Ergodic properties of invariant measures for piecewise monotonic transformations

1982 ◽  
Vol 180 (1) ◽  
pp. 119-140 ◽  
Author(s):  
Franz Hofbauer ◽  
Gerhard Keller
Keyword(s):  
Invariant Measures ◽  
Ergodic Properties ◽  
Piecewise Monotonic

scholarly journals Equilibrium states for piecewise monotonic transformations

1982 ◽  
Vol 2 (1) ◽  
pp. 23-43 ◽  
Author(s):  
Franz Hofbauer ◽  
Gerhard Keller
Keyword(s):  
Periodic Orbits ◽  
Bounded Variation ◽  
Critical Points ◽  
Equilibrium States ◽  
Absolutely Continuous ◽  
Specification Property ◽  
Ergodic Properties ◽  
Piecewise Monotonic

AbstractWe show that equilibrium states μ of a function φ on ([0,1], T), where T is piecewise monotonic, have strong ergodic properties in the following three cases:(i) sup φ — inf φ <htop(T) and φ is of bounded variation.(ii) φ satisfies a variation condition and T has a local specification property.(iii) φ = —log |T′|, which gives an absolutely continuous μ, T is C2, the orbits of the critical points of T are finite, and all periodic orbits of T are uniformly repelling.

Local dimension for piecewise monotonic maps on the interval

1995 ◽  
Vol 15 (6) ◽  
pp. 1119-1142 ◽  
Author(s):  
Franz Hofbauer
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measures ◽  
Positive Characteristic ◽  
Characteristic Exponent ◽  
Local Dimension ◽  
Pressure Function ◽  
Piecewise Monotonic ◽  
Almost Everywhere ◽  
Conformal Measures

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.

Ergodic Properties and Invariant Measures

1993 ◽  
pp. 327-436
Author(s):  
Welington Melo ◽  
Sebastian Strien
Keyword(s):  
Invariant Measures ◽  
Ergodic Properties

Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points

2000 ◽  
Vol 20 (5) ◽  
pp. 1519-1549 ◽  
Author(s):  
ROLAND ZWEIMÜLLER
Keyword(s):  
Limit Theorem ◽  
Fixed Points ◽  
Limit Theorems ◽  
Invariant Measures ◽  
Transfer Operator ◽  
Large Collection ◽  
Interval Maps ◽  
Infinite Measure ◽  
Ergodic Properties ◽  
Markov Extensions

We consider piecewise twice differentiable maps $T$ on $[0,1]$ with indifferent fixed points giving rise to infinite invariant measures, and we study their behaviour on ergodic components. As we do not assume the existence of a Markov partition but only require the first image of the fundamental partition to be finite, we use canonical Markov extensions to first prove pointwise dual-ergodicity, which, together with an identification of wandering rates, leads to distributional limit theorems. We show that $T$ satisfies Rohlin's formula and prove a variant of the Shannon–McMillan–Breiman theorem. Moreover, we give a stronger limit theorem for the transfer operator providing us with a large collection of uniform and Darling–Kac sets. This enables us to apply recent results from fluctuation theory.

scholarly journals Ergodic properties of invariant measures for C1+α non-uniformly hyperbolic systems

2012 ◽  
Vol 33 (2) ◽  
pp. 560-584 ◽  
Author(s):  
CHAO LIANG ◽  
WENXIANG SUN ◽  
XUETING TIAN
Keyword(s):  
Invariant Measures ◽  
Hyperbolic Systems ◽  
Hyperbolic Case ◽  
Compact Spaces ◽  
Ergodic Properties ◽  
Fixed Set ◽  
Uniformly Hyperbolic Systems ◽  
Time Averages ◽  
Support Sets ◽  
Dirac Measures

AbstractFor every ergodic hyperbolic measure ω of a C1+α diffeomorphism, there is an ω-full-measure set $\tilde {\Lambda }$ (the union of $\tilde \Lambda _l=\mathrm {supp}( \omega |_{\Lambda _{l}})$, the support sets of ω on each Pesin block Λl, l=1,2,…) such that every non-empty, compact and connected subset $V\subseteq \mathrm {Closure}(\mathcal {M}_{\mathrm {inv}}(\tilde \Lambda ))$ coincides with Vf(x), where $\mathcal {M}_{\mathrm {inv}}(\tilde {\Lambda })$ denotes the space of invariant measures supported on $\tilde {\Lambda }$ and Vf(x) denotes the accumulation set of time averages of Dirac measures supported at one orbit of some point x. For each fixed set V, the points with the above property are dense in the support supp (ω) . In particular, points satisfying $V_f(x)=\mathrm {Closure}(\mathcal {M}_{\mathrm {inv}}(\tilde \Lambda ))$ are dense in supp (ω) . Moreover, if supp (ω) is isolated, the points satisfying $V_f(x)\supseteq \mathrm {Closure}(\mathcal {M}_{\mathrm {inv}}(\tilde \Lambda ))$ form a residual subset of supp (ω) . These extend results of K. Sigmund [On dynamical systems with the specification property. Trans. Amer. Math. Soc. 190 (1974), 285–299] (see also M. Denker, C. Grillenberger and K. Sigmund [Ergodic Theory on Compact Spaces (Lecture Notes in Mathematics, 527). Springer, Berlin, Ch. 21]) from the uniformly hyperbolic case to the non-uniformly hyperbolic case. As a corollary, irregular + points form a residual set of supp (ω) .

scholarly journals Equilibrium states for partially hyperbolic horseshoes

2010 ◽  
Vol 31 (1) ◽  
pp. 179-195 ◽  
Author(s):  
R. LEPLAIDEUR ◽  
K. OLIVEIRA ◽  
I. RIOS
Keyword(s):  
Invariant Measures ◽  
Ergodic Measure ◽  
Equilibrium States ◽  
One Dimensional ◽  
Hyperbolic Diffeomorphisms ◽  
Ergodic Properties ◽  
Central Direction ◽  
Positive Exponent ◽  
Continuous Potential ◽  
Partially Hyperbolic

AbstractWe study ergodic properties of invariant measures for the partially hyperbolic horseshoes, introduced in Díaz et al [Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergod. Th. & Dynam. Sys. 29 (2009), 433–474]. These maps have a one-dimensional center direction Ec, and are at the boundary of the (uniformly) hyperbolic diffeomorphisms (they are constructed bifurcating hyperbolic horseshoes via heterodimensional cycles). We prove that every ergodic measure is hyperbolic, but the set of Lyapunov exponents in the central direction has gap: all ergodic invariant measures have negative exponent, with the exception of one ergodic measure with positive exponent. As a consequence, we obtain the existence of equilibrium states for any continuous potential. We also prove that there exists a phase transition for the smooth family of potentials given by ϕt=t log ∣DF∣Ec∣.

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