Expanding maps on compact metric spaces

Reddy, William L.

doi:10.1016/0166-8641(82)90040-2

Cited by 44 publications

(19 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then there exist a metric p for X and constants ô > 0 and X > 1 such that if p(x, y) < ô then p(f(x), f(y)) > kp(x, y). This is proved in the same way as in [10] (notice that / is not always surjective). For e > 0 and x G X, let Uf(x) = {y e X: p(x, y) < e} and denote by Cf(x) the connected component of x in U((x).…”

mentioning

confidence: 63%

Nonexistence of positively expansive maps on compact connected manifolds with boundary

Hiraide¹

1990

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

mentioning

confidence: 63%

Nonexistence of positively expansive maps on compact connected manifolds with boundary

Hiraide¹

1990

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…For a proof, see [21]. In the proof, compactness is used only to obtain (1.1) and not used in the other place except (1.1).…”

Section: In Particular DL Is Uniformly Equivalent To Dmentioning

confidence: 97%

“…The following was proved in [5]: an open map of a compact metric space is positively expansive (this notion was given by WILLIAMS [27] and EISENBERG [9]) if and only if it is expanding in the sense of Ruelle. Further REDDV showed in [21] that local expansion exploited by ROSENHOLTZ [24], short path expanding and positive expansivity could be mutually equivalent on compact path connected metric spaces.…”

Section: Introductionmentioning

confidence: 98%

Expanding maps of solenoids

Aoki

1988

Monatshefte f�r Mathematik

View full text Add to dashboard Cite

Abstract. We prove that compact metric groups which admit expanding maps must be solenoidal groups, and that every expanding map on a solenoidal group is topologically conjugate to a positively expansive group endomorphism. This first was studied by Shub for expanding differentiable maps of tori and by Manning for Anosov diffeomorphisms of tori. IntroductionA part of a general theory of expanding maps was developed by DUVAL and HuscH [6] and COVEN and REDDY [5] in accordance with the theory of expandiag differentiable maps, which was worked out by SHUB [26] as a specM example of Anosov endomorphisms. Anosov endomorphisms were studied by MA~E and PUSH [17] and PRZYTYCKI [19].In order to capture Shub's notion of expanding maps in a topological setting, a notion of short path expanding maps was given in [6]. After that RUELLE introduced in [22] a topological notion of expanding maps sufficient to recover the ergodic properties of smooth expanding maps. The following was proved in [5]: an open map of a compact metric space is positively expansive (this notion was given by WILLIAMS [27] and EISENBERG [9]) if and only if it is expanding in the sense of Ruelle. Further REDDV showed in [21] that local expansion exploited by ROSENHOLTZ [24], short path expanding and positive expansivity could be mutually equivalent on compact path connected metric spaces.Let M be a branched manifold (see WILLIAMS [29,30] for the definition). Every torus is an example of a branched manifold that has no branch points (every torus dealt with here is finite-dimensional).For a continuous map g: M-o M, we let Mg = {(xn): x,~M and g(x,+l) = x, for all n >/0}, g((x~)) = (g(x,)) and :r((x,)) = x 0. Then g is called the shift map determined by g and :r o g = g o ~r holds. Remark

show abstract

“…We say that f is positively expansive [12] if there is an admissible metric d for X and a positive number c > 0 such that if x, y ∈ X and x = y, then there is a natural number n 0 such that d( f n (x), f n (y)) > c. Note that this property is independent of the choice of metrics for X . We say that f is a Ruelle expanding map [13] if f is positively expansive and an open onto map.…”

Section: Introductionmentioning

confidence: 95%

Fractal metrics of Ruelle expanding maps and expanding ratios

Fujita

Kato

Matsumoto

2010

Topology and its Applications

View full text Add to dashboard Cite

In the theory of dynamical systems, it is well known that if f : X → X is a surjective equicontinuous map of a compactum X, then there is an admissible metric d for X such that f : (X, d) → (X, d) is an isometry. In Reddy (1982) [12], Reddy proved that if f : X → X is a positively expansive map of a compactum X, then f expands small distances. In this paper, we will study the similar properties of Ruelle expanding maps and admissible metrics. By use of the construction of the Alexandroff-Urysohn's metrization theorem we prove the following theorem which is a more precise result in case of Ruelle expanding maps (= positively expansive open maps): If f : X → X is a Ruelle expanding map of a compactum X and any positive number s > 1, then there exist an admissible metric d for X and positive numbers > 0, λ (1 < λ < s) such that if x, y ∈ X and d(x, y), y). For a case of graphs, we prove that if f : X → X is a positively expansive map of a graph X (= 1-dimensional compact polyhedron), then the same conclusion holds. In these cases, the metrics d satisfy the following equality:where dim H (X, d), D d (X) and D d (X) denote the Hausdorff dimension, the lower boxcounting dimension and the upper box-counting dimension of the compact metric space (X, d) respectively, and h( f ) is the topological entropy of f . This implies that such a metric d is a "fractal" metric for X. In fact, we can consider that the compact metric space (X, d) has some sort of local self-similarity with respect to the inverse f −1 of f and the similarity ratio 1/λ. Also, we prove that if f : X → X is an expanding homeomorphism of a noncompact metric space X, then there exist an admissible metric d for X and a positive number λ > 1 such that if x, y ∈ X, then d( f (x), f (y)) = λd(x, y).

show abstract

Expanding maps on compact metric spaces

Cited by 44 publications

References 8 publications

Nonexistence of positively expansive maps on compact connected manifolds with boundary

Nonexistence of positively expansive maps on compact connected manifolds with boundary

Expanding maps of solenoids

Fractal metrics of Ruelle expanding maps and expanding ratios

Contact Info

Product

Resources

About