Triangular Maps Research Articles

Full analogy of Sharkovsky's theorem for lower semicontinuous maps

A full analogy of the celebrated Sharkovsky cycle coexistence theorem is established for lower semicontinuous (multivalued) maps on metrizable linear continua. This result is further extended to triangular maps.

Estimates for Sobolev norms of triangular mappings

An increasing triangular mapping T on the n-dimensional cube Θ = [0, 1]n transforming a measure μ to a measure ν is considered, where μ and ν are absolutely continuous Borel probability measures having densities ρμ and ρν. It is shown that if there exist positive constants ɛ and M such that ɛ 1 such that p = αβ(n − 1)−1 (α + β)−1 > 1 and ρμ ∈ W1,α (Θ), ρν ∈ W1,β) (Θ), where W1,q denotes a Sobolev class, then the mapping T belongs to the class W1,p (Θ).

A Fixed Point Theorem for Triangular Mappings

In this paper, we give a xed point theorem for triangular mappings whose derivatives are bounded.

Subadditive pressure for triangular maps

We investigate properties of the zero of the subadditive pressure used by Falconer, Barreira and Zhang to estimate the box and Hausdorff dimension of a non-conformal repeller. In the conformal case, and in Falconer's 1-bunched non-conformal case, the contraction rates satisfy bounded distortion and so this zero is insensitive to where on each cylinder the contraction is evaluated. We study some nonlinear two-dimensional examples which do not satisfy bounded distortion but do exhibit the same insensitivity. Here the contraction rate fails to specify ellipses that can be used to cover cylinders.

Strange distributionally chaotic triangular maps III

In the class T of triangular maps of the square we consider the strongest notion of distributional chaos, DC1, originally introduced by Schweizer and Smítal [Trans Amer Math Soc 1994;344:737–854] for continuous maps of the interval. We show that a map F ∈ T is DC1 if F has a periodic orbit with period ≠ 2 n , for any n ⩾ 0. Consequently, a map in T is DC1 if it has a homoclinic trajectory. This result is important since in general systems like T , positive topological entropy itself does not imply DC1. It contributes to the solution of a long-standing open problem of A. N. Sharkovsky concerning classification of triangular maps of the square.

Li–Yorke sensitive minimal maps

Let Q be the Cantor middle third set and S the circle, and let τ : Q → Q be an adding machine (i.e. odometer). Let X = Q × S be equipped with (a metric equivalent to) the Euclidean metric. We show that there are continuous triangular maps Fi: X → X, Fi: (x, y) ↦ (τ(x), gi(x, y)), i = 1, 2, with the following properties.(i) Both (X, F1) and (X, F2) are minimal systems, without weak mixing factors (i.e. neither of them is semiconjugate to a weak mixing system).(ii) (X, F1) is spatio-temporally chaotic but not Li–Yorke sensitive.(iii) (X, F2) is Li–Yorke sensitive.This disproves conjectures of Akin and Kolyada (2003 Nonlinearity 16 1421–33.)

Convergence of Triangular Transformations of Measures

We prove that if a Borel probability measure $\mu$ on a countable product of Souslin spaces satisfies a certain condition of atomlessness, then for every Borel probability measure $\nu$ on this product, there exists a triangular mapping $T_{\mu,\nu}$ that takes $\mu$ to $\nu$. It is shown that in the case of metrizable spaces one can choose triangular mappings in such a way that convergence in variation of measures $\mu_n$ to $\mu$ and of measures $\nu_n$ to $\nu$ implies convergence of the mappings $T_{\mu_n,\nu_n}$ to $T_{\mu,\nu}$ in measure $\mu$.

Recurrence equals uniform recurrence does not imply zero entropy for triangular maps of the square

In 1992, S. Kolyada asked the question whether for triangular maps of the square zero topological entropy is equivalent to the fact that every recurrent point is uniformly recurrent. One of the implications was answered negatively by G.-L. Forti, L. Paganoni and J. Smital in 1995. They showed that a zero entropy triangular map may have a recurrent point which is not uniformly recurrent. In this paper we show that neither the converse implication is true by constructing a triangular map of the square with positive topological entropy and with every chain recurrent point uniformly recurrent. In fact we first construct an appropriate minimal positive entropy system whose phase space is a nonhomogeneous subset of the square and then we extend it to a triangular map with required properties in the square.

Nonlinear Transformations of Convex Measures

Given a uniformly convex measure $\mu$ on ${\bf R}^\infty$ that is equivalent to its translation to the vector $(1,0,0,\ldots)$ and a probability measure $\nu$ that is absolutely continuous with respect to $\mu$, we show that there is a Borel mapping $T=(T_k)_{k=1}^\infty$ of ${\bf R}^\infty$ transforming $\mu$ into $\nu$ and having the form $T(x)=x+F(x)$, where F has values in $l^2$. Moreover, if $\mu$ is a product-measure, then T can be chosen triangular in the sense that each component $T_k$ is a function of $x_1,\ldots,x_k$. In addition, for any uniformly convex measure $\mu$ on ${\bf R}^\infty$ and any probability measure $\nu$ with finite entropy Ent$_\mu(\nu)$ with respect to $\mu$, the canonical triangular mapping $T=I+F$ transforming $\mu$ into $\nu$ satisfies the inequality $\|F\|_{L^2(\mu,l^2)}^2\le C(\mu)$Ent$_\mu (\nu)$. Several inverse assertions are proved. Our results apply, in particular, to the standard Gaussian product-measure. As an application we obtain a new sufficient condition for ...

On topological entropy of transitive triangular maps

In the paper of Alsedà, Kolyada, Llibre and Snoha [L. Alsedà, S.F. Kolyada, J. Llibre, L'. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc. 351 (1999) 1551–1573] there was—among others—proved that a nonminimal continuous transitive map f of a compact metric space ( X , ρ ) can be extended to a triangular map F on X × I (i.e., f is the base for F) in such a way that F is transitive and has the same entropy as f. The presented paper shows that under certain conditions the extension of minimal maps is guaranteed, too: Let ( X , f ) be a solenoidal dynamical system. Then there exist a transitive triangular map F such that h ( F ) = h ( f ) .

Strange distributionally chaotic triangular maps

Strange distributionally chaotic triangular maps

A triangular map with homoclinic orbits and no infinite ω-limit set containing periodic points

Recently, Forti, Paganoni and Smítal constructed an example of a triangular map of the unite square, F ( x , y ) = ( f ( x ) , g ( x , y ) ) , possessing periodic orbits of all periods and such that no infinite ω-limit set of F contains a periodic point. In this note we show that the above quoted map F has a homoclinic orbit. As a consequence, we answer in the negative the problem presented by A.N. Sharkovsky in the eighties whether, for a triangular map of the square, existence of a homoclinic orbit implies the existence of an infinite ω-limit set containing a periodic point. It is well known that, for a continuous map of the interval, the answer is positive.

The triangular maps with closed sets of periodic points

In a recent paper we provided a characterization of triangular maps of the square, i.e., maps given by F ( x , y ) = ( f ( x ) , g x ( y ) ) , satisfying condition (P1) that any chain recurrent point is periodic. For continuous maps of the interval, there is a list of 18 other conditions equivalent to (P1), including (P2) that there is no infinite ω-limit set, (P3) that the set of periodic points is closed and (P4) that any regularly recurrent point is periodic, for instance. We provide an almost complete classification among these conditions for triangular maps, improve a result given by C. Arteaga [C. Arteaga, Smooth triangular maps of the square with closed set of periodic points, J. Math. Anal. Appl. 196 (1995) 987–997] and state an open problem concerning minimal sets of the triangular maps. The paper solves partially a problem formulated by A.N. Sharkovsky in the eighties. The mentioned open problem, the validity of (P4) ⇒ (P3), is related to the question whether some regularly recurrent point lies in the fibres over an f-minimal set possessing a regularly recurrent point. We answered this question in the positive for triangular maps with nondecreasing fiber maps. Consequently, the classification is completed for monotone triangular maps.

Triangular transformations of measures

A new identity for the entropy of a non-linear image of a measure on is obtained, which yields the well-known Talagrand's inequality. Triangular mappings on and are studied, that is, mappings such that the th coordinate function depends only on the variables . With the help of such mappings the well-known open problem on the representability of each probability measure that is absolutely continuous with respect to a Gaussian measure on an infinite dimensional space as the image of under a map of the form where takes values in the Cameron-Martin space of the measure is solved in the affirmative. As an application, a generalized logarithmic Sobolev inequality is also proved.

Triangular maps with all periods and no infinite ω-limit set containing periodic points

Our main result is an example of a triangular map of the unite square, F ( x , y ) = ( f ( x ) , g x ( y ) ) , possessing periodic orbits of all periods and such that no infinite ω-limit set of F contains a periodic point. We also show that there is a triangular map F of type 2 ∞ monotone on the fibres such that any recurrent point of F is uniformly recurrent and F restricted to the set of its recurrent points is chaotic in the sense of Li and Yorke. For a continuous map φ of the interval there is a long list of more than 50 conditions characterizing zero topological entropy, including, e.g., conditions (i) φ is of type 2 ∞ , (ii) every recurrent point of φ is uniformly recurrent, (iii) φ restricted to the set of chain recurrent points is not chaotic in the sense of Li and Yorke, (iv) no infinite ω-limit set contains a cycle. The problem presented by A.N. Sharkovsky in the eighties is to decide which of these conditions remain equivalent in the class of triangular maps. Our second example completes the results obtained, e.g., by Forti et al. (1999), Kočan (2003) and Šindelářová (2003), concerning triangular maps monotone on the fibres. The first example, with a more sophisticated proof, contributes to a more difficult problem of classification of general triangular maps, which is still not completely solved; the main partial results have been obtained by Kolyada (1992).

Triangular Maps Non-Decreasing on the Fibers

There is a list of about 50 properties which characterize continuous maps of the interval with zero topological entropy. Most of them were proved by A.N. Sharkovsky [cf., e.g., Sharkovsky et al., Dynamics of One-Dimensional Mappings, Kluwer 1997]. It is also well known that only a few of these properties remain equivalent for continuous maps of the square. Recall, e.g., the famous Kolyada's example of a triangular map of type $2^\infty$ with positive topological entropy. In 1989 Sharkovsky formulated the problem to classify these conditions in a special case of triangular maps of the square. The present paper is a step toward the solution. In particular, we give a classification of 23 conditions in the case of triangular maps of the square which are non-decreasing on the fibers. We show that the weakest is"no homoclinic trajectory", the two strongest, mutually incomparable, are "map restricted to the set of chain recurrent points is not Li and Yorke chaotic" and "every $\omega$-limit set contains a unique minimal set"

Power Linear Keller Maps of Nilpotency Index Two#

Let K be a field. A polynomial map K n → K n is power linear of degree d if it is of the form X + H = (X 1 + H 1,…, X n + H n ), where is a linear form in X 1,…, X n and d > 1. In this paper it is proved that if K is a field of characteristic not dividing d and F is a power linear polynomial map of degree d with nilpotency index two, i.e., (JH)2 = 0, then there exists a linear invertible polynomial map Φ such that Φ−1 FΦ is a triangular power linear map of degree d.

From a Floyd–Auslander minimal system to an odd triangular map

In 1989 A.N. Sharkovsky asked the question which of the properties characterizing continuous maps of the interval with zero topological entropy remain equivalent for triangular maps of the square. The problem is difficult and only partial results are known. However, in the case of triangular maps with nondecreasing fibres there are only few gaps in a classification (given by Z. Kočan) of a set of 24 of these conditions. In the present paper we remove these gaps by giving an example of a triangular map in the square with the following properties: (1) all fibre maps are nondecreasing, (2) all recurrent points of the map are uniformly recurrent, and (3) the restriction of the map to the set of recurrent points has an uncountable scrambled set (and so is Li–Yorke chaotic). The example is obtained by taking an appropriate Floyd–Auslander minimal system and then taking its appropriate continuous extension to a triangular map of the square.

Complete Coverage Navigation of Cleaning Robots Using Triangular-Cell-Based Map

This paper presents a novel approach for navigation of cleaning robots in an unknown workspace. To do so, we propose a new map representation method as well as a complete coverage navigation method. First, we discuss a triangular cell map representation which makes the cleaning robot navigate with a shorter path and increased flexibility than a rectangular cell map representation. Then, we propose the complete coverage navigation and map construction methods which enable the cleaning robot to navigate the complete workspace without complete information about the environment. Finally, we evaluate the performance of our proposed triangular cell map via the existing distance-transform-based path-planning method comparing it to that of the rectangular cell map. Also, we verify the effectiveness of the proposed methods through computer simulations.

Distributional chaos for triangular maps

In this paper we exhibit a triangular map F of the square with the following properties: (i) F is of type 2 ∞ but has positive topological entropy; we recall that similar example was given by Kolyada in 1992, but our argument is much simpler. (ii) F is distributionally chaotic in the wider sense, but not distributionally chaotic in the sense introduced by Schweizer and Smítal [Trans. Amer. Math. Soc. 344 (1994) 737]. In other words, there are lower and upper distribution functions Φ xy and Φ xy ∗ generated by F such that Φ xy ∗≡1 and Φ xy (0 +)<1, and no distribution functions Φ uv , and Φ uv ∗ such that Φ uv ∗≡1 and Φ uv ( t)=0 whenever 0< t< ϵ, for some ϵ>0. We also show that the two notions of distributional chaos used in the paper, for continuous maps of a compact metric space, are invariants of topological conjugacy.