Download A homology theory for Smale spaces by Ian F. Putnam PDF

By Ian F. Putnam

The writer develops a homology conception for Smale areas, which come with the fundamentals units for an Axiom A diffeomorphism. it's in keeping with parts. the 1st is a far better model of Bowen's end result that each such process is similar to a shift of finite style lower than a finite-to-one issue map. the second one is Krieger's size team invariant for shifts of finite sort. He proves a Lefschetz formulation which relates the variety of periodic issues of the process for a given interval to track information from the motion of the dynamics at the homology teams. The lifestyles of this type of concept used to be proposed by way of Bowen within the Nineteen Seventies

Show description

Read or Download A homology theory for Smale spaces PDF

Best topology books

Art Meets Mathematics in the Fourth Dimension (2nd Edition)

To work out items that dwell within the fourth size we people would have to upload a fourth measurement to our three-d imaginative and prescient. An instance of such an item that lives within the fourth size is a hyper-sphere or “3-sphere. ” the search to visualize the elusive 3-sphere has deep historic roots: medieval poet Dante Alighieri used a 3-sphere to exhibit his allegorical imaginative and prescient of the Christian afterlife in his Divine Comedy.

Algebraic L-theory and topological manifolds

This booklet offers the definitive account of the purposes of this algebra to the surgical procedure category of topological manifolds. The critical result's the id of a manifold constitution within the homotopy form of a Poincaré duality area with an area quadratic constitution within the chain homotopy form of the common disguise.

Differential Forms in Algebraic Topology

The guideline during this publication is to exploit differential varieties as an reduction in exploring a few of the much less digestible elements of algebraic topology. Accord­ ingly, we flow essentially within the realm of tender manifolds and use the de Rham thought as a prototype of all of cohomology. For functions to homotopy idea we additionally speak about when it comes to analogy cohomology with arbitrary coefficients.

Additional info for A homology theory for Smale spaces

Example text

Let (X, ϕ) and (Y, ψ) be Smale spaces and let π : (Y, ψ) → (X, ϕ) be a map. We say that π is s-bijective (or u-bijective) if, for any y in Y , its restriction to Y s (y) (or Y u (y), respectively) is a bijection to X s (π(y)) (or X u (π(y)), respectively). It is relatively easy to find an example of a map which is s-resolving, but not s-bijective and we will give one in a moment. However, one important distinction between the two cases should be pointed out at once. The image of a Smale space under an s-resolving map is not necessarily a Smale space.

Let π : Y → X be a continuous map and let x0 be in X with π −1 {x0 } = {y1 , y2 , . . , yN } finite. For any > 0, there exists δ > 0 such that π −1 (X(x0 , δ)) ⊂ ∪N n=1 Y (yn , ). Proof. If there is no such δ, we may construct a sequence xk , k ≥ 1 in X converging to x0 and a sequence y k , k ≥ 1 with π(y k ) = xk and y k not in ∪N n=1 Y (yn , ). Passing to a convergent subsequence of the y k , let y be the limit point. Then y is k k not in ∪N n=1 Y (yn , ), since that set is open, while π(y) = limk π(y ) = limk x = x0 .

Similarly, X s (x0 ) = ∪l≥0 ϕ−l (X s (x0 , δ)) and the topology is the inductive limit topology. It follows at once that π is a homeomorphism from the former to the latter. Now we turn to arbitrary point y in Y and x = π(y) in X and show that π : Y s (y) → X s (x) is onto. We choose x0 and {y1 , . . , yN } to be periodic points as above so that π : Y s (yn ) → X s (x0 ) are homeomorphisms. By replacing x0 by another point in its orbit (which will satisfy the same condition), we may assume that x is in the closure of X s (x0 ).

Download PDF sample

Rated 4.46 of 5 – based on 24 votes