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

2008 ◽  
Vol 60 (3) ◽  
pp. 658-684 ◽  
Author(s):  
Eugen Mihailescu ◽  
Mariusz Urbański
Keyword(s):  
Hausdorff Dimension ◽  
Arbitrary Point ◽  
Base Point ◽  
Topological Pressure ◽  
Axiom A ◽  
Hyperbolic Diffeomorphisms ◽  
Base Points ◽  
Basic Set ◽  
Fixed Positive Number ◽  
Stable Dimension

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.

INVERSE TOPOLOGICAL PRESSURE WITH APPLICATIONS TO HOLOMORPHIC DYNAMICS OF SEVERAL COMPLEX VARIABLES

2004 ◽  
Vol 06 (04) ◽  
pp. 653-679 ◽  
Author(s):  
EUGEN MIHAILESCU ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Technical Condition ◽  
Topological Pressure ◽  
Holomorphic Dynamics ◽  
Critical Set ◽  
Axiom A ◽  
Type Definition ◽  
Basic Set ◽  
Unique Zero ◽  
Stable Dimension ◽  
The One

In this paper, we introduce a few notions of inverse topological pressure [Formula: see text], defined in terms of backward orbits (prehistories) instead of forward orbits. This inverse topological pressure has some properties similar to the regular (forward) pressure but, in general, if the map is not a homeomorphism, they do not coincide. In fact, there are several ways to define inverse topological pressure; for instance, we show that the Bowen type definition coincides with the one using spanning sets. Then we consider the case of a holomorphic map [Formula: see text] which is Axiom A and such that its critical set does not intersect a particular basic set of saddle type Λ. We will prove that, under a technical condition, the Hausdorff dimension of the intersection between the local stable manifold and the basic set is equal to ts, i.e. [Formula: see text], for all points x belonging to Λ. Here ts represents the unique zero of the function t→P-(tϕs), with P- denoting the inverse topological pressure and [Formula: see text], y∈Λ. In general, [Formula: see text] will be estimated above by ts and below by [Formula: see text], where [Formula: see text] is the unique zero of the map t→P_(tϕs). As a corollary we obtain that, if the stable dimension is non-zero, then Λ must be a non-Jordan curve, and also, if f|Λ happens to be a homeomorphism (like in the examples from [13]), then the stable dimension cannot be zero.

Metric properties of some fractal sets and applications of inverse pressure

2009 ◽  
Vol 148 (3) ◽  
pp. 553-572 ◽  
Author(s):  
EUGEN MIHAILESCU
Keyword(s):  
Hausdorff Dimension ◽  
General Setting ◽  
Topological Pressure ◽  
Box Dimension ◽  
Stable Manifolds ◽  
Fractal Sets ◽  
Metric Properties ◽  
Basic Set ◽  
Unstable Set ◽  
Unstable Manifolds

AbstractWe consider iterations of smooth non-invertible maps on manifolds of real dimension 4, which are hyperbolic, conformal on stable manifolds and finite-to-one on basic sets. The dynamics of non-invertible maps can be very different than the one of diffeomorphisms, as was shown for example in [4,7,12,17,19], etc. In [13] we introduced a notion of inverse topological pressureP−which can be used for estimates of the stable dimension δs(x) (i.e the Hausdorff dimension of the intersection between the local stable manifoldWsr(x) and the basic set Λ,x∈ Λ). In [10] it is shown that the usual Bowen equation is not always true in the case of non-invertible maps. By using the notion of inverse pressureP−, we showed in [13] that δs(x) ≤ts(ϵ), wherets(ϵ) is the unique zero of the functiont→P−(tφs, ϵ), for φs(y):= log|Dfs(y)|,y∈ Λ and ϵ > 0 small. In this paper we prove that if Λ is not a repellor, thents(ϵ) < 2 for any ϵ > 0 small enough. In [11] we showed that a holomorphic s-hyperbolic map on2has a global unstable set with empty interior. Here we show in a more general setting than in [11], that the Hausdorff dimension of the global unstable setWu() is strictly less than 4 under some technical derivative condition. In the non-invertible case we may have (infinitely) many unstable manifolds going through a point in Λ, and the number of preimages belonging to Λ may vary. In [17], Qian and Zhang studied the case of attractors for non-invertible maps and gave a condition for a basic set to be an attractor in terms of the pressure of the unstable potential. In our case the situation is different, since the local unstable manifolds may intersect both inside and outside Λ and they do not form a foliation like the stable manifolds. We prove here that the upper box dimension ofWsr(x) ∩ Λ is less thants(ϵ) for any pointx∈ Λ. We give then an estimate of the Hausdorff dimension ofWu() by a different technique, using the Holder continuity of the unstable manifolds with respect to their prehistories.

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

2018 ◽  
Vol 40 (4) ◽  
pp. 1083-1107
Author(s):  
WEISHENG WU
Keyword(s):  
Hausdorff Dimension ◽  
Tangent Space ◽  
Anosov Diffeomorphism ◽  
Hyperbolic Diffeomorphisms ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms ◽  
Dense Orbits ◽  
Set Of Points ◽  
Partially Hyperbolic Diffeomorphism ◽  
Stable Subspace

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.

scholarly journals Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180761 ◽  
Author(s):  
Matteo Petrera ◽  
Jennifer Smirin ◽  
Yuri B. Suris
Keyword(s):  
Vector Field ◽  
Base Point ◽  
Hamiltonian Vector ◽  
Geometric Condition ◽  
Hamiltonian Vector Field ◽  
Cubic Curves ◽  
Birational Map ◽  
Base Points ◽  
Finite Base ◽  
Pass Through

Kahan discretization is applicable to any quadratic vector field and produces a birational map which approximates the shift along the phase flow. For a planar quadratic canonical Hamiltonian vector field, this map is known to be integrable and to preserve a pencil of cubic curves. Generically, the nine base points of this pencil include three points at infinity (corresponding to the asymptotic directions of cubic curves) and six finite points lying on a conic. We show that the Kahan discretization map can be represented in six different ways as a composition of two Manin involutions, corresponding to an infinite base point and to a finite base point. As a consequence, the finite base points can be ordered so that the resulting hexagon has three pairs of parallel sides which pass through the three base points at infinity. Moreover, this geometric condition on the base points turns out to be characteristic: if it is satisfied, then the cubic curves of the corresponding pencil are invariant under the Kahan discretization of a planar quadratic canonical Hamiltonian vector field.

On the canonical systems of certain algebraic surfaces

1937 ◽  
Vol 33 (3) ◽  
pp. 311-314
Author(s):  
D. Pedoe
Keyword(s):  
Algebraic Surface ◽  
Base Point ◽  
Canonical System ◽  
Algebraic Surfaces ◽  
Simple Point ◽  
Canonical Systems ◽  
Base Points ◽  
Complete Linear System ◽  
Pass Through ◽  
Simple Points

A complete linear system of curves on an algebraic surface may have assigned base points. The canonical system, from its definition, has no assigned base points at simple points of the surface. But we may construct surfaces on which, all the same, the canonical system has “accidental base points” at simple points of the surface. The classical example, due to Castelnuovo, is a quintic surface with two tacnodes. On this surface the canonical system is cut out by the planes passing through the two tacnodes. These planes also pass through the simple point in which the join of the two tacnodes meets the surface again. This point is the accidental base point of the canonical system on the quintic surface.

scholarly journals A relation between Lyapunov exponents, Hausdorff dimension and entropy

1981 ◽  
Vol 1 (4) ◽  
pp. 451-459 ◽  
Author(s):  
Anthony Manning
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Lyapunov Exponents ◽  
Unstable Manifold ◽  
Positive Lyapunov Exponent ◽  
Axiom A ◽  
Generic Points

AbstractFor an Axiom A diffeomorphism of a surface with an ergodic invariant measure we prove that the entropy is the product of the positive Lyapunov exponent and the Hausdorff dimension of the set of generic points in an unstable manifold.

scholarly journals Bowen’s equation in the non-uniform setting

2010 ◽  
Vol 31 (4) ◽  
pp. 1163-1182 ◽  
Author(s):  
VAUGHN CLIMENHAGA
Keyword(s):  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Metric Space ◽  
Periodic Points ◽  
Topological Pressure ◽  
Conformal Map ◽  
Arbitrary Subset ◽  
Compact Metric Space ◽  
Symbolic Space ◽  
Contraction Condition

AbstractWe show that Bowen’s equation, which characterizes the Hausdorff dimension of certain sets in terms of the topological pressure of an expanding conformal map, applies in greater generality than has been heretofore established. In particular, we consider an arbitrary subset Z of a compact metric space and require only that the lower Lyapunov exponents be positive on Z, together with a tempered contraction condition. Among other things, this allows us to compute the dimension spectrum for Lyapunov exponents for maps with parabolic periodic points, and to relate the Hausdorff dimension to the topological entropy for arbitrary subsets of symbolic space with the appropriate metric.

scholarly journals Homology for one-dimensional solenoids

2017 ◽  
Vol 121 (2) ◽  
pp. 219 ◽  
Author(s):  
Massoud Amini ◽  
Ian F. Putnam ◽  
Sarah Saeidi Gholikandi
Keyword(s):  
Finite Type ◽  
Homology Theory ◽  
One Dimensional ◽  
Axiom A ◽  
Group Invariant ◽  
Dimension Group ◽  
Basic Set ◽  
The One ◽  
Topological Dynamical Systems ◽  
Smale Spaces

Smale spaces are a particular class of hyperbolic topological dynamical systems, defined by David Ruelle. The definition was introduced to give an axiomatic description of the dynamical properties of Smale's Axiom A systems when restricted to a basic set. They include Anosov diffeomeorphisms, shifts of finite type and various solenoids constructed by R. F. Williams. The second author constructed a homology theory for Smale spaces which is based on (and extends) Krieger's dimension group invariant for shifts of finite type. In this paper, we compute this homology for the one-dimensional generalized solenoids of R. F. Williams.

scholarly journals Base Point Split Algorithm for Generating Polygon Skeleton Lines on the Example of Lakes

2020 ◽  
Vol 9 (11) ◽  
pp. 680
Author(s):  
Elżbieta Lewandowicz ◽  
Paweł Flisek
Keyword(s):  
Base Point ◽  
Vector Data ◽  
Triangulated Irregular Network ◽  
Base Points ◽  
Complex Polygon ◽  
Selection Of

This article presents the Base Point Split (BPSplit) algorithm to generate a complex polygon skeleton based on sets of vector data describing lakes and rivers. A key feature of the BPSplit algorithm is that it is dependent on base points representing the source or mouth of a river or a stream. The input values of base points determine the shape of the resulting skeleton of complex polygons. Various skeletons can be generated with the use of different base points. Base points are applied to divide complex polygon boundaries into segments. Segmentation supports the selection of triangulated irregular network (TIN) edges inside complex polygons. The midpoints of the selected TIN edges constitute a basis for generating a skeleton. The algorithm handles complex polygons with numerous holes, and it accounts for all holes. This article proposes a method for modifying a complex skeleton with numerous holes. In the discussed approach, skeleton edges that do not meet the preset criteria (e.g., that the skeleton is to be located between holes in the center of the polygon) are automatically removed. An algorithm for smoothing zigzag lines was proposed.

The ambient structure of basic sets

1995 ◽  
Vol 15 (6) ◽  
pp. 1183-1188
Author(s):  
A. Löffler
Keyword(s):  
Riemannian Manifold ◽  
Compact Riemannian Manifold ◽  
Product Structure ◽  
Axiom A ◽  
Basic Set ◽  
Local Product ◽  
Basic Sets

AbstractLet Λ be a basic set of an Axiom A diffeomorphism of a compact Riemannian manifold M without boundary. If ε is small enough one can find by local product structure that for x ε Λ there is a neighborhood V(x) in M such that V ∩ Λ is homeomorphic to . The author proves that this homeomorphism can be extended to a homeomorphism of V onto .

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