Related papers: Nonstandard Expansiveness

The dynamical degrees of a rational map $f:X\dashrightarrow X$ are fundamental invariants describing the rate of growth of the action of iterates of $f$ on the cohomology of $X$. When $f$ has nonempty indeterminacy set, these quantities can…

Let $(X,T)$ be a topological dynamical system, $n\geq 2$ and $\mathcal{F}$ be a Furstenberg family of subsets of $\mathbb{Z}_+$. $(X,T)$ is called broken $\mathcal{F}$-$n$-sensitive if there exist $\delta>0$ and $F\in\mathcal{F}$ such that…

We investigate symmetry properties of positive solutions for fully nonlinear uniformly elliptic systems, such as $$ F_i \,(x,Du_i,D^2u_i) +f_i \,(x,u_1, \ldots , u_n,Du_i)=0, \;\; 1 \leq i \leq n, $$ in a bounded domain $\Omega$ in…

For a digraph D and three parameters x, y, z in {0,1,+,-} we define the digraph D^(x,y,z) and call it the (x,y,z)-transformation of D. We show that for every r-regular digraph D the adjacency characteristic polynomial A(t, D^(x,y,z)) of…

A length-$n$ random sequence $X_1,\ldots,X_n$ in a space $S$ is finitely exchangeable if its distribution is invariant under all $n!$ permutations of coordinates. Given $N > n$, we study the extendibility problem: when is it the case that…

Let $f$ be a postcritically finite rational map. We prove that, as $n$ large enough, there exists an $f^n$-invariant (finite connected) graph on $\widehat{\mathbb{C}}$ such that it contains the postcritical set of $f$.

In this work we study the class of mostly expanding partially hyperbolic diffeomorphisms. We prove that such class is $C^r$-open, $r>1$, among the partially hyperbolic diffeomorphisms (in the narrow sense) and we prove that the mostly…

\textit{Non-statistical dynamics} are those for which a set of points with positive measure (w.r.t. a reference probability measure which is in most examples the Lebesgue on a manifold) do not have a convergent sequence of empirical…

Let X_i, i\in N, be i.i.d. B-valued random variables, where B is a real separable Banach space. Let \Phi be a smooth enough mapping from B into R. An asymptotic evaluation of Z_n=E(\exp (n\Phi (\sum_{i=1}^nX_i/n))), up to a factor (1+o(1)),…

This is a revision of the paper archived previously on August 22, 2002. It corrects a mistake in Sec. 8 concerning eccentricities of graphs. From any given sequence of finite or infinite graphs, a nonstandard graph is constructed. The…

We consider linear dynamical systems with a structure of a multigraph. The vertices are associated to linear spaces and the edges correspond to linear maps between those spaces. We analyse the asymptotic growth of trajectories (associated…

This note has several aims. Firstly, it portrays a non-standard analysis as a functor, namely a functor * that maps any set A to the set *A of its non-standard elements. That functor, from the category of sets to itself, is postulated to be…

Let $(X,d,f)$ be a dynamical system, where $(X,d)$ is a compact metric space and $f:X\rightarrow X$ is a continuous map. Using the concepts of \textit{g-almost product property} and \textit{uniform separation property} introduced by Pfister…

In this paper we study the dynamics of a general non-autonomous dynamical system generated by a family of continuous self maps on a compact space $X$. We derive necessary and sufficient conditions for the system to exhibit complex dynamical…

Let $f: M \to M$ be a partially hyperbolic diffeomorphism with conformality on unstable manifolds. Consider a set of points with nondense forward orbit: $E(f, y) := \{ z\in M: y\notin \overline{\{f^k(z), k \in \mathbb{N}\}}\}$ for some $y…

We study the ergodicity of non-autonomous discrete dynamical systems with non-uniform expansion. As an application we get that any uniformly expanding finitely generated semigroup action of $C^{1+\alpha}$ local diffeomorphisms of a compact…

It is known that if a compact metric space X admits a minimal expansive homeomorphism then X is totally disconnected. In this note we give a short proof of this result and we analyze its extension to expansive flows.

In this article, we develop a functional-analytic framework to establish existence, uniqueness, regularity of disintegration, and statistical properties of equilibrium states for a broad class of dynamical systems, potentially discontinuous…

Deterministic chaos is commonly associated with spectral criticality: exponential sensitivity is expected when Jacobian eigenvalues exceed unity in parts of the attractor, producing the local expansion that offsets contraction elsewhere. We…

We introduce the notion of a generalized $(C, \lambda)$-structure, which generalizes hyperbolicity to nonlinear dynamics in Banach spaces. The main novelties are that we allow the hyperbolic splitting to be discontinuous, and that in the…