We show that every transitive dynamically coherent partially hyperbolic diffeomorphism with a one-dimensional center foliation $\W^c$ satisfying that $f(W)=W$ for every leaf $W\in \W^c$ is a discretized Anosov flow.
We show a flexibility result in the context of generalized entropy. The space of dynamical systems we work with is, homeomorphisms on the sphere whose non-wandering set consist in only one fixed point.
Let (X,Z) be a minimal dynamical system on a compact metric X and k an integer such that mdim X< k. We show that (X,Z) admits an equivariant embedding in the shift (D^k)^Z where D is a superdendrite.
This note is to show that the position-space embedding in \cite{ESP2021embedding} in the position and occupation bases can be obtained by considering the dynamics of Dirac delta function $$δ(\mathbf{x}- \mathbf{z}(t)) = δ(x_1-z_1(t))\cdots δ(x_d-z_d(t)),$$ where $\mathbf{z}(t)\in \mathbb{R}^d$ is the solution of a nonlinear dynamical system and $\mathbf{x}\in \mathbb{R}^d$ is a variable in the position space.
Here we study the behaviour of the horocyclic orbit of a vector on the unit tangent bundle of a geometrically infinite surface with variable negative curvature, when the corresponding geodesic ray is almost minimizing and the injectivity radius is finite.
In this article, for degree $d\geq 1$, we construct an embedding $Φ_d $ of the connectedness locus $\mathcal{M}_{d+1}$ of the polynomials $z^{d+1}+c$ into the connectedness locus of degree $2d+1$ bicritical odd polynomials.
We propose a criterion, referred to as order-n transversality, for transitivity of area preserving partially hyperbolic endomorphisms. Besides, we also give a further answer to the Gan's problem, as proposed in the work of Baolin He.
This article gives a precise description of the Fatou sets and Julia sets of matrix-valued polynomials in $\mathcal{M}(2,\mathbb{C})$ in terms of the corresponding polynomials in $\mathbb{C}$. Further, we construct Green functions and Böttcher-type functions for these matrix-valued polynomials.
The local return rates have been introduced by Hirata, Saussol and Vaienti \cite{HSV99} as a tool for the study of the asymptotic distribution of the return times to cylinders. We give formulas for these rates in Sturmian subshifts.
We prove that for every hyperbolic component of the Mandelbrot set, any two limbs with equal denominators are homeomorphic so that the homeomorphism preserves periods of hyperbolic components. This settles a conjecture on the Mandelbrot set that goes back to 1994.
We consider a homeomorphism on a totally disconnected, compact metric space and define a binary relation on the family of clopen subsets. We will show that the comparability of any clopen sets with respect to the relation is equivalent to the unique ergodicity of the homeomorphism.
We consider the Hausdorff measure of Julia sets and escaping sets of exponential maps with respect to certain gauge functions. We give conditions on the growth of the gauge function which imply that the measure is zero or infinity, respectively.
We present an example of a fibred quadratic polynomial admitting an attracting invariant 2-curve. By an unfolding construction we obtain an example of a fibred quadratic polynomial admitting two attracting invariant curves. This phenomena can not occur in the non-fibred setting.
For each piecewise linear Lorenz map that expand on average, we show that it admits a dichotomy: it is either periodic renormalizable or prime. As a result, such a map is conjugate to a $β$-transformation.