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On a certain map of a triangle

Grzegorz Świrszcz (1998)

Fundamenta Mathematicae

The paper answers some questions asked by Sharkovski concerning the map F:(u,v) ↦ (u(4-u-v),uv) of the triangle Δ = u,v ≥ 0: u+v ≤ 4. We construct an absolutely continuous σ-finite invariant measure for F. We also prove the following strange phenomenon. The preimages of side I = Δ ∩ v=0 form a dense subset ∪ F - n ( I ) of Δ and there is another dense set Λ consisting of points whose orbits approach the interval I but are not attracted by I.

On a one-dimensional analogue of the Smale horseshoe

Ryszard Rudnicki (1991)

Annales Polonici Mathematici

We construct a transformation T:[0,1] → [0,1] having the following properties: 1) (T,|·|) is completely mixing, where |·| is Lebesgue measure, 2) for every f∈ L¹ with ∫fdx = 1 and φ ∈ C[0,1] we have ∫ φ ( T n x ) f ( x ) d x → ∫ φ d μ , where μ is the cylinder measure on the standard Cantor set, 3) if φ ∈ C[0,1] then n - 1 ∑ i = 0 n - 1 φ ( T i x ) → ∫ φ d μ for Lebesgue-a.e. x.

On approximation of homeomorphisms of a Cantor set

Konstantin Medynets (2007)

Fundamenta Mathematicae

We continue the study of topological properties of the group Homeo(X) of all homeomorphisms of a Cantor set X with respect to the uniform topology τ, which was started by Bezuglyi, Dooley, Kwiatkowski and Medynets. We prove that the set of periodic homeomorphisms is τ-dense in Homeo(X) and deduce from this result that the topological group (Homeo(X),τ) has the Rokhlin property, i.e., there exists a homeomorphism whose conjugacy class is τ-dense in Homeo(X). We also show that for any homeomorphism...

On fixed points of C 1 extensions of expanding maps in the unit disc

Roberto Tauraso (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Using a result due to M. Shub, a theorem about the existence of fixed points inside the unit disc for C 1 extensions of expanding maps defined on the boundary is established. An application to a special class of rational maps on the Riemann sphere and some considerations on ergodic properties of these maps are also made.

On invariant measures for the tend map.

Francesc Bofill (1988)

Stochastica

The bifurcation structure of a one parameter dependent piecewise linear population model is described. An explicit formula is given for the density of the unique invariant absolutely continuous probability measure mub for each parameter value b. The continuity of the map b --> mub is established.

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