Thermodynamic formalism for certain nonhyperbolic maps

1999 ◽  
Vol 19 (5) ◽  
pp. 1365-1378 ◽  
Author(s):  
MICHIKO YURI
Keyword(s):  
Number Theory ◽  
Gibbs Measure ◽  
Existence And Uniqueness ◽  
Thermodynamic Formalism ◽  
Periodic Points ◽  
Two Dimensional ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Conformal Measures ◽  
One Dimensional Maps

We establish a generalized thermodynamic formalism for certain nonhyperbolic maps with countably many preimages. We study existence and uniqueness of conformal measures and statistical properties of the equilibrium states absolutely continuous with respect to the conformal measures. We will see that such measures are not Gibbs but satisfy a version of Gibbs property (weak Gibbs measure). We apply our results to a one-parameter family of one-dimensional maps and a two-dimensional nonconformal map related to number theory. Both of them admit indifferent periodic points.

MULTIDIMENSIONAL PERTURBATIONS OF ONE-DIMENSIONAL MAPS AND STABILITY OF SARKOVSKĬ ORDERING

1999 ◽  
Vol 09 (09) ◽  
pp. 1867-1876 ◽  
Author(s):  
PIOTR ZGLICZYŃSKI
Keyword(s):  
Infinite Number ◽  
Periodic Points ◽  
One Dimensional ◽  
One Dimensional Maps

In this paper we present Sarkovskĭ theorem for multidimensional perturbations of one-dimensional maps. We outline a proof that if the unperturbed one-dimensional map has a point of period n, then sufficiently close multidimensional perturbations of this map have periodic points of all periods which are allowed by Sarkovskĭ theorem. We describe also how the approach from the proof of this theorem is used to show the existence of an infinite number of periodic points for Rössler equations.

On the full periodicity kernel for one-dimensional maps

1999 ◽  
Vol 19 (1) ◽  
pp. 101-126
Author(s):  
M. CARME LESEDUARTE ◽  
JAUME LLIBRE
Keyword(s):  
Topological Space ◽  
Periodic Points ◽  
One Dimensional ◽  
Fixed Set ◽  
Branching Point ◽  
One Dimensional Maps

Let $\bpropto$ be the topological space obtained by identifying the points 1 and 2 of the segment $[0,3]$ to a point. Let $\binfty$ be the topological space obtained by identifying the points 0, 1 and 2 of the segment $[0,2]$ to a point. An $\bpropto$ (respectively $\binfty$) map is a continuous self-map of $\bpropto$ (respectively $\binfty$) having the branching point fixed. Set $E\in\{\bpropto,\binfty\}$. Let $f$ be an $E$ map. We denote by $\Per(f)$ the set of periods of all periodic points of $f$. The set $K \subset{\mathbb N}$ is the full periodicity kernel of $E$ if it satisfies the following two conditions: (1) if $f$ is an $E$ map and $K\subset \Per(f)$, then $\Per(f)={\mathbb N}$; (2) for each $k\in K$ there exists an $E$ map $f$ such that $\Per(f)={\mathbb N}\setminus\{ k\}$. In this paper we compute the full periodicity kernel of $\bpropto$ and $\binfty$.

Sharkovskii's theorem for multidimensional perturbations of one-dimensional maps

1999 ◽  
Vol 19 (6) ◽  
pp. 1655-1684 ◽  
Author(s):  
PIOTR ZGLICZYŃSKI
Keyword(s):  
Periodic Points ◽  
One Dimensional ◽  
One Dimensional Maps

We present Sharkovskii's theorem for multidimensional perturbations of one-dimensional maps. We show that if an unperturbed one-dimensional map has a point of period $n$, then sufficiently close multidimensional perturbations of this map have periodic points of all periods which are allowed by Sharkovskii's theorem.

Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps

1992 ◽  
Vol 12 (1) ◽  
pp. 13-37 ◽  
Author(s):  
Michael Benedicks ◽  
Lai-Sang Young
Keyword(s):  
Positive Measure ◽  
Invariant Measures ◽  
Random Perturbations ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Absolutely Continuous Invariant Measure ◽  
Small Random Perturbations ◽  
One Dimensional Maps ◽  
Absolutely Continuous Invariant

AbstractWe study the quadratic family and show that for a positive measure set of parameters the map has an absolutely continuous invariant measure that is stable under small random perturbations.

scholarly journals Large deviation principles of one-dimensional maps for Hölder continuous potentials

2014 ◽  
Vol 36 (1) ◽  
pp. 127-141 ◽  
Author(s):  
HUAIBIN LI
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Weak Form ◽  
Periodic Points ◽  
Hölder Continuous ◽  
One Dimensional ◽  
Deviation Principle ◽  
Large Deviation Principles ◽  
One Dimensional Maps ◽  
Level 2

We show some level-2 large deviation principles for real and complex one-dimensional maps satisfying a weak form of hyperbolicity. More precisely, we prove a large deviation principle for the distribution of iterated preimages, periodic points, and Birkhoff averages.

Periodic points and topological entropy of one dimensional maps

1980 ◽  
pp. 18-34 ◽  
Author(s):  
Louis Block ◽  
John Guckenheimer ◽  
Michal Misiurewicz ◽  
Lai Sang Young
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
One Dimensional ◽  
One Dimensional Maps

scholarly journals Stationary states of two-dimensional magnetohydrodynamic turbulence: non-dissipative limit

1979 ◽  
Vol 22 (3) ◽  
pp. 385-396 ◽  
Author(s):  
Ronald Calinon ◽  
Danilo Merlini
Keyword(s):  
Gibbs Measure ◽  
Existence And Uniqueness ◽  
Canonical Ensemble ◽  
Two Dimensions ◽  
Navier Stokes ◽  
Stationary States ◽  
Two Dimensional ◽  
Magnetohydrodynamic Equations ◽  
Magnetohydrodynamic Turbulence ◽  
Statistical Solutions

A class of exact stationary statistical states for the inviscid magnetohydrodynamic equations in two dimensions and in various geometries is found and the corresponding fluctuation spectra are calculated. Some solutions agree with previous computations in the canonical ensemble while other solutions are found. In particular, the Navier—Stokes limit is recovered and maximum cross helicity solutions exist in two dimensions. The difficulty of proving existence and uniqueness of statistical solutions for non-dissipative two-dimensional turbulence is quoted in terms of rugged constants and associated Gibbs measure.

On Some Properties of Invariant Sets of Two-Dimensional Noninvertible Maps

1997 ◽  
Vol 07 (06) ◽  
pp. 1167-1194 ◽  
Author(s):  
Christos E. Frouzakis ◽  
Laura Gardini ◽  
Ioannis G. Kevrekidis ◽  
Gilles Millerioux ◽  
Christian Mira
Keyword(s):  
Periodic Points ◽  
Invariant Sets ◽  
Two Dimensional ◽  
S Locus ◽  
One Dimensional ◽  
Critical Curves ◽  
Invariant Circles ◽  
Saddle Type ◽  
Unstable Manifolds ◽  
Noninvertible Maps

We study the nature and dependence on parameters of certain invariant sets of noninvertible maps of the plane. The invariant sets we consider are unstable manifolds of saddle-type fixed and periodic points, as well as attracting invariant circles. Since for such maps a point may have more than one first-rank preimages, the geometry, transitions, and general properties of these sets are more complicated than the corresponding sets for diffeomorphisms. The critical curve(s) (locus of points having at least two coincident preimages) as well as its antecedent(s), the curve(s) where the map is singular (or "curve of merging preimages") play a fundamental role in such studies. We focus on phenomena arising from the interaction of one-dimensional invariant sets with these critical curves, and present some illustrative examples.

Sign in / Sign up

Export Citation Format

Share Document