Any continuous, transitive, piecewise monotonic map is determined up to a binary choice by its dimension module with the associated finite sequence of generators. The dimension module by itself determines the topological entropy of any transitive piecewise monotonic map, and determines any transitive unimodal map up to conjugacy. For a transitive piecewise monotonic map which is not essentially injective, the associated dimension group is a direct sum of simple dimension groups, each with a unique state.
We state that for continuous interval maps the existence of a non empty closed invariant subset which is transitive and sensitive to initial conditions is implied by positive topological entropy and implies chaos in the sense of Li-Yorke, and we exhibit examples showing that these three notions are distinct.
We study shadowing property for random infinite pseudotrajectories of a continuous map $f$ of a compact metric space. For the cases of transitive maps and transitive attractors we prove a dichotomy: either $f$ satisfies shadowing property or random pseudotrajectory is shadowable with probability 0.
We improve previous results by exhibiting a construction that contains all known examples. A suficient condition for the existence of robustly transitive maps displaying singularities on a certain large class of compact manifolds is given.
We show that every dendrite satisfying the condition that no subtree of it contains all free arcs admits a transitive, even exactly Devaney chaotic map with arbitrarily small entropy. This gives a partial answer to a question of Baldwin from 2001.
In this article we intend to contribute in the understanding of the ergodic properties of the set RT of robustly transitive local diffeomorphisms on a compact manifold M without boundary. We prove that there exists a C^1 residual subset R_0 of RT such that any f in R_0 has a residual subset of M with dense pre-orbits. Moreover, C^1 generically in the space of local diffeomorphisms with no splitting and all points with dense pre-orbit, there are uncountably many ergodic expanding invariant measures with full support and exhibiting exponential decay of correlations. In particular, these results …
For a continuous map $f$ from the real line (half-open interval $[0,1)$) into itself let ent(f) denote the supremum of topological entropies of $f|_K$, where $K$ runs over all compact $f$-invariant subsets of $\mathbb{R}$ ($[0,1)$, respectively). It is proved that if $f$ is topologically transitive, then the best lower bound of ent(f) is $\log\sqrt{3}$ ($\log 3$, respectively) and it is not attained. This solves a problem posed by C{á}novas [Dyn. Syst. \textbf{24} (2009), no. 4, 473--483].
We show that time-one maps of transitive Anosov flows of compact manifolds are accumulated by diffeomorphisms robustly satisfying the following dichotomy: either all of the measures of maximal entropy are non-hyperbolic, or there are exactly two ergodic measures of maximal entropy, one with a positive central exponent and the other with a negative central exponent. We establish this dichotomy for certain partially hyperbolic diffeomorphisms isotopic to the identity whenever both of their strong foliations are minimal. Our proof builds on the approach developed by Margulis for Anosov flows wher…
For every group $G$, we show that either $G$ has a topologically transitive action on the line $\mathbb R$ by orientation-preserving homeomorphisms, or every orientation-preserving action of $G$ on $\mathbb R$ has a wandering interval. According to this result, all groups are divided into two types: transitive type and wandering type, and the types of several groups are determined. We also show that every finitely generated orderable group of wandering type is indicable. As a corollary, we show that if a higher rank lattice $Γ$ is orderable, then $Γ$ is of transitive type.
We show that a uniformly continuous random perturbation of a transitive map defines an aperiodic Harris chain which also satisfies Doeblin's condition. As a result, we get exponential decay of correlations for suitable random perturbations of such systems. We also prove that, for transitive maps, the limiting distribution for Extreme Value Laws (EVLs) and Hitting/Return Time Statistics (HTS/RTS) is standard exponential. Moreover, we show that the Rare Event Point Process (REPP) converges in distribution to a standard Poisson process.
Let $C(\mathbf I)$ be the set of all continuous self-maps from ${\mathbf I}=[0,1]$ with the topology of uniformly convergence. A map $f\in C({\mathbf I})$ is called a transitive map if for every pair of non-empty open sets $U,V$ in $\mathbf{I}$, there exists a positive integer $n$ such that $U\cap f^{-n}(V)\not=\emptyset.$ We note $T(\mathbf{I})$ and $\overline{T(\mathbf{I})}$ to be the sets of all transitive maps and its closure in the space $C(\mathbf I)$. In this paper, we show that $T(\mathbf{I})$ and $\overline{T(\mathbf{I})}$ are homeomorphic to the separable Hilbert space $\ell_2$.
We study the existence of transitive exchange maps with flips defined on the unit circle. We provide a complete answer to the question of whether there exists a transitive exchange map of the unit circle defined on n subintervals and having f flips.
We address the problem of necessary conditions and topological obstructions for the existence of robustly transitive maps on surfaces. Concretely, we show that partial hyperbolicity is a necessary condition in order to have $C^1$ robustly transitive endomorphisms with critical points on surfaces, and the only surfaces that admits robustly transitive maps are either the torus or the klein bottle. Moreover, we show that every robustly transitive endomorphism is homotopic to a linear map having at least one eigenvalue with modulus larger than one.
In this note, continuous transitive maps $f$ on the interval $I$ are re-addressed, where $I$ denotes one of the intervals: $(-\infty, \infty)$, $(-\infty, a]$, $[b, \infty)$, $[a, b]$, where $a < b$ are real numbers. Such maps must have a fixed point, say $z$, in the interior of $I$. Some well-known properties of such maps are re-proved in a systematic way according to the following : (1) $f$ moves some point $c \ne z$ away from $z$, i.e., fo some point $c \ne z$, we have $f(c) \le c < z$ or $z < c \le f(c)$; (2) $f$ moves some point $\hat c \ne z$ towards but not "over" $z$, i.e., for some po…
We investigate the properties of chain recurrent, chain transitive, and chain mixing maps (generalizations of the well-known notions of non-wandering, topologically transitive, and topologically mixing maps). We describe the structure of chain transitive maps. These notions of recurrence are defined using $\ep$-chains, and the minimal lengths of these $\ep$-chains give a way to measure recurrence time (chain recurrence and chain mixing times). We give upper and lower bounds for these recurrence times and relate the chain mixing time to topological entropy.
For a continuous map f on a compact metric space (X,d), a subset D of X is internally chain transitive if for every x and y in D and every delta > 0 there is a sequence of points {x=x_0,x_1, ...,x_n=y} such that d(f(x_i),x_{i+1}) < delta for i=0,1, ...,n-1. It is known that every omega-limit set is internally chain transitive; in earlier work it was shown that for X a shift of finite type, a closed subset D of X is internally chain transitive if and only if D is an omega-limit set for some point in X, and that the same is also true for the tent map with slope equal to 2. In this paper, we prov…
With each piecewise monotonic map of the unit interval, a dimension triple is associated. The dimension triple, viewed as a Z[t, t^{-1}] module, is finitely generated, and generators are identified. Dimension groups are computed for Markov maps, unimodal maps, multimodal maps, and interval exchange maps. It is shown that the dimension group defined here is isomorphic to K_0(A), where A is a C*-algebra (an "AI-algebra") defined in dynamical terms.
We describe Markov interval maps via branching systems and develop the theory of relative branching systems, characterizing when the associated representations of relative graph C*-algebras are faithful. When the Markov interval maps $f$ have escape sets, we use our results to characterize injectivity of the associated relative graph algebra representations, improving on previous work by the first, third, and fourth authors.
For each irrational $α\in[0,1)$ we construct a continuous function $f\: [0,1)\to \R$ such that the corresponding cylindrical transformation $[0,1)\times\R \ni (x,t) \mapsto (x+α, t+ f(x)) \in [0,1)\times\R$ is transitive and the Hausdorff dimension of the set of points whose orbits are discrete is 2. Such cylindrical transformations are shown to display a certain chaotic behaviour of Devaney-like type.
We produce and study a family of representations of relative graph algebras on Hilbert spaces that arise from the orbits of points of one dimensional dynamical systems, where the underlying Markov interval maps $f$ have escape sets. We identify when such representations are faithful in terms of the transitions to the escape subintervals.
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 $β$-transformations, and reduces a complete description of irregular sets of transitive piecewise monotonic maps to Hofbauer-Raith p…
It is well known that a continuous piecewise monotone interval map with positive topological entropy is semiconjugate to a map of a constant slope and the same entropy, and if it is additionally transitive then this semiconjugacy is actually a conjugacy. We generalize this result to piecewise continuous piecewise monotone interval maps, and as a consequence, get it also for piecewise monotone graph maps. We show that assigning to a continuous transitive piecewise monotone map of positive entropy a map of constant slope conjugate to it defines an operator, and show that this operator is not con…
For any infinite transitive sofic shift $X$ we construct a reversible cellular automaton (i.e. an automorphism of the shift $X$) which breaks any given finite point of the subshift into a finite collection of gliders traveling into opposing directions. This shows in addition that every infinite transitive sofic shift has a reversible CA which is sensitive with respect to all directions. As another application we prove a finitary Ryan's theorem: the automorphism group aut$(X)$ contains a two-element subset whose centralizer consists only of shift maps. We also show that in the class of $S$-gap …
For each piecewise monotonic map tau of [0,1], we associate a pair of C*-algebras F_tau and O_tau and calculate their K-groups. The algebra F_tau is an AI-algebra. We characterize when F_tau and O_τare simple. In those cases, F_tau has a unique trace, and O_tau is purely infinite with a unique KMS-state. In the case that tau is Markov, these algebras include the Cuntz-Krieger algebras O_A, and the associated AF-algebras F_A. Other examples for which the K-groups are computed include tent maps, quadratic maps, multimodal maps, interval exchange maps, and beta-transformations. For the case of in…
We study dynamics of continuous maps on compact metrizable spaces containing a free interval (i.e., an open subset homeomorphic to an open interval). A special attention is paid to relationships between topological transitivity, weak and strong topological mixing, dense periodicity and topological entropy as well as to the topological structure of minimal sets. In particular, a trichotomy for minimal sets and a dichotomy for transitive maps are proved.
We survey, and extend, results on the adjoint action of the homeomorphism group $H(X)$ on the space of surjective continuous maps, $C_s(X)$, where $X$ is a Cantor set. We look also at the restriction of the action to various dynamically defined subsets of $C_s(X)$, e. g. the sets of topologically transitive maps, chain transitive maps, chain mixing maps, etc. In each case, we consider whether there exist elements with a dense conjugacy class and if so, what the generic elements look like.
We prove that the group of diffeomorphisms of the interval $[0,1]$ contains surface groups whose action on $(0,1)$ has no global fix point, is topologically transitive, and such that only countably many points of the interval $(0,1)$ have non-trivial stabiliser.
In this paper we consider a class of skew products over transitive subshifts of finite type with interval fibers. For a natural class of 1-parameter families we prove that for all but countably many parameter values the nonwandering set (in particular, the union of all attractors and repellers) has zero measure. As a consequence, the same holds for a residual subset of the space of skew products.
We describe all possible bimodal over-twist patterns. In particular, we give an algorithm allowing one to determine what the left endpoint of the over-rotation interval of a given bimodal map is. We then define a new class of polymodal interval maps called well behaved, and generalize the above results onto well behaved maps.
We prove that the Möbius disjointness conjecture holds for graph maps and for all monotone local dendrite maps. We further show that this also hold for continuous map on certain class of dendrites. Moreover, we see that there is a transitive dendrite map with zero entropy for which Möbius disjointness holds.
A map $f{:}\,[0,1)\to [0,1)$ is a {\it piecewise contraction of $n$ intervals} ($n$-PC) if there exist $0<λ<1$ and a partition of $[0,1)$ into intervals $I_1,\ldots,I_n$ such that $f\vert_{I_i}$ is $λ$-Lipschitz for every $1\le i\le n$. An infinite word $θ=θ_0θ_1\ldots$ over the alphabet $\mathcal{A}=\{1,\ldots,n\}$ is a {\it natural coding of} $f$ if there exists $x\in I$ such that $θ_k=i$ if and only if $f^k(x)\in I_i$. We prove that if $θ$ is a natural coding of an injective $n$-PC, then some infinite subword of $θ$ is either periodic or isomorphic to a natural coding of a topologically tra…
We study the expanding properties of random perturbations of regular interval maps satisfying the summability condition of exponent one. Under very general conditions on the interval maps and perturbation types, we prove strong stochastic stability.
Let $\{f_t\}_{t\in(1,2]}$ be the family of core tent maps of slopes $t$. The parameterized Barge-Martin construction yields a family of disk homeomorphisms $Φ_t\colon D^2\to D^2$, having transitive global attractors $Λ_t$ on which $Φ_t$ is topologically conjugate to the natural extension of $f_t$. The unique family of absolutely continuous invariant measures for $f_t$ induces a family of ergodic $Φ_t$-invariant measures $ν_t$, supported on the attractors $Λ_t$. We show that this family $ν_t$ varies weakly continuously, and that the measures $ν_t$ are physical with respect to a weakly continuou…
We prove that for continuous maps on the interval, the existence of an n-cycle, implies the existence of n-1 points which interwind the original ones and are permuted by the map. We then use this combinatorial result to show that piecewise affine maps (with no zero slope) cannot be infinitely renormalizable.
We show that the connectedness of the set of parameters for which the over-rotation interval of a bimodal interval map is constant. In other words, the over-rotation interval is a monotone function of a bimodal interval map.
We prove that the flow generated by any interval map with zero topological entropy is minimally mean-attractable (MMA) and minimally mean-L-stable (MMLS). One of the consequences is that any oscillating sequence is linearly disjoint with all flows generated by interval maps with zero topological entropy. In particular, the Möbius function is orthogonal to all flows generated by interval maps with zero topological entropy (Sarnak's conjecture for interval maps). Another consequence is a non-trivial example of a flow having the discrete spectrum.
We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and study its properties. Further, we study how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear~naturally.
In this paper, we study the Hausdorff dimension of the generalized intrinsic level set with respect to the given ergodic meausre in a class of non-uniformly hyperbolic interval maps with finitely many branches.
In this paper we generalize the recently introduced concept of fair measure (M. Misiurewicz and A. Rodrigues, Counting preimages. Ergod. Th. & Dynam. Sys. 38 (2018), no. 5, 1837 -- 1856). We study transitive countable state Markov shift maps and extend our results to a particular class of interval maps, Markov and mixing interval maps. Finally, we move beyond the interval and look for fair measures for graph maps.
We study the jump of topological entropy for $C^r$ interval or circle maps. We prove in particular that the topological entropy is continuous at any $f \in C^r([0; 1])$ with $h_{top}(f) > \frac{\log^+ \|f'\|_\infty}{r}$. To this end we study the continuity of the entropy of the Buzzi-Hofbauer diagrams associated to $C^r$ interval maps.
For piecewise $C^1$ interval maps possibly containing critical points and discontinuities with negative Schwarzian derivative, under two summability conditions on the growth of the derivative and recurrence along critical orbits, we prove the nonexistence of wandering intervals, the existence of absolutely continuous invariant measures, and the bounded backward contraction property. The proofs are based on the method of proving the existence of absolutely continuous invariant measures of unimodal map, developed by Nowicki and van Strien.
In this short note we describe a simple but remarkably effective method for rigorously estimating Lyapunov exponents for expanding maps of the interval. We illustrate the applicability of this method with some standard examples.
In this paper we will develop a very general approach which shows that critical relations of holomorphic maps on the complex plane unfold transversally in a positively oriented way. We will mainly illustrate this approach to obtain transversality for a wide class of one-parameter families of interval maps, for example maps with flat critical points, piecewise linear maps, maps with discontinuities but also for families of maps with complex analytic extensions such as certain polynomial-like maps.
For uniformly expanding maps on the interval, analogous versions of the Berry-Esséen theorem are known but only with an unexplicit upper bound in $O(1/\sqrt{n})$ without any constants being specified. In this paper, we use the recent complex cone technique to prove an explicit Berry-Esséen estimate with a reasonable constant for these maps. Our method is not limited to maps on the interval however and should apply to many situations.
Consider the two-parameter family of real analytic maps $F_{a,b}:x \mapsto x+ a+{b\over 2π} \sin(2πx)$ which are lifts of degree one endomorphisms of the circle. The purpose of this paper is to provide a proof that for any closed interval $I$, the set of maps $F_{a,b}$ whose rotation interval is $I$, form a contractible set.
We study some special classes of piecewise continuous maps on a finite smooth partition of a compact manifold and look for invariant measures for such maps. We show that in the simplest one-dimensional case (so-called interval translation maps) a Borel probability non-atomic invariant measure exists for any map. We use this result to demonstrate that any interval translation map endowed with such a measure is metrically equivalent to an interval exchange map. Finally, we study the general case of piecewise continuous maps and prove a simple result on existence of an invariant measure provided …
We use the concept of Baire Ergodicity and Ergodic Formalism introduced to study topological and statistical attractors for interval maps, even with discontinuities. For that we also analyze the {\em wandering intervals attractors}. As a result, we establish the finiteness of the non-periodic topological attractors for piecewise $C^2$ maps with discontinuities. For $C^2$ interval maps without discontinuities, we show the coincidence of the statistical attractors with the topological ones and we calculate the upper Birkhoff averages of continuous functions for generic points, even when the map …
This survey article concerns inducing schemes in the context of interval maps. We explain how the study of these induced systems allows for the fine description of, not only, the thermodynamic formalism for certain multimodal maps, but also of its multifractal structure.
We prove that the solutions of the cohomological equation for Roth type interval exchange maps are Hölder continuous provided that the datum is of class $C^r$ with $r>1$ and belongs to a finite-codimension linear subspace.
