Nonuniformly Expanding 1D Maps

Nonuniformly Expanding 1D Maps

This paper attempts to make accessible a body of ideas surrounding the following result: Typical families of (possibly multi-model) 1-dimensional maps passing through “Misiurewicz points” have invariant densities for positive measure sets of parameters. The simplest paradigms of chaotic behavior in dynamical systems are found in uniformly expanding and uniformly hyperbolic (or Anosov) maps. All...

Nonuniformly Expanding 1D Maps With Logarithmic Singularity

Let fa,L : R→ R be such that (1) fa,L : θ 7→ θ + a + L ln |Φ(θ)| where a ∈ [0, 1], L ∈ R are real parameters and Φ(θ) is such that Φ(θ + 1) = Φ(θ). We assume that Φ(θ) is a Morse function and the graph of y = Φ(θ) is transversal to the θ-axis. The functions fa,L induce a two parameter family of 1D maps from S1 to S1 where S1 = R/Z is the unit circle. In this paper we prove that there exists an ...

Infinitely Many Sinks around Nonuniformly Expanding 1d Maps

In this paper we study families of multi-modal 1D maps following the setting of Wang and Young [20]. Under a mild combinatoric assumption, we prove that for generic one parameter families of 1D maps containing a Misiurewicz map, parameters of non-uniformly expanding maps, the measure abundance of which was proved previously in [20], are accumulation points of paramaters admitting super-stable p...

Stochastic-like Behaviour in Nonuniformly Expanding Maps

Topological Entropy for Nonuniformly Continuous Maps

The literature contains several extensions of the standard definitions of topological entropy for a continuous self-map f : X→X from the case when X is a compact metric space to the case when X is allowed to be noncompact. These extensions all require the space X to be totally bounded, or equivalently to have a compact completion, and are invariants of uniform conjugacy. When themap f is unifor...