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A self-consistent dynamical system with multiple absolutely continuous invariant measures

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*The author was supported by ERC grant No 787304 and NKFIH OTKA grant K123782

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  • In this paper we study a class of self-consistent dynamical systems, self-consistent in the sense that the discrete time dynamics is different in each step depending on current statistics. The general framework admits popular examples such as coupled map systems. Motivated by an example of [9], we concentrate on a special case where the dynamics in each step is a $ \beta $-map with some $ \beta \geq 2 $. Included in the definition of $ \beta $ is a parameter $ \varepsilon > 0 $ controlling the strength of self-consistency. We show such a self-consistent system which has a unique absolutely continuous invariant measure (acim) for $ \varepsilon = 0 $, but at least two for any $ \varepsilon > 0 $. With a slight modification, we transform this system into one which produces a phase transition-like behavior: it has a unique acim for $ 0< \varepsilon < \varepsilon^* $, and multiple for sufficiently large values of $ \varepsilon $. We discuss the stability of the invariant measures by the help of computer simulations employing the numerical representation of the self-consistent transfer operator.

    Mathematics Subject Classification: Primary: 37A05, 37A10, 37E05; Secondary: 37M25, 65P99.

    Citation:

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  • Figure 1.  The choice of $ \beta_k $, $ k = 3 $ is pictured

    Figure 2.  Approximation of $ \psi^{\varepsilon} $ with ergodic averages, $ F(x) = x $. $ P_{x, N}(\beta) $ is plotted for $ \beta \in [2, \beta_1] $ with gridsize $ \Delta $ and for $ x $ drawn uniform randomly from $ [0, 1] $. The line $ x = y $ is plotted

    Figure 3.  Approximation of $ \psi^{\varepsilon} $ with ergodic averages. $ P_{x, N}(\beta) $ is plotted for $ \beta \in [2, 3] $ with gridsize $ \Delta = 10^{-4} $, $ N = 10^6 $ and for $ x $ drawn uniform randomly from $ [0, 1] $, $ \varepsilon = 0.8 $. The line $ x = y $ is plotted

    Figure 4.  Approximation of $ \psi^{\varepsilon} $ with ergodic averages, $ F(x) = x^2 $. $ P_{x, N}(\beta) $ is plotted for $ \beta \in [2, \beta_1] $ with gridsize $ \Delta = 10^{-4} $, $ N = 10^6 $ and for $ x $ drawn uniform randomly from $ [0, 1] $. The line $ x = y $ is plotted

    Figure 5.  Approximation of $ \psi^{\varepsilon} $ with ergodic averages, $ P_{x, N}(\beta) $ is plotted for $ \beta \in [2, 3] $ with gridsize $ \Delta = 10^{-4} $, $ N = 10^6 $ and for $ x $ drawn uniform randomly from $ [0, 1] $. The line $ x = y $ is plotted

    Figure 6.  Total variation for each $ f_t^i $ as a function of time, $ F(x) = x $, $ T = 100 $, $ K_1\times K_2 = 100 $

    Figure 7.  Associated slope to each $ f_t^i $ as a function of time, $ F(x) = x $, $ T = 100 $, $ K_1\times K_2 = 100 $

    Figure 8.  Approximation of the invariant density of the self-consistent system (4) with $ F(x) = x $ by a high iterate of an appropriate initial density

    Figure 9.  For each $ \varepsilon = k \cdot 10^{-3} $, $ k = 1, \dots, 10^3 $ the values $ \overline{\text{var}}(t) $ and $ \overline{\beta}(t) $ are plotted for $ t = 150, \dots, 200 $, $ F(x) = x $, $ K_1 \times K_2 = 100 $

    Figure 10.  For each $ \varepsilon = k \cdot \Delta $, $ k = 1, \dots, E/\Delta $ the values $ \overline{\text{var}}(t) $ are plotted for $ t = 150, \dots, 200 $, $ K_1 \times K_2 = 100 $

    Table 1.  Computation of the mean total variation $ \overline{\text{var}} $ and mean slope for $ F(x) = x $. $ T = 100 $, $ M = K_1 = K_2 = 10 $. Left hand side: $ \varepsilon = 0.1 $, right hand side: $ \varepsilon = 0.2 $

    $t $ $\overline{\text{var}}$ $\overline{\beta}$ $t $ $\overline{\text{var}}$ $\overline{\beta}$
    0 7.1815 2.0065 0 5.1121 1.9993
    5 1.1544 2.0039 5 0.9983 2.0130
    10 0.9559 2.0020 10 0.9415 2.0117
    15 0.9856 2.0015 15 0.9490 2.0134
    20 0.9901 2.0012 20 0.9429 2.0149
    25 0.9917 2.0011 25 0.9371 2.0160
    30 0.9928 2.0010 30 0.9341 2.0165
    35 0.9934 2.0009 35 0.9323 2.0168
    40 0.9940 2.0008 40 0.9330 2.0171
    45 0.9944 2.0008 45 0.9320 2.0174
    50 0.9947 2.0007 50 0.9309 2.0176
    55 0.9949 2.0007 55 0.9298 2.0178
    60 0.9950 2.0007 60 0.9289 2.0179
    65 0.9952 2.0007 65 0.9280 2.0180
    70 0.9953 2.0007 70 0.9274 2.0181
    75 0.9954 2.0007 75 0.9270 2.0181
    80 0.9954 2.0007 80 0.9269 2.0181
    85 0.9955 2.0007 85 0.9269 2.0181
    90 0.9955 2.0006 90 0.9269 2.0181
    95 0.9956 2.0006 95 0.9270 2.0181
    100 0.9956 2.0006 100 0.9270 2.0181
     | Show Table
    DownLoad: CSV

    Table 2.  Computation of the mean total variation $ \overline{\text{var}} $ and mean slope for $ F(x) = x $, $ \text{var}(f^i_0) < 10^{-4} $. $ T = 300 $, $ M = K_1 = K_2 = 10 $. Left hand side: $ \varepsilon = 0.1 $, right hand side: $ \varepsilon = 0.2 $

    $t $ $\overline{\text{var}}$ $\overline{\beta}$ $t $ $\overline{\text{var}}$ $\overline{\beta}$
    0 0.0000 2.0000 0 0.0000 2.0000
    25 1.0000 2.0000 25 1.0000 2.0000
    50 1.0000 2.0000 50 0.9993 2.0003
    75 1.0000 2.0000 75 0.9783 2.0072
    100 0.9998 2.0000 100 0.9362 2.0160
    125 0.9995 2.0001 125 0.9307 2.0177
    150 0.9987 2.0002 150 0.9268 2.0181
    175 0.9978 2.0003 175 0.9269 2.0181
    200 0.9969 2.0005 200 0.9269 2.0181
    225 0.9966 2.0005 225 0.9269 2.0181
    250 0.9966 2.0005 250 0.9269 2.0181
    275 0.9964 2.0005 275 0.9269 2.0181
    300 0.9962 2.0006 300 0.9269 2.0181
     | Show Table
    DownLoad: CSV

    Table 3.  Computation of $ \overline{\text{int}}_{f_*(\varepsilon)} $ and mean slope for $ F(x) = x $, $ T = 200 $, $ \bar{T} = 5000 $. $ M = K_1 = K_2 = 10 $

    $t$ $\overline{\text{int}}_{f_*(0.1)}$ $\overline{\text{int}}_{f_*(0.2)}$
    0 0.4096 0.5018
    10 0.0101 0.0321
    20 0.0049 0.0174
    30 0.0028 0.0097
    40 0.0017 0.0064
    50 0.0011 0.0040
    60 0.0007 0.0021
    70 0.0005 0.0007
    80 0.0004 0.0003
    90 0.0003 0.0002
    100 0.0002 0.0001
    110 0.0002 0.0001
    120 0.0002 0.0001
    130 0.0001 0.0001
    140 0.0001 0.0001
    150 0.0001 0.0001
    160 0.0001 0.0001
    170 0.0001 0.0001
    180 0.0001 0.0001
    190 0.0001 0.0001
    200 0.0001 0.0001
     | Show Table
    DownLoad: CSV

    Table 4.  Computation of the mean total variation $ \overline{\text{var}} $. $ T = 100 $, $ M = K_1 = K_2 = 10 $. Left hand side: $ F(x) = x^2 $, center: $ F(x) = x^4 $, right hand side: $ F(x) = x^6 $

    $t $ $ \varepsilon=1 $ $ \varepsilon=2.5$ $t $ $\varepsilon=1 $ $ \varepsilon=35$ $t$ $\varepsilon=1$ $ \varepsilon=400 $
    0 6.4694 5.0341 0 5.3766 6.4806 0 7.7582 5.1444
    5 1.2483 1.1951 5 1.2370 1.1785 5 1.3852 1.0419
    10 0.9808 0.8849 10 0.9882 0.9788 10 0.5224 0.5990
    15 0.9914 0.8471 15 0.6884 0.9873 15 0.0210 0.1435
    20 0.9976 0.8447 20 0.0474 0.5994 20 0.0007 0.0026
    25 0.9994 0.8275 25 0.0015 0.1093 25 0.0000 0.0002
    30 0.9999 0.8418 30 0.0000 0.0347 30 0.0000 0.0000
    35 1.0000 0.8305 35 0.0000 0.0023 35 0.0000 0.0000
    40 1.0000 0.8468 40 0.0000 0.0001 40 0.0000 0.0000
    45 1.0000 0.8321 45 0.0000 0.0000 45 0.0000 0.0000
    50 0.9531 0.8358 50 0.0000 0.0000 50 0.0000 0.0000
    55 0.5357 0.8382 55 0.0000 0.0000 55 0.0000 0.0000
    60 0.1802 0.7651 60 0.0000 0.0000 60 0.0000 0.0000
    65 0.0128 0.6366 65 0.0000 0.0000 65 0 0.0000
    70 0.0004 0.5117 70 0.0000 0.0000 70 0 0
    75 0.0000 0.4535 75 0 0.0000 75 0 0
    80 0.0000 0.4269 80 0 0.0000 80 0 0
    85 0.0000 0.4255 85 0 0 85 0 0
    90 0.0000 0.4015 90 0 0 90 0 0
    95 0.0000 0.3966 95 0 0 95 0 0
    100 0.0000 0.3813 100 0 0 100 0 0
     | Show Table
    DownLoad: CSV
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