrix has more than one eigenvalue outside the unit circle. Also, Bellon and Viallet conjectured that the algebraic entropy of every rational map is the logarithm of an algebraic integer, and Theorem 6.2 establishes this for monomial maps. However, a simple example using a monomial map shows that a stronger conjecture of Bellon and Viallet is incorrect, in that the sequence of algebraic degrees of the iterates of a rational map of projective space need not satisfy a linear recurrence relation with constant coefficients.

Statistical properties of some spatially discretized dynamical systems

Computer simulations of dynamical systems are {spatial discretizations} in which the finite machine arithmetic space replaces the continuum state space of the original system. Any trajectory of a spatial discretization of a dynamical system is thus eventually periodic, so the dynamical behaviour of such computations are essentially determined by the cycles of the discretized map. Such dynamical behaviour depends seemingly randomly on the fineness of the discretization mesh. In this paper statistical properties of the maximal cycles of spatial discretizations are investigated for some systems such as the tent map, rotations on a circle and toral endomorphisms.

Topological properties of differentiable maps derived from 2-toral endomorphisms

We consider one-parameter families of C r regular maps starting at hyperbolic toral endomorphisms on the 2-torus T 2. We show that some perturbations of them satisfy Axiom A as in the case of diffeomorphisms. We also prove that the Hausdorff dimension of the nonwandering set is continuous across the bifurcation.

S-integer dynamical systems: periodic points.

We associate via duality a dynamical system to each pair (RS,x), where RS is the ring of S-integers in an A-field k, and x is an element of RS\{0}. These dynamical systems include the circle doubling map, certain solenoidal and toral endomorphisms, full one- and two-sided shifts on prime power alphabets, and certain algebraic cellular automata. In the arithmetic case, we show that for S finite the systems have properties close to hyperbolic systems: the growth rate of periodic points exists and the periodic points are uniformly distributed with respect to Haar measure. The dynamical zeta function is in general irrational however. For S infinite the systems exhibit a wide range of behaviour. Using Heath-Brown's work on the Artin conjecture, we exhibit examples in which S is infinite but the upper growth rate of periodic points is positive.

A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions

In this paper we establish the complete multifractal formalism for equilibrium measures for Holder continuous conformal expanding maps andexpanding Markov Moran-like geometric constructions. Examples include Markov maps of an interval, beta transformations of an interval, rational maps with hyperbolic Julia sets, and conformal toral endomorphisms. We also construct a Holder continuous homeomorphism of a compact metric space with an ergodic invariant measure of positive entropy for which the dimension spectrum is not convex, and hence the multifractal formalism fails.

Measures of full dimension on affine-invariant sets

AbstractWe determine the Hausdorff and Minkowski dimensions of some self-affine Sierpinski sponges, extending results of McMullen and Bedford. This result is used to show that every compact set invariant under an expanding toral endomorphism which is a direct sum of conformal endomorphisms supports an invariant measure of full dimension.

On 4-Lacunary Sequences Generated by Ergodic Toral Endomorphisms

On 4-Lacunary Sequences Generated by Ergodic Toral Endomorphisms

On exact toral endomorphisms

A necessary and sufficient condition for the exactness of a toral endomorphism is given in terms of the characteristic polynomial of the matrix which induces it.

On 4-lacunary sequences generated by ergodic toral endomorphisms

It is proved that if φ \varphi is an ergodic endomorphism of T k {\mathbb {T}^k} and f f is a sufficiently regular complex-valued function on T k {\mathbb {T}^k} with the Haar integral zero, then ( f ∘ φ n ) (f \circ {\varphi ^n}) is a 4 4 -lacunary sequence. Within the class of ergodic toral endomorphisms and sufficiently regular complex-valued functions, applications are given to the convergence of series, a generalization of the ergodic theorem, the existence of solutions of a generalized cohomology equation, and the convergence of moments in the central limit theorem.

Periodic points and invariant pseudomeasures for toral endomorphisms

AbstractExtending a result of Livsic [10] it is proved that the coboundary equation f(Tx)−f(x) = g(x) admits a C∞ solution f for C∞g when T is an ergodic toral endomorphism and g sums to zero over every periodic orbit. The same statement is false with C1 in place of C∞, in contrast to the Livsic (hyperbolic) theorem. In one dimension the ‘Lip α’ case leads to questions relating to the generalized Riemann hypothesis.

Shorter Notes: The Zeta Function of Toral Endomorphisms

Shorter Notes: The Zeta Function of Toral Endomorphisms

The zeta function of toral endomorphisms

In this note we give a completely elementary way to compute the number of fixed points of a certain class of toral endomorphisms. This, in turn, gives the zeta function of these endomorphisms.