This paper constructs a continuous decomposition of the Sierpiński curve into acyclic continua one of which is an arc. This decomposition is then used to construct another continuous decomposition of the Sierpiński curve. The resulting decomposition space is homeomorphic to the continuum obtained from taking the Sierpiński curve and identifying two points on the boundary of one of its complemen...

We consider a family of one-dimensional continuous piecewise smooth maps with monotone increasing and monotone decreasing branches. It is associated with a credit cycle model introduced by Matsuyama, under the assumption of the Cobb-Douglas production function. We offer a detailed analysis of the dynamics of this family. In particular, using the skew tent map as a border collision normal form w...

We introduce a new concept of general $G$-$eta$-monotone operator generalizing the general $(H,eta)$-monotone operator cite{arvar2, arvar1}, general $H-$ monotone operator cite{xiahuang} in Banach spaces, and also generalizing $G$-$eta$-monotone operator cite{zhang}, $(A, eta)$-monotone operator cite{verma2}, $A$-monotone operator cite{verma0}, $(H, eta)$-monotone operator cite{fanghuang}...

We consider the genealogy tree for a critical branching process conditioned on non-extinction. We enumerate vertices in each generation of the tree so that for each two generations one can define a monotone map describing the ancestor–descendant relation between their vertices. We show that under appropriate rescaling this family of monotone maps converges in distribution in a special topology ...

In connection with the Entropy Conjecture it is known that the topological entropy of a continuous graph map is bounded from below by the spectral radius of the induced map on the first homology group. We show that in the case of a piecewise monotone graph map, its topological entropy is equal precisely to the maximum of the mentioned spectral radius and the exponential growth rate of the numbe...

We propose a new approach to the issue of existence and uniqueness of solutions to the Bellman equation, exploiting an emerging class of methods, called monotone map methods, pioneered in the work of Krasnosel’skii (1964) and Krasnosel’skii-Zabreiko (1984). The approach is technically simple and intuitive. It is derived from geometric ideas related to the study of fixed points for monotone conc...

Here x is a periodic configuration variable (usually computed modulo 1), y ∈R is the momentum variable, and the parameter k represents the strength of a nonlinear kick. This map was first proposed in 1968 by Bryan Taylor and then independently obtained by Boris Chirikov to describe the dynamics of magnetic field lines. The standard map and Hénon’s area-preserving quadratic map are extensively s...

We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.

We prove some generalizations of the first Fredholm theorem for Hammerstein operator equations in Banach spaces and study the number of their solutions using a projection like method.The linear part is assumed to be either selfadjoint or nonseladjoint while the nonlinearities are such that the corresponding map is (pseudo) A-proper. In particular, the nonlinearities can be either of monotone ty...

The allowed patterns of a map on a one-dimensional interval are those permutations that are realized by the relative order of the elements in its orbits. The set of allowed patterns is completely determined by the minimal patterns that are not allowed. These are called basic forbidden patterns. In this paper we study basic forbidden patterns of several functions. We show that the logistic map L...