Preimages Of Points Research Articles

Reeb graphs of circle-valued functions: A survey and basic facts

The Reeb graph of a circle-valued function is a topological space obtained by contracting connected components of level sets (preimages of points) to points. For some smooth functions, the Reeb graph has the structure of a finite graph. This notion finds numerous applications in the theory of dynamical systems, as well as in the topological classification of circle-valued functions and the study of their homotopy properties. However, important theoretical facts on the topological properties of the Reeb graphs of circle-valued functions are scattered across numerous papers on different topics, according to the specific needs of the corresponding application. In this paper, we systematize the existing results on the Reeb graphs of circle-valued functions and generalize some of them to wider classes of functions or spaces. We also show how some results can be carried out from real-valued functions. Finally, we adapt some facts from the theory of foliations to the Reeb graphs of circle-valued functions. In particular, we analyze the cycle rank of the Reeb graph and address the problem of realization of a finite graph as a Reeb graph.

Number and location of pre-images under harmonic mappings in the plane

We derive a formula for the number of pre-images under a non-degenerate harmonic mapping $f$, using the argument principle. This formula reveals a connection between the pre-images and the caustics. Our results allow to deduce the number of pre-images under $f$ geometrically for every non-caustic point. We approximately locate the pre-images of points near the caustics. Moreover, we apply our results to prove that for every $k = n, n+1, \ldots, n^2$ there exists a harmonic polynomial of degree $n$ with $k$ zeros.

Distribution of Typical Orbits for Random Dynamics Generated by Finitely Many Rational Maps

In this paper, we consider the dynamics of a skew-product map defined on the Cartesian product of the symbolic one-sided shift space on N symbols and the complex sphere where we allow N rational maps, $$R_{1}, R_{2}, \ldots , R_{N}$$ , each with degree $$d_{i};\ 1 \le i \le N$$ and with at least one $$R_{i}$$ in the collection whose degree is at least 2. We obtain results regarding the distribution of pre-images of points and the periodic points in a subset of the product space (where the skew-product map does not behave normally). We further explore the ergodicity of the Sumi-Urbanski (equilibrium) measure associated to some real-valued Holder continuous function defined on the Julia set of the skew-product map and obtain estimates on the mean deviation of the behaviour of typical orbits, violating such ergodic necessities.

Approximating resolutions by cell-like maps with codimension-three point inverses

Let X be a resolvable generalized n-manifold and M be a (PL) n-manifold, where n≥6. We claim every resolution from M to X can be approximated, arbitrarily closely, by cell-like maps with (n−3)-dimensional point preimages. The key instrument of the proof is the disjoint Pontryagin disks property.

CANNON–THURSTON MAPS FOR KLEINIAN GROUPS

We show that Cannon–Thurston maps exist for degenerate free groups without parabolics, that is, for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon–Thurston maps for surface groups, we show that Cannon–Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon–Thurston maps for degenerate free groups without parabolics correspond to endpoints of leaves of an ending lamination in the Masur domain, whenever a point has more than one pre-image. This proves a conjecture of Otal. We also prove a similar result for point pre-images under Cannon–Thurston maps for arbitrary finitely generated Kleinian groups without parabolics.

Ending Laminations and Cannon–Thurston Maps

In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian surface groups. In this paper we prove that pre-images of points are precisely end-points of leaves of the ending lamination whenever the Cannon-Thurston map is not one-to-one. In particular, the Cannon-Thurston map is finite-to-one. This completes the proof of the conjectural picture of Cannon-Thurston maps for surface groups.

UVk-mappings on homology manifolds

We prove a strong controlled generalization of a theorem of Bestvina and Walsh, which states that a (k+1)–connected map from a topological n–manifold to a polyhedron, 2k+3≤n, is homotopic to a UVk–map, that is, a surjection whose point preimages are, in some sense, k–connected. One consequence of our main result is that a compact ENR homology n–manifold, n≥5, having the disjoint disks property satisfies the linear UV⌊(n−3)∕2⌋–approximation property for maps to compact ANRs. The method of proof is general enough to show that any compact ENR satisfying the disjoint (k+1)–disks property has the linear UVk–approximation property.

Large entropy measures for endomorphisms of ℂℙ k

Let f be a holomorphic endomorphism of ℂℙ k . We construct by using coding techniques a class of ergodic measures as limits of non-uniform probability measures on preimages of points. We show that they have large metric entropy, close to log d k . We establish for them strong stochastic properties and prove the positivity of their Lyapunov exponents. Since they have large entropy, those measures are supported in the support of the maximal entropy measure of f. They in particular provide lower bounds for the Hausdorff dimension of the Julia set.

Preservation of the Borel class under open-LC functions

Let $X$ be a Borel subset of the Cantor set \textbf{C} of additive or multiplicative class ${\alpha},$ and $f: X \to Y$ be a continuous function with compact preimages of points onto $Y \subset \textbf{C}.$ If the image $f(U)$ of every clopen set $U$ is the intersection of an open and a closed set, then $Y$ is a Borel set of the same class. This result generalizes similar results for open and closed functions.

Connectedness of levels for moment maps on various classes of loop groups

The space $\Omega(G)$ of all based loops in a compact semisimple simply connected Lie group $G$ has an action of the maximal torus $T\subset G$ (by pointwise conjugation) and of the circle $S^1$ (by rotation of loops). Let $\mu : \Omega(G)\to (\t\times i\mathbb{R})^*$ be a moment map of the resulting $T\times S^1$ action. We show that all levels (that is, pre-images of points) of $\mu$ are connected subspaces of $\Omega(G)$ (or empty). The result holds if in the definition of $\Omega(G)$ loops are of class $C^{\infty}$ or of any Sobolev class $H^s$, with $s\ge 1$ (for loops of class $H^1$, connectedness of regular levels has been proved by Harada, Holm, Jeffrey, and the author).

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 .

Relations between stable dimension and the preimage counting function on basic sets with overlaps

In this paper we study non-invertible hyperbolic maps f and the relation between the stable dimension (that is, the Hausdorff dimension of the intersection between local stable manifolds of f and a given basic set Λ) and the preimage counting function of the map f restricted to the fractal set Λ. The case of diffeomorphisms on surfaces was considered in [A. Manning and H. McCluskey, ‘Hausdorff dimension for horseshoes’, Ergodic Theory Dynam. Systems 3 (1983) 251–260], where thermodynamic formalism was used to study the stable/unstable dimensions. In the case of endomorphisms, the non-invertibility generates new phenomena and new difficulties due to the overlappings coming from the different preimages of points, and also due to the variations of the number of preimages belonging to Λ (when compared with [E. Mihailescu and M. Urbanski, ‘Estimates for the stable dimension for holomorphic maps’, Houston J. Math. 31 (2005) 367–389]). We show that, if the number of preimages belonging to Λ of any point is less than or equal to a continuous function ω(·) on Λ, then the stable dimension at every point is greater than or equal to the zero of the pressure function t → P(tΦs−log ω(·)). As a consequence we obtain that, if d is the maximum value of the preimage counting function on Λ and if there exists x ∈ Λ with the stable dimension at x equal to the zero td of the pressure function t → P(t Φs − log d), then the number of preimages in Λ of any point y is equal to d, and the stable dimension is td everywhere on Λ. This has further consequences to estimating the stable dimension for non-invertible skew products with overlaps in fibers.

Inverse Pressure Estimates and the Independence of Stable Dimension for Non-Invertible Maps

AbstractWe study the case of an Axiom A holomorphic non-degenerate (hence non-invertible) mapf: ℙ2ℂ → ℙ2ℂ, where ℙ2ℂ stands for the complex projective space of dimension 2. Letδs(x)denote a basic set for f of unstable index 1, and x an arbitrary point of Λ; we denote byδs(x)the Hausdorff dimension of∩ Λ, whereris some fixed positive number andis the local stable manifold atxof sizer;δs(x)is calledthe stable dimension at x. Mihailescu and Urba ńnski introduced a notion of inverse topological pressure, denoted by P−, which takes into consideration preimages of points. Manning and McCluskey studied the case of hyperbolic diffeomorphisms on real surfaces and give formulas for Hausdorff dimension. Our non-invertible situation is different here since the local unstable manifolds are not uniquely determined by their base point, instead they depend in general on whole prehistories of the base points. Hence our methods are different and are based on using a sequence of inverse pressures for the iterates off, in order to give upper and lower estimates of the stable dimension. We obtain an estimate of the oscillation of the stable dimension on Λ. When each pointxfrom Λ has the same numberd′of preimages in Λ, then we show thatδs(x)is independent of x; in factδs(x)is shown to be equal in this case with the unique zero of the mapt → P(tϕs−log d′). We also prove the Lipschitz continuity of the stable vector spaces over Λ; this proof is again different than the one for diffeomorphisms (however, the unstable distribution is not always Lipschitz for conformal non-invertible maps). In the end we include the corresponding results for a real conformal setting.

Arc-smooth generalized universal covering spaces

Let Y be a path-connected subset of a CAT(0) space Z, allowing for a map f : Y → X to a 1-dimensional separable metric space X, such that the nontrivial point preimages of f form a null sequence of convex subsets of Z. Such Y need not be homotopy equivalent to a 1-dimensional space. We prove that Y admits a generalized universal covering space, which we equip with an arc-smooth structure by consistently and continuously selecting one tight representative from each path homotopy class of Y. It follows that all homotopy groups of Y vanish in dimensions greater than 1.

Trees and hairs for some hyperbolic entire maps of finite order

Let f be an entire transcendental map of finite order, such that all the singularities of f−1 are contained in a compact subset of the immediate basin B of an attracting fixed point. It is proved that there exist geometric coding trees of preimages of points from B with all branches convergent to points from \({\hat {\mathbb C}}\). This implies that the Riemann map onto B has radial limits everywhere. Moreover, the Julia set of f consists of disjoint curves (hairs) tending to infinity, homeomorphic to a half-line, composed of points with a given symbolic itinerary and attached to the unique point accessible from B (endpoint of the hair). These facts generalize the corresponding results for exponential maps.

Coding trees and boundaries of attracting basins for some entire maps

Let f be an entire transcendental map, such that all the singularities of f−1 are contained in a compact subset of the immediate basin B(z0) of an attracting fixed point z0. We study the structure of the Julia set of f, which is equal to the boundary of B(z0), and the behaviour of the Riemann mapping φ onto B(z0) using the technique of geometric coding trees of preimages of points from B(z0). We show that for a given symbolic itinerary, if codes of the tracts of f are bounded and codes of the fundamental domains grow no faster than the iterates of an exponential function, then there exists a point in the Julia set with this itinerary. Moreover, we determine cluster sets for φ and show that φ has an unrestricted limit equal to ∞ at points of a dense uncountable set in the unit circle.

Equidistribution quantitative des points de petite hauteur sur la droite projective

We introduce a new class of adelic heights on the projective line. We estimate their essential minimum and prove a result of equidistribution (at every place) for points of small height with estimates on the speed of convergence. To each rational function R in one variable and defined over a number field K, is associated a normalized height on the algebraic closure of K. We show that these dynamically defined heights are adelic in our sense, and deduce from this equidistribution results for preimages of points under R at every place of K. Our approach follows that of Bilu, and relies on potential theory in the complex plane, as well as in the Berkovich space associated to the projective line over , for each prime p.

Suites d’Applications Méromorphes Multivaluées et Courants Laminaires

Let Fn: X1 → X2 be a sequence of (multivalued) meromorphic maps between compact Kahler manifolds. We study the asymptotic distribution of preimages of points by Fn and, for multivalued self-maps of a compact Riemann surface, the asymptotic distribution of repelling fixed points.

2-groups and approximate fibrations

This paper strives for instant detection of approximate fibrations. A wide class of homotopy types sustains this effort, in the sense that those maps between manifolds for which all point preimages have the homotopy type of a fixed object in this class necessarily are approximate fibrations. Often, as exemplified here, the fundamental group or merely the first homology group carries the determining feature. The main result promises that the closed manifold N is a codimension-2 fibrator, meaning that it has the desired effect on proper maps from an ( n+2)-manifold M onto a metric space B whose fibers all have the homotopy type of N, if π 1( N) is an Abelian 2-group. Under more restrictive conditions, the same conclusion about N holds if either H 1( N) is a cyclic 2-group or if H 1( N) is an arbitrary finite cyclic group and no element of π 1( N) has finite order.

Dynamique des applications rationnelles des espaces multiprojectifs

We study the dynamics of rational mappings f of C k by compactifying them in multiprojective spaces P n1 P ns. We focus on maps of the surface P 1 P 1 . We follow the ap- proach of (Si 99) and associate to any algebraically stablef an in- variant positive closedN1; 1O current. We then consider the exis- tence of anf invariant measure using the theory of pluripositive currents, and relates it to the measure of Russsakovskii-ShiVman describing the distribution of preimages of points. Our point of view enables us to treat new classes of examples: we consider in particular polynomial skew products with varying degrees, and birational polynomial mappings ofC 2 . We also describe the com- pact convex set off invariant currents for monomial and bira- tional maps of C 2 .