Piecewise Monotonic Transformations Research Articles

On subshifts with slow forbidden word growth

AbstractIn this work, we treat subshifts, defined in terms of an alphabet $\mathcal {A}$ and (usually infinite) forbidden list $\mathcal {F}$ , where the number of n-letter words in $\mathcal {F}$ has ‘slow growth rate’ in n. We show that such subshifts are well behaved in several ways; for instance, they are boundedly supermultiplicative in the sense of Baker and Ghenciu [Dynamical properties of S-gap shifts and other shift spaces. J. Math. Anal. Appl.430(2) (2015), 633–647] and they have unique measures of maximal entropy with the K-property and which satisfy Gibbs bounds on large (measure-theoretically) sets. The main tool in our proofs is a more general result, which states that bounded supermultiplicativity and a sort of measure-theoretic specification property together imply uniqueness of the measure of maximum entropy and our Gibbs bounds. We also show that some well-known classes of subshifts can be treated by our results, including the symbolic codings of $x \mapsto \alpha + \beta x$ (the so-called $\alpha $ - $\beta $ shifts of Hofbauer [Maximal measures for simple piecewise monotonic transformations. Z. Wahrsch. verw. Geb.52(3) (1980), 289–300]) and the bounded density subshifts of Stanley [Bounded density shifts. Ergod. Th. & Dynam. Sys.33(6) (2013), 1891–1928].

The recurrence dimension for piecewise monotonic maps of the interval

We investigate a weighted version of Hausdorff dimension introduced by V. Afraimovich, where the weights are determined by recurrence times. We do this for an ergodic invariant measure with positive entropy of a piecewise monotonic transformation on the interval [0, 1], giving first a local result and proving then a formula for the dimension of the measure in terms of entropy and characteristic exponent. This is later used to give a relation between the dimension of a closed invariant subset and a pressure function. Mathematics Subject Classification (2000): 37E05 (primary); 37C45, 28A80, 37B40, 28A78 (secondary).

CONDITIONALLY INVARIANT MEASURES FOR POSITION DEPENDENT RANDOM MAPS OF AN INTERVAL WITH HOLES

We study position dependent random maps on the unit interval with holes where the possible laws of motion are piecewise monotonic transformations. The main result of this note is proving the existence of absolutely continuous conditionally invariant measures.

The Rényi dimension of a conformal measure for a piecewise monotonic map of the interval

We consider a continuous piecewise monotonic transformation T on the interval, which is expanding, and an e - g -conformal measure m . Set A =supp m and p =-log | T '|. For each s≥ 0 there is a unique t =t( s ) such that the pressure p ( T │A, sg + t p) equals zero. For the Renyi dimension R s with s ∈ R + b{1} we show under certain assumptions that R s ( m ) = t( s )/(1- s ).

Computing the Topological Entropy for Piecewise Monotonic Maps on the Interval

A new method for computing the topological entropy of a piecewise monotonic transformation on the interval is presented. It uses a transition matrix associated with the transformation. For this matrix we give a spectral theorem. This can be used for an estimation of the accuracy of the algorithm.

Ulam's method for random interval maps

We consider the approximation of absolutely continuous invariant measures (ACIMs) of systems defined by random compositions of piecewise monotonic transformations. Convergence of Ulam's finite approximation scheme in the case of a single transformation was dealt with by Li (1976 J. Approx. Theory 17 177-86). We extend Ulam's construction to the situation where a family of piecewise monotonic transformations are composed according to either an iid or Markov law, and prove an analogous convergence result. In addition, we obtain a convergence rate for our approximations to the unique ACIM, and provide rigorous bounds for the L1 error of the Ulam approximation.

Conformal measure and decay of correlation for covering weighted systems

We show that for a large class of piecewise monotonic transformations on a totally ordered, compact set one can construct conformal measures and obtain the exponential mixing rate for the associated equilibrium state. The method is based on the study of the Perron–Frobenius operator. The conformal measure, the density of the invariant measure and the rate of mixing are deduced by using an appropriate Hilbert metric, without any compactness arguments, even in the case of a countable to one transformation.

A new approach to controlling chaotic systems

Given a piecewise monotonic transformation τ which is strongly transitive and preserves an invariant probability density function f, a construction is described which modifies τ very slightly but which changes f significantly in order to possess desired properties. This construction permits control of the global long term behavior of chaotic dynamical systems.

The box dimension of completely invariant subsets for expanding piecewise monotonic transformations

We consider completely invariant subsetsA of expanding piecewise monotonic transformationsT on [0, 1]. An estimate of the box dimension of such setsA in terms of a certain pressure function is given, which implies equality of box dimension and Hausdorff dimension ofA.

Some Ergodic Theorems for Piecewise Monotonic Transformations

LetS: [0,1]→[0,1] be a piecewise monotonic transformation satisfying some conditions. We show that time averages (1/n) ∑n−1j=0f∘Sjconverge strongly inLp(0,1) and pointwisely almost everywhere for anyf∈Lp(0,1), where 1≤p<∞.

Local dimension for piecewise monotonic maps on the interval

AbstractThe local dimension of invariant and conformal measures for piecewise monotonic transformations on the interval is considered. For ergodic invariant measures m with positive characteristic exponent χm we show that the local dimension exists almost everywhere and equals hm/χm For certain conformal measures we show a relation between a pressure function and the Hausdorff dimension of sets, on which the local dimension is constant.

Fractal Properties of Invariant Subsets for Piecewise Monotonic Maps on the Interval

Let T be a piecewise monotonic transformation on [0, 1] and let A be a T-invariant subset, which has positive topological entropy and satisfies the Darboux property. A general existence theorem for conformal measures on A is proved. This is then used to show equality of the dynamical dimension of A and the minimal zero of a certain pressure function.

Fractal properties of invariant subsets for piecewise monotonic maps on the interval

Let T be a piecewise monotonic transformation on [0, 1] and let A be a T-invariant subset, which has positive topological entropy and satisfies the Darboux property. A general existence theorem for conformal measures on A is proved. This is then used to show equality of the dynamical dimension of A and the minimal zero of a certain pressure function.

Compactness of Invariant Densities for Families of Expanding, Piecewise Monotonic Transformations

Let I = [0,1] and let be the space of all integrable functions on I, where m denotes Lebesque measure on I. Let ∥ ∥1 be the ℒ-1-norm and let be a measurable, nonsingular transformation on I. Let denote the space of densities. The probability measure μ is invariant under τ if for all measurable sets A, The measure μ is absolutely continuous if there exists an such that for any measurable set A We refer to ƒ* as the invariant density of τ (with respect to m). It is well-known that ƒ * is a fixed point of the Frobenius-Perron operator defined by

Generic properties of invariant measures for continuous piecewise monotonic transformations

We endow the set of all invariant measures of topologically transitive subsetsL withhtop (L)>0 of a continuous piecewise monotonic transformation on [0, 1] with the weak topology. We show that the set of periodic orbit measures is dense, that the sets of ergodic, of nonatomic, and of measures with supportL are denseGδ-sets, that the set of strongly mixing measures is of first category, and that the set of measures with zero entropy contains a denseGδ-set.

Ergodic properties of computer orbits for simple piecewise monotonic transformations

Using a general model for normalized floating point arithmetic, we study the computer generated orbit of simple piecewise monotonic transformations on [0, 1] of the form τ( x) = kx(mod 1). Conditions under which the computer orbit is eventually exact are established. Moreover it is shown that for sufficiently high precision of computation the computer orbits display the same ergodic properties as the exact orbits of the transformations.

Generic properties of invariant measures for simple piecewise monotonic transformations

We endow the set of all invariant measures of topologically transitive subsetsL of certain piecewise monotonic transformations on [0, 1] with the weak topology. We show that the set of periodic orbit measures is dense, that the sets of ergodic, of nonatomic, and of measures with supportL are dense-sets, that the se of strongly mixing measures is of first category, and that the set of measures with zero entropy contains a denseGin/gd-set.

Periodic points for piecewise monotonic transformations

AbstractPeriodic points of piecewise monotonic transformations are investigated using an oriented graph, whose closed paths represent the periodic orbits of the transformation. In the first part it is shown that the inverse of the ζ-function is a kind of characteristic polynomial of this graph, which is a power series if the graph is infinite. In the second part, the sets are determined, which can occur as{withandwhereTis a monotonic mod one transformation.

Generalized bounded variation and applications to piecewise monotonic transformations

We prove the quasi-compactness of the Perron-Frobenius operator of piecewise monotonic transformations when the inverse of the derivative is Hölder-continuous or, more generally, of bounded p-variation.

On the supports of absolutely continuous ergodic measures for piecewise monotonic transformations

On the supports of absolutely continuous ergodic measures for piecewise monotonic transformations