Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous map with the specification property, and $φ: X \mapsto \IR$ a continuous function. We consider the set of points for which the Birkhoff average of $φ$ does not exist (which we call the irregular set for $φ$) and show that this set is either empty or carries full topological pressure (in the sense of Pesin and Pitskel). We formulate various equivalent natural conditions on $φ$ that completely describe when the latter situation holds and give examples of interesting systems to which our results apply but were not previously known. As an application, we show that for a suspension flow over a continuous map with specification, the irregular set carries full topological entropy.
Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous map satisfying a property we call almost specification (which is slightly weaker than the $g$-almost product property of Pfister and Sullivan), and $ϕ$ be a continuous function on $X$. We show that the set of points for which the Birkhoff average of $ϕ$ does not exist (which we call the irregular set) is either empty or has full topological entropy. Every $β$-shift satisfies almost specification and we show that the irregular set for any $β$-shift or $β$-transformation is either empty or has full topological entropy and Hau…
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 and Sullivan in \cite{Pfister2007}, we give a variational principle for certain non-compact with relation the asymptotically additive topological pressure. We also study the set of points that are irregular for an collection finite or infinite of asymptotically additive sequences and we show that carried the full asymptotically additive topological pressure…
In this article we prove that for a diffeomorphism on a compact Riemannian manifold, if there is a nontrival homoclinic class that is not uniformly hyperbolic or the diffeomorphism is a $C^{1+α}$ and there is a hyperbolic ergodic measure whose support is not uniformly hyperbolic, then we find a type of strongly distributional chaos which is stronger than usual distributional chaos and Li-Yorke chaos in the set of irregular orbits that are not uniformly hyperbolic. Meanwhile, we prove that various fractal sets are strongly distributional chaotic, such as irregular sets, level sets, several recu…
In this paper we mainly study the dynamical complexity of Birkhoff ergodic average under the simultaneous observation of any number of continuous functions. These results can be as generalizations of [6,35] etc. to study Birkhorff ergodic average from one (or finite) observable function to any number of observable functions from the dimensional perspective. For any topological dynamical system with $g-$almost product property and uniform separation property, we show that any {\it jointly-irregular set}(i.e., the intersection of a series of $ϕ-$irregular sets over several continuous functions) …
We study the Bowen topological entropy of generic and irregular points for certain dynamical systems. We define the topological entropy of noncompact sets for flows, analogous to Bowen's definition. We show that this entropy coincides with the Bowen topological entropy of the time-1 map on any set. We also show a Bowen's inequality for flows; namely, that the metric entropy with respect to every invariant measure for a continuous flow is an upper bound for the topological entropy of the set of generic points with respect to the same measure, and the equality is always true if the measure is er…
In this article we prove estimates for the topological pressure of the set of points whose Birkhoff time averages are far from the space averages corresponding to the unique equilibrium state that has a weak Gibbs property. In particular, if $f$ has an expanding repeller and $ϕ$ is an Hölder continuous potential we prove that the topological pressure of the set of points whose accumulation values of Birkhoff averages belong to some interval $I\subset \mathbb R$ can be expressed in terms of the topological pressure of the whole system and the large deviations rate function. As a byproduct we de…
It follows from Oseledec Multiplicative Ergodic Theorem (or Kingmans Subadditional Ergodic Theorem) that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any dynamical system f with exponential specification property and a Holder continuous matrix cocycle A, we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of A displays distributional chaos of type 1.
This paper discusses the thermodynamic properties for certain time-dependent dynamical systems. In particular, we are interested in time-dependent dynamical systems with the specification property. We show that each time-dependent dynamical system given by a sequence of surjective continuous self maps of a compact metric space with the specification property has positive topological entropy and all points are entropy point. In particular, it is proved that these systems are topologically chaotic. We will treat the dynamics of uniformly Ruelle-expanding time-dependent dynamical systems on compa…
Let $(X,d)$ be a compact metric space, $f:X \mapsto X$ be a continuous map with the specification property, and $φ: X \mapsto \IR$ be a continuous function. We prove a variational principle for topological pressure (in the sense of Pesin and Pitskel) for non-compact sets of the form \[ \{x \in X : \lim_{n \ra \infty} \frac{1}{n} \sum_{i = 0}^{n-1} φ(f^i (x)) = α\}. \] Analogous results were previously known for topological entropy. As an application, we prove multifractal analysis results for the entropy spectrum of a suspension flow over a continuous map with specification and the dimension s…
In this paper we introduce the notions of topological entropy and topological pressure for non-autonomous iterated function systems (or NAIFSs for short) on countably infinite alphabets. NAIFSs differ from the usual (autonomous) iterated function systems, they are given \cite{LGMU} by a sequence of collections of continuous maps on a compact topological space, where maps are allowed to vary between iterations. Several basic properties of topological pressure and topological entropy of NAIFSs are provided. Especially, we generalize the classical Bowen's result to NAIFSs ensures that the topolog…
In this paper we introduce a notion of free energy and large deviations rate function for asymptotically additive sequences of potentials via an approximation method by families of continuous potentials. We provide estimates for the topological pressure of the set of points whose non-additive sequences are far from the limit described through Kingman's sub-additive ergodic theorem and give some applications in the context of Lyapunov exponents for diffeomorphisms and cocycles, and Shannon-McMillan-Breiman theorem for Gibbs measures.
In their recent paper [KNS2019], the first author, S. Kiriki, and T. Soma introduced a concept of pointwise emergence to measure the complexity of irregular orbits. They constructed a residual subset of the full shift with high pointwise emergence. In this paper we consider the set of points with high pointwise emergence for topologically mixing subshifts of finite type. We show that this set has full topological entropy, full Hausdorff dimension, and full topological pressure for any Hölder continuous potential. Furthermore, we show that this set belongs to a certain class of sets with large …
We show that a shift space on a finite alphabet with a non-uniform specification property can be modeled by a strongly positive recurrent countable-state Markov shift to which every equilibrium state lifts. In addition to uniqueness of the equilibrium state, this gives strong statistical properties including the Bernoulli property, exponential decay of correlations, central limit theorem, and analyticity of pressure, which are new even for uniform specification. We give applications to shifts of quasi-finite type, synchronised and coded shifts, and factors of beta-shifts and S-gap shifts.
It follows from Oseledec Multiplicative Ergodic Theorem that the Lyapunov-irregular set of points for which the Oseledec averages of a given continuous cocycle diverge has zero measure with respect to any invariant probability measure. In strong contrast, for any dynamical system $f:X\rightarrow X$ with exponential specification property and a H$\ddot{\text{o}}$lder continuous matrix cocycle $A:X\rightarrow G (m,\mathbb{R})$, we show here that if there exist ergodic measures with different Lyapunov spectrum, then the Lyapunov-irregular set of $A$ is residual (i.e., containing a dense $G_δ$ set…
There are lots of results to study dynamical complexity on irregular sets and level sets of ergodic average from the perspective of density in base space, Hausdorff dimension, Lebesgue positive measure, positive or full topological entropy (and topological pressure) etc.. However, it is unknown from the viewpoint of chaos. There are lots of results on the relationship of positive topological entropy and various chaos but it is known that positive topological entropy does not imply a strong version of chaos called DC1 so that it is non-trivial to study DC1 on irregular sets and level sets. In t…
By an appropriate definition, we divide the irregular set into level sets. Then we characterize the multifractal spectrum of these new pieces by calculating their entropies. We also compute the entropies of various intersections of the level sets of regular and irregular set which is rarely studied in the literature. Moreover, our conclusions also hold for the topological pressure. Finally, we consider the continuous case and use our results to give a description for the suspension flow.
In the present paper we study the thermodynamical properties of finitely generated continuous subgroup actions. We address a notion of topological entropy and pressure functions that does not depend on the growth rate of the semigroup and introduce strong and orbital specification properties, under which, the semigroup actions have positive topological entropy and all points are entropy points. Moreover, we study the convergence and Lipschitz regularity of the pressure function and obtain relations between topological entropy and exponential growth rate of periodic points in the context of sem…
In this paper we deal with an invariant ergodic hyperbolic measure $μ$ for a diffeomorphism $f,$ assuming that $f$ it is either $C^{1+α}$ or $f$ is $C^1$ and the Oseledec splitting of $μ$ is dominated. We show that this system $(f,μ)$ satisfies a weaker and non-uniform version of specification, related with notions studied in several recent papers, including \cite{STV,Y, PS, T,Var, Oli}. Our main results have several consequences: as corollaries, we are able to improve the results about quantitative Poincaré recurrence, removing the assumption of the non-uniform specification property in the m…
