Full Topological Entropy Research Articles

On multifractal analysis and large deviations of singular hyperbolic attractors

In this paper we study the multifractal analysis and large derivations for singular hyperbolic attractors, including the geometric Lorenz attractors. For each singular hyperbolic homoclinic class whose periodic orbits are all homoclinically related and such that the space of ergodic probability measures is connected, we prove that: (i) level sets associated to continuous observables are dense in the homoclinic class and satisfy a variational principle; (ii) irregular sets are either empty or are Baire generic and carry full topological entropy. The assumptions are satisfied by C 1-generic singular hyperbolic attractors and Cr -generic geometric Lorenz attractors . Finally we prove level-2 large deviations bounds for weak Gibbs measures, which provide a large deviations principle in the special case of Gibbs measures. The main technique we apply is the horseshoe approximation property.

Entropy of Irregular Points for Some Dynamical Systems

We derive sufficient conditions for a dynamical systems to have a set of irregular points with full topological entropy. Such conditions are verified for nonuniformly hyperbolic systems such as positive entropy surface diffeomorphisms and rational functions on the Riemann sphere.

Non-dense orbits of systems with the approximate product property

We show that for any topological dynamical system with the approximate product property, the set of points whose forward orbits do not accumulate to any point in a large set Z carries full topological entropy, as well as full topological pressure for any continuous potential. For instance, the set Z can include a finite union of the basins of any given invariant measures.

Topological entropy and Hausdorff dimension of irregular sets for non-hyperbolic dynamical systems

We systematically investigate examples of non-hyperbolic dynamical systems having irregular sets of full topological entropy and full Hausdorff dimension. The examples include some partially hyperbolic systems and geometric Lorenz flows. We also pose interesting questions for future research.

Irregular Sets for Piecewise Monotonic Maps

For any transitive piecewise monotonic map for which the set of periodic measures is dense in the set of ergodic invariant measures (such as monotonic mod one transformations and piecewise monotonic maps with two monotonic pieces), we show that the set of points for which the Birkhoff average of a continuous function does not exist (called the irregular set) is either empty or has full topological entropy. This generalizes Thompson's theorem for irregular sets of $\beta$-transformations, and reduces a complete description of irregular sets of transitive piecewise monotonic maps to the Hofbauer-Raith problem on the density of periodic measures.

Highly irregular orbits for subshifts of finite type: large intersections and emergence

In their recent paper (2019 arXiv:1904.03424), the first author, Kiriki and 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 intersection property. This is a natural generalization of Färm and Persson (2011 Nonlinearity 24 1291–1309) to pointwise emergence and Carathéodory dimension.

Almost additive multifractal analysis for flows * *Partially supported by FCT/Portugal through UID/MAT/04459/2019. CH was supported by FCT/Portugal through the Grant PD/BD/135523/2018.

Building on the construction of equilibrium measures, we establish a conditional variational principle for the multifractal spectra of an almost additive family with respect to a continuous flow Φ such that the entropy map μ ↦ h μ (Φ) is upper-semicontinuous. We also show that the spectrum is continuous and that in the case of hyperbolic flows the corresponding irregular sets have full topological entropy. More generally, we consider the spectrum for the u-dimension and obtain corresponding results.

A construction of multiply Xiong chaotic set with full topological entropy for β-transformation

We construct a multiply Xiong chaotic set with full topological entropy everywhere for positively expansive systems with specification property. We also show that every β-transformation contains a multiply Xiong chaotic set with full topological entropy, although β-transformation is neither continuous nor satisfies specification property for all β>1.

Distributional chaos in multifractal analysis, recurrence and transitivity

There is much research on the dynamical complexity on irregular sets and level sets of ergodic average from the perspective of density in base space, the Hausdorff dimension, Lebesgue positive measure, positive or full topological entropy (and topological pressure), etc. However, this is not the case from the viewpoint of chaos. There are many results on the relationship of positive topological entropy and various chaos. However, positive topological entropy does not imply a strong version of chaos, called DC1. Therefore, it is non-trivial to study DC1 on irregular sets and level sets. In this paper, we will show that, for dynamical systems with specification properties, there exist uncountable DC1-scrambled subsets in irregular sets and level sets. Meanwhile, we prove that several recurrent level sets of points with different recurrent frequency have uncountable DC1-scrambled subsets. The major argument in proving the above results is that there exists uncountable DC1-scrambled subsets in saturated sets.

Transitively-saturated property, Banach recurrence and Lyapunov regularity

The topological entropy of upper recurrent, lower recurrent points and Birkhoff regularity were considered in Tian (2016 Adv. Math. 288 464–526) but Banach upper recurrence points are not considered. In this paper we mainly consider whether the points that are Banach upper recurrent but not upper recurrent can carry full topological entropy. Moreover, we use statistical -limit sets to obtain entropy results on the refined orbit distribution of Banach recurrence. In this process, we strengthen Pfister and Sullivan’s result of Pfister and Sullivan (2007 Ergod. Theor. Dynam. Syst. 27 929–56) from saturated property to transitively-saturated property (and from single-saturated property to transitively-convex-saturated property). We also establish an abstract framework on the combination of Banach recurrence and multifractal analysis of general observable functions and in particular, how these results can be applied in the combination of Banach recurrence Lyapunov regularity to obtain a generalized and refined theory of Barreira and Doutor (2009 J. Math. Pures Appl. 92 1–17); Barreira and Gelfert (2006 Commun. Math. Phys. 267 393–418); Barreira and Saussol (2001 Trans. Am. Math. Soc. 353 3919–44); Feng and Huang (2010 Commun. Math. Phys. 297 1–43); Feng (2003 J. Math. 138 353–76); Feng (2009 Israel J. Math. 170 355–94); Feng and Lau (2002 Math. Res. Lett. 9 363–78) etc, on mixed Lyapunov multifractal analysis such as irregular sets and level sets of Lyapunov exponents.

Topological entropy on points without physical-like behaviour

We study a class of asymptotically entropy-expansive $C^1$ diffeomorphisms with dominated splitting on a compact manifold $M$, that satisfy the specification property. This class includes, in particular, transitive Anosov diffeomorphisms and time-one maps of transitive Anosov flows. We consider the nonempty set of physical-like measures that attracts the empirical probabilities (i.e. the time averages) of Lebesgue-almost all the orbits. We define the set $I_f \cap \Gamma_f \subset M$ of irregular points without physical-like behaviour. We prove that, if not all the invariant measures of $f$ satisfy Pesin Entropy Formula (for instance in the Anosov case), then $I_f \cap \Gamma_f$ has full topological entropy. We also obtain this result for some class of asymptotically entropy-expansive continuous maps on a compact metric space, if the set of physical-like measures are equilibrium states with respect to some continuous potential. Finally, we prove that also the set $(M \setminus I_f) \cap \Gamma_f$ of regular points without physical-like behaviour, has full topological entropy.

Topological pressure for the completely irregular set of birkhoff averages

It is well-known that for certain dynamical systems (satisfying specification or its variants), the set of irregular points w.r.t. a continuous function $\phi$ (i.e. points with divergent Birkhoff ergodic averages observed by $\phi$ ) either is empty or carries full topological entropy (or pressure, see [6,17,36,37] etc. for example). In this paper we study the set of irregular points w.r.t. a collection $D$ of finite or infinite continuous functions (that is, points with divergent Birkhoff ergodic averages simultaneously observed by all $\phi∈D$ ) and obtain some generalized results. As consequences, these results are suitable for systems such as mixing shifts of finite type, uniformly hyperbolic diffeomorphisms, repellers and $β-$ shifts.

Exceptional sets for nonuniformly expanding maps

Given a rational map of the Riemann sphere and a subset A of its Julia set, we study the A-exceptional set, that is, the set of points whose orbit does not accumulate at A. We prove that if the topological entropy of A is less than the topological entropy of the full system then the A-exceptional set has full topological entropy. Furthermore, if the Hausdorff dimension of A is smaller than the dynamical dimension of the system then the Hausdorff dimension of the A-exceptional set is larger than or equal to the dynamical dimension, with equality in the particular case when the dynamical dimension and the Hausdorff dimension coincide.We also discuss the case of a general conformal dynamical system and, in particular, certain multimodal interval maps on their Julia set.

Different asymptotic behavior versus same dynamical complexity: Recurrence & (ir)regularity

For any dynamical system T:X→X of a compact metric space X with g-almost product property and uniform separation property, under the assumptions that the periodic points are dense in X and the periodic measures are dense in the space of invariant measures, we distinguish various periodic-like recurrences and find that they all carry full topological entropy and so do their gap-sets. In particular, this implies that any two kind of periodic-like recurrences are essentially different. Moreover, we coordinate periodic-like recurrences with (ir)regularity and obtain lots of generalized multi-fractal analyses for all continuous observable functions. These results are suitable for all β-shifts (β>1), topological mixing subshifts of finite type, topological mixing expanding maps or topological mixing hyperbolic diffeomorphisms, etc.Roughly speaking, we combine many different “eyes” (i.e., observable functions and periodic-like recurrences) to observe the dynamical complexity and obtain a Refined Dynamical Structure for Recurrence Theory and Multi-fractal Analysis.

Divergence points in systems satisfying the specification property

Let $f$ be a continuous transformation of a compact metric space $(X,d)$ and $\varphi$ any continuous function on $X$. In this paper, under the hypothesis that $f$ satisfies the specification property, we determine the topological entropy of the following sets:$$K_{I}=\Big\{x\in X: A\big(\frac{1}{n}\sum_{i=0}^{n-1}\varphi(f^{i}(x))\big)=I\Big\},$$where $I$ is a closed subinterval of $\mathbb{R}$ and $A(a_{n})$ denotes the set of accumulation points of the sequence $\{a_{n}\}_{n}$. Our result generalizes the classical result of Takens and Verbitskiy ( Ergod. Th. Dynam. Sys., 23 (2003), 317-348 ). As an application, we present another concise proof of the fact that the irregular set has full topological entropy if $f$ satisfies the specification property.

The irregular set for maps with the specification property has full topological pressure

Let (X, d) be a compact metric space, f : X ↦ X be a continuous map with the specification property and ϕ : X ↦ ℝ 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 see, for example [Y.B. Pesin, Dimension Theory in Dimensional Systems: Contemporary Views and Applications, University of Chicago Press, Chicago, 1997]). 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.

Higher order Birkhoff averages

There are well-known examples of dynamical systems for which the Birkhoff averages with respect to a given observable along some or all of the orbits do not converge. It has been suggested that such orbits could be classified using higher order averages. In the case of a bounded observable, we show that a classical result of G.H. Hardy implies that if the Birkhoff averages do not converge, then neither do the higher order averages. If the Birkhoff averages do not converge then we may denote by [α k ,β k ] the limit set of the k-th order averages. The sequence of intervals thus generated is nested: [α k+1,β k+1]⊂[α k ,β k ]. We can thus make a distinction among non-convergent Birkhoff averages; either: B1. is a point B ∞, or, B2. is an interval [α∞,β∞]. We give characterizations of the types B1 and B2 in terms of how slowly they oscillate and we give examples that exhibit both behaviours B1 and B2 in the context of full shifts on finite symbols and ‘Bowen's example’. For finite full shifts, we show that the set of orbits with type B2 behaviour has full topological entropy.

Topological entropy for divergence points

Let X be a compact metric space, f a continuous transformation on X, and Y a vector space with linear compatible metric. Denote by M(X) the collection of all the probability measures on X. For a positive integer n, define the nth empirical measure as\[ L_{n}x=\frac{1}{n}\sum _{k=0}^{n-1}\delta_{f^{k}x}, \]where is continuous and affine with respect to the weak topology on M(X). We think of the composite\[ \Xi\circ L_{n}:X \xrightarrow{L_n} M(X)\xrightarrow{\Xi} Y \]as a continuous and affine deformation of the empirical measure Ln. The set of divergence points of such a deformation is defined as\[ D(f,\Xi)=\{x\in X\mid \mbox{the limit of }\Xi L_n x \mbox{ does not exist}\}. \]In this paper we show that for a continuous transformation satisfying the specification property, if is not a singleton, then the set of divergence points has full topological entropy, i.e.\[ h_{\rm top}(D(f,\Xi))=h_{\rm top}(f). \]

Sets of “Non-typical” points have full topological entropy and full Hausdorff dimension

For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension. This follows from a much stronger statement formulated for a class of symbolic dynamical systems which includes subshifts with the specification property. Our proofs strongly rely on the multifractal analysis of dynamical systems and constitute a non-trivial mathematical application of this theory.

Multifractal Analysis of Hyperbolic Flows

We establish the multifractal analysis of hyperbolic flows and of suspension flows over subshifts of finite type. A non-trivial consequence of our results is that for every Holder continuous function non-cohomologous to a constant, the set of points without Birkhoff average has full topological entropy.