Combinatorics of distance doubling maps

Karsten Keller; Steffen Winter

Displaying similar documents to “Combinatorics of distance doubling maps”

Kneading sequences for double standard maps

Michael Benedicks, Ana Rodrigues (2009)

Fundamenta Mathematicae

Similarity:

We investigate the symbolic dynamics for the double standard maps of the circle onto itself, given by $f_{a, b} (x) = 2 x + a + (b / π) s i n (2 π x) (m o d 1)$ , where b = 1 and a is a real parameter, 0 ≤ a < 1.

Mod 2 normal numbers and skew products

Geon Ho Choe, Toshihiro Hamachi, Hitoshi Nakada (2004)

Studia Mathematica

Similarity:

Let E be an interval in the unit interval [0,1). For each x ∈ [0,1) define dₙ(x) ∈ 0,1 by $d ₙ (x) : = \sum_{i = 1}^{n} 1_{E} (2^{i - 1} x) (m o d 2)$ , where t is the fractional part of t. Then x is called a normal number mod 2 with respect to E if $N^{- 1} \sum_{n = 1}^{N} d ₙ (x)$ converges to 1/2. It is shown that for any interval E ≠(1/6, 5/6) a.e. x is a normal number mod 2 with respect to E. For E = (1/6, 5/6) it is proved that $N^{- 1} \sum_{n = 1}^{N} d ₙ (x)$ converges a.e. and the limit equals 1/3 or 2/3 depending on x.

Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Artur Avila, Mikhail Lyubich, Weixiao Shen (2011)

Journal of the European Mathematical Society

Similarity:

We study the parameter space of unicritical polynomials $f_{c} : z \mapsto z^{d} + c$ . For complex parameters, we prove that for Lebesgue almost every $c$ , the map $f_{c}$ is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every $c$ , the map $f_{c}$ is either hyperbolic, or Collet–Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the “principal nest” of parapuzzle pieces.

On Surjective Bing Maps

Hisao Kato, Eiichi Matsuhashi (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

In [7], M. Levin proved that the set of all Bing maps of a compact metric space to the unit interval is a dense $G_{δ}$ -subset of the space of all maps. In [6], J. Krasinkiewicz independently proved that the set of all Bing maps of a compact metric space to an n-dimensional manifold (n ≥ 1) is a dense $G_{δ}$ -subset of the space of maps. In [9], J. Song and E. D. Tymchatyn, solving some problems of J. Krasinkiewicz ([6]), proved that the set of all Bing maps of a compact metric space to a nondegenerate...

Persistence of fixed points under rigid perturbations of maps

Salvador Addas-Zanata, Pedro A. S. Salomão (2014)

Fundamenta Mathematicae

Similarity:

Let f: S¹ × [0,1] → S¹ × [0,1] be a real-analytic diffeomorphism which is homotopic to the identity map and preserves an area form. Assume that for some lift f̃: ℝ × [0,1] → ℝ × [0,1] we have Fix(f̃) = ℝ × 0 and that f̃ positively translates points in ℝ × 1. Let $f ̃_{ϵ}$ be the perturbation of f̃ by the rigid horizontal translation (x,y) ↦ (x+ϵ,y). We show that $F i x (f ̃_{ϵ}) = \emptyset$ for all ϵ > 0 sufficiently small. The proof follows from Kerékjártó’s construction of Brouwer lines for orientation preserving...

On the quartic character of quadratic units

Zhi-Hong Sun (2013)

Acta Arithmetica

Similarity:

Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of integers m and n. Let p be a prime of the form 4k+1 and p = c²+d² with c,d ∈ ℤ, $d = 2^{r} d ₀$ and c ≡ d₀ ≡ 1 (mod 4). In the paper we determine ${(b + \sqrt (b ² + 4^{α}) / 2)}^{(p - 1) / 4)} (m o d p)$ for p = x²+(b²+4α)y² (b,x,y ∈ ℤ, 2∤b), and ${(2 a + \sqrt 4 a ² + 1)}^{(p - 1) / 4} (m o d p)$ for p = x²+(4a²+1)y² (a,x,y∈ℤ) on the condition that (c,x+d) = 1 or (d₀,x+c) = 1. As applications we obtain the congruence for $U_{(p - 1) / 4} (m o d p)$ and the criterion for $p | U_{(p - 1) / 8}$ (if p ≡ 1 (mod 8)), where Uₙ is the Lucas sequence given by U₀ = 0, U₁ = 1 and...

Exact $^{\infty}$ covering maps of the circle without (weak) limit measure

Roland Zweimüller (2002)

Colloquium Mathematicae

Similarity:

We construct $^{\infty}$ maps T on the interval and on the circle which are Lebesgue exact preserving an absolutely continuous infinite measure μ ≪ λ, such that for any probability measure ν ≪ λ the sequence ${(n^{- 1} \sum_{k = 0}^{n - 1} ν \circ T^{- k})}_{n \geq 1}$ of arithmetical averages of image measures does not converge weakly.

More on tie-points and homeomorphism in ℕ*

Alan Dow, Saharon Shelah (2009)

Fundamenta Mathematicae

Similarity:

A point x is a (bow) tie-point of a space X if X∖x can be partitioned into (relatively) clopen sets each with x in its closure. We denote this as $X = A ⋈_{x} B$ where A, B are the closed sets which have a unique common accumulation point x. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of βℕ = ℕ* (by Veličković and Shelah Steprans) and in the recent study (by Levy and Dow Techanie) of precisely 2-to-1 maps on ℕ*. In these cases the tie-points have been the unique...

On the congruences $σ (n) \equiv a (m o d n)$ and $n \equiv a (m o d φ (n))$

Carl Pomerance (1975)

Acta Arithmetica

Similarity:

On R. Chapman's "evil determinant": case p ≡ 1 (mod 4)

Maxim Vsemirnov (2013)

Acta Arithmetica

Similarity:

For p ≡ 1 (mod 4), we prove the formula (conjectured by R. Chapman) for the determinant of the (p+1)/2 × (p+1)/2 matrix $C = (C_{i j})$ with $C_{i j} = ((j - i) / p)$ .

On sums of binomial coefficients modulo p²

Zhi-Wei Sun (2012)

Colloquium Mathematicae

Similarity:

Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum $\binom{\sum_{k = 0}^{p^{a} - 1} (h p^{a} - 1}{\binom{k) (2 k}{k) / m^{k} (m o d p ²)}}$ , where h and m are p-adic integers with m ≢ 0 (mod p). For example, we show that if h ≢ 0 (mod p) and $p^{a} > 3$ , then $\binom{\sum_{k = 0}^{p^{a} - 1} (h p^{a} - 1}{\binom{k) (2 k}{{k) (- h / 2)}^{k} \equiv ((1 - 2 h) / (p^{a})) (1 + h ({(4 - 2 / h)}^{p - 1} - 1)) (m o d p ²)}}$ , where (·/·) denotes the Jacobi symbol. Here is another remarkable congruence: If $p^{a} > 3$ then $\binom{\sum_{k = 0}^{p^{a} - 1} (p^{a} - 1}{\binom{k) (2 k}{{k) (- 1)}^{k} \equiv 3^{p - 1} (p^{a} / 3) (m o d p ²)}}$ .

Strong bifurcation loci of full Hausdorff dimension

Thomas Gauthier (2012)

Annales scientifiques de l'École Normale Supérieure

Similarity:

In the moduli space $ℳ_{d}$ of degree $d$ rational maps, the bifurcation locus is the support of a closed $(1, 1)$ positive current $T_{bif}$ which is called the bifurcation current. This current gives rise to a measure $μ_{bif} : = {(T_{bif})}^{2 d - 2}$ whose support is the seat of strong bifurcations. Our main result says that $supp (μ_{bif})$ has maximal Hausdorff dimension $2 (2 d - 2)$ . As a consequence, the set of degree $d$ rational maps having $(2 d - 2)$ distinct neutral cycles is dense in a set of full Hausdorff dimension.

A classification of inverse limit spaces of tent maps with periodic critical points

Lois Kailhofer (2003)

Fundamenta Mathematicae

Similarity:

We work within the one-parameter family of symmetric tent maps, where the slope is the parameter. Given two such tent maps $f_{a}$ , $f_{b}$ with periodic critical points, we show that the inverse limit spaces $(_{a}, f_{a})$ and $(_{b}, g_{b})$ are not homeomorphic when a ≠ b. To obtain our result, we define topological substructures of a composant, called “wrapping points” and “gaps”, and identify properties of these substructures preserved under a homeomorphism.

Real $C^{k}$ Koebe principle

Weixiao Shen, Michael Todd (2005)

Fundamenta Mathematicae

Similarity:

We prove a $C^{k}$ version of the real Koebe principle for interval (or circle) maps with non-flat critical points.

On the structural result on normal plane maps

Tomás Madaras, Andrea Marcinová (2002)

Discussiones Mathematicae Graph Theory

Similarity:

We prove the structural result on normal plane maps, which applies to the vertex distance colouring of plane maps. The vertex distance-t chromatic number of a plane graph G with maximum degree Δ(G) ≤ D, D ≥ 12 is proved to be upper bounded by $6 + [(2 D + 12) / (D - 2)] ({(D - 1)}^{(t - 1)} - 1)$ . This improves a recent bound $6 + [(3 D + 3) / (D - 2)] ({(D - 1)}^{t - 1} - 1)$ , D ≥ 8 by Jendrol’ and Skupień, and the upper bound for distance-2 chromatic number.

Some q-supercongruences for truncated basic hypergeometric series

Victor J. W. Guo, Jiang Zeng (2015)

Acta Arithmetica

Similarity:

For any odd prime p we obtain q-analogues of van Hamme’s and Rodriguez-Villegas’ supercongruences involving products of three binomial coefficients such as $\sum_{k = 0}^{(p - 1) / 2} {[\binom{2 k}{k}]}_{q ²}^{3} (q^{2 k}) / ((- q ²; q ²) ²_{k} (- q; q) ²_{2 k} ²) \equiv 0 (m o d [p] ²)$ for p≡ 3 (mod 4), $\sum_{k = 0}^{(p - 1) / 2} {[\binom{2 k}{k}]}_{q ³} ({(q; q ³)}_{k} {(q ²; q ³)}_{k} q^{3 k}) ({(q ⁶; q ⁶)}_{k} ²) \equiv 0 (m o d [p] ²)$ for p≡ 2 (mod 3), where $[p] = 1 + q + \dots + q^{p - 1}$ and $(a; q) ₙ = (1 - a) (1 - a q) \dots (1 - a q^{n - 1})$ . We also prove q-analogues of the Sun brothers’ generalizations of the above supercongruences. Our proofs are elementary in nature and use the theory of basic hypergeometric series and combinatorial q-binomial identities including a new q-Clausen type summation formula. ...

$α$ -continuous multivalued maps

J. Eweret (1989)

Matematički Vesnik

Similarity:

Infinite Iterated Function Systems Depending on a Parameter

Ludwik Jaksztas (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia-Lavaurs sets $J_{0, σ}$ for the map f₀(z) = z²+1/4 on the parameter σ. Using homographies, we imitate the construction of the iterated function system (IFS) whose limit set is a subset of $J_{0, σ}$ , given by Urbański and Zinsmeister. The closure of the limit set of our IFS $ϕ_{σ, α}^{n, k}$ is the closure of some family of circles, and if the parameter σ varies, then the behavior of the limit set is similar to the behavior of...

Some remarks providing discontinuous maps on some $C_{p} (X)$ spaces

S. Moll (2008)

Banach Center Publications

Similarity:

Let X be a completely regular Hausdorff topological space and $C_{p} (X)$ the space of continuous real-valued maps on X endowed with the pointwise topology. A simple and natural argument is presented to show how to construct on the space $C_{p} (X)$ , if X contains a homeomorphic copy of the closed interval [0,1], real-valued maps which are everywhere discontinuous but continuous on all compact subsets of $C_{p} (X)$ .

Defining complete and observable chaos

Víctor Jiménez López (1996)

Annales Polonici Mathematici

Similarity:

For a continuous map f from a real compact interval I into itself, we consider the set C(f) of points (x,y) ∈ I² for which $l i m i n f_{n \to \infty} | f^{n} (x) - f^{n} (y) | = 0$ and $l i m s u p_{n \to \infty} | f^{n} (x) - f^{n} (y) | > 0$ . We prove that if C(f) has full Lebesgue measure then it is residual, but the converse may not hold. Also, if λ² denotes the Lebesgue measure on the square and Ch(f) is the set of points (x,y) ∈ C(f) for which neither x nor y are asymptotically periodic, we show that λ²(C(f)) > 0 need not imply λ²(Ch(f)) > 0. We use these results to propose some plausible...

On power integral bases for certain pure number fields defined by $x^{18} - m$

Lhoussain El Fadil (2022)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let $K = ℚ (α)$ be a number field generated by a complex root $α$ of a monic irreducible polynomial $f (x) = x^{18} - m$ , $m \neq \mp 1$ , is a square free rational integer. We prove that if $m \equiv 2$ or $3 (\mod 4)$ and $m \neg \equiv \mp 1 (\mod 9)$ , then the number field $K$ is monogenic. If $m \equiv 1 (\mod 4)$ or $m \equiv 1 (\mod 9)$ , then the number field $K$ is not monogenic.

3x+1 inverse orbit generating functions almost always have natural boundaries

Jason P. Bell, Jeffrey C. Lagarias (2015)

Acta Arithmetica

Similarity:

The 3x+k function $T_{k} (n)$ sends n to (3n+k)/2, resp. n/2, according as n is odd, resp. even, where k ≡ ±1 (mod 6). The map $T_{k} (\cdot)$ sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of $T_{k} (\cdot)$ . We consider the generating functions $f_{k, m} (z) = \sum_{n > 0, n \to m} z^{n}$ , which are holomorphic in the unit disk. We give sufficient conditions on (k,m) for the functions $f_{k, m} (z)$ to have the unit circle |z|=1 as a natural boundary to analytic continuation. For the 3x+1 function these conditions hold...

One-sided discrete square function

A. de la Torre, J. L. Torrea (2003)

Studia Mathematica

Similarity:

Let f be a measurable function defined on ℝ. For each n ∈ ℤ we consider the average $A ₙ f (x) = 2^{- n} \int_{x}^{x + 2 ⁿ} f$ . The square function is defined as $S f (x) = (\sum_{n = - \infty}^{\infty} | A ₙ f (x) - A_{n - 1} f (x) {| ²)}^{1 / 2}$ . The local version of this operator, namely the operator $S ₁ f (x) = (\sum_{n = - \infty}^{0} | A ₙ f (x) - A_{n - 1} f (x) {| ²)}^{1 / 2}$ , is of interest in ergodic theory and it has been extensively studied. In particular it has been proved [3] that it is of weak type (1,1), maps $L^{p}$ into itself (p > 1) and $L^{\infty}$ into BMO. We prove that the operator S not only maps $L^{\infty}$ into BMO but it also maps BMO into BMO. We also prove that the $L^{p}$ boundedness...

On Bressan's conjecture on mixing properties of vector fields

Stefano Bianchini (2006)

Banach Center Publications

Similarity:

In [9], the author considers a sequence of invertible maps $T_{i} : S ¹ \to S ¹$ which exchange the positions of adjacent intervals on the unit circle, and defines as Aₙ the image of the set 0 ≤ x ≤ 1/2 under the action of Tₙ ∘ ... ∘ T₁, (1) Aₙ = (Tₙ ∘ ... ∘ T₁)x₁ ≤ 1/2. Then, if Aₙ is mixed up to scale h, it is proved that (2) $\sum_{i = 1}^{n} (T o t . V a r . (T_{i} - I) + T o t . V a r . (T_{i}^{- 1} - I)) \geq C l o g 1 / h$ . We prove that (1) holds for general quasi incompressible invertible BV maps on ℝ, and that this estimate implies that the map Tₙ ∘ ... ∘ T₁ belongs to the Besov space $B^{0, 1, 1}$ , and its...

On the Lucas sequence equations Vₙ = kVₘ and Uₙ = kUₘ

Refik Keskin, Zafer Şiar (2013)

Colloquium Mathematicae

Similarity:

Let P and Q be nonzero integers. The sequences of generalized Fibonacci and Lucas numbers are defined by U₀ = 0, U₁ = 1 and $U_{n + 1} = P U ₙ - Q U_{n - 1}$ for n ≥ 1, and V₀ = 2, V₁ = P and $V_{n + 1} = P V ₙ - Q V_{n - 1}$ for n ≥ 1, respectively. In this paper, we assume that P ≥ 1, Q is odd, (P,Q) = 1, Vₘ ≠ 1, and $V_{r} \neq 1$ . We show that there is no integer x such that $V ₙ = V_{r} V ₘ x ²$ when m ≥ 1 and r is an even integer. Also we completely solve the equation $V ₙ = V ₘ V_{r} x ²$ for m ≥ 1 and r ≥ 1 when Q ≡ 7 (mod 8) and x is an even integer. Then we show that when P ≡ 3 (mod 4) and...

Structure of the McMullen domain in the parameter planes for rational maps

Robert L. Devaney (2005)

Fundamenta Mathematicae

Similarity:

We show that, for the family of functions $F_{λ} (z) = z ⁿ + λ / z ⁿ$ where n ≥ 3 and λ ∈ ℂ, there is a unique McMullen domain in parameter space. A McMullen domain is a region where the Julia set of $F_{λ}$ is homeomorphic to a Cantor set of circles. We also prove that this McMullen domain is a simply connected region in the plane that is bounded by a simple closed curve.

Canonical Banach function spaces generated by Urysohn universal spaces. Measures as Lipschitz maps

Piotr Niemiec (2009)

Studia Mathematica

Similarity:

It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure $C F L (_{r})$ of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space $_{r}$ of diameter r, is (isometrically if r = +∞) isomorphic to the space $L I P (_{r})$ of equivalence classes of all real-valued Lipschitz maps on $_{r}$ . The space of all signed (real-valued) Borel measures on $_{r}$ is isometrically embedded in the dual space of $C F L (_{r})$ and it is shown that the image...

Bifurcation theorems for nonlinear problems with lack of compactness

Francesca Faraci, Roberto Livrea (2003)

Annales Polonici Mathematici

Similarity:

We deal with a bifurcation result for the Dirichlet problem ⎧ $- Δ_{p} u = μ / {| x |}^{p} {| u |}^{p - 2} u + λ f (x, u)$ a.e. in Ω, ⎨ ⎩ $u_{| \partial Ω} = 0$ . Starting from a weak lower semicontinuity result by E. Montefusco, which allows us to apply a general variational principle by B. Ricceri, we prove that, for μ close to zero, there exists a positive number $λ *_{μ}$ such that for every $λ \in] 0, λ *_{μ} [$ the above problem admits a nonzero weak solution $u_{λ}$ in $W ₀^{1, p} (Ω)$ satisfying $l i m_{λ \to 0 ⁺} | | u_{λ} | | = 0$ .

Non-Typical Points for β-Shifts

David Färm, Tomas Persson (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

We study sets of non-typical points under the map $f_{β} \mapsto β x$ mod 1 for non-integer β and extend our results from [Fund. Math. 209 (2010)] in several directions. In particular, we prove that sets of points whose forward orbit avoid certain Cantor sets, and the set of points for which ergodic averages diverge, have large intersection properties. We observe that the technical condition β > 1.541 found in the above paper can be removed.

Congruences for $q^{[p / 8]} (m o d p)$

Zhi-Hong Sun (2013)

Acta Arithmetica

Similarity:

Let ℤ be the set of integers, and let (m,n) be the greatest common divisor of the integers m and n. Let p ≡ 1 (mod 4) be a prime, q ∈ ℤ, 2 ∤ q and p=c²+d²=x²+qy² with c,d,x,y ∈ ℤ and c ≡ 1 (mod 4). Suppose that (c,x+d)=1 or (d,x+c) is a power of 2. In this paper, by using the quartic reciprocity law, we determine $q^{[p / 8]} (m o d p)$ in terms of c,d,x and y, where [·] is the greatest integer function. Hence we partially solve some conjectures posed in our previous two papers.